Talk:Meantone temperament

Early material replaced
This is what was in the article before I edited. I don't really understand it, and what I do understand doesn't seem strictly relevant to meantone, so I'm moving it here. I've replaced it with a stubby entry for now, but I'll be expanding it in time. --Camembert

Original article text follows

In a modern 12 note equal temperament the semitone is exactly half a tone logarithmically.

If the semitone is allowed to be more than half a tone, one can get a better approximation to just intonation.

If the semitone is a rational fraction of a tone logarithmically, one gets a finite number of notes in an octave.

Semitone/Tone Notes 1/2       12      4/7        43      3/5        31      5/8        50      2/3        19

One can make some intervals perfectly tuned. In Pythagorean tuning, 1:2 (the Octave) and 2:3 (the Fifth) are perfect and 9:10 is approximated by 8:9. This gives a semitone of 243:256, a tone of 8:9, and a major Third of 64:81 (1.265625). Quarter comma meantone tunes 1:2 and 4:5 (the major Third) perfectly, giving a tone ratio 2:sqrt(5), a semitone ratio 5^(5/4):8, and a Fifth of 5^(1/4) (1.495349).


 * The preceding comment and removal of material was made by User:Camembert at some unspecified date. I've taken the liberty of adding a section heading so this shows up in the Contents list for the talk page.  yoyo (talk) 19:42, 21 March 2018 (UTC)

Notation and the wolf
The page as it stands has a nonstandard notation for accidentals (black notes). The usual is to notate all accidentals on the flatward side of the wolf as sharps, and vice versa. That is, if the wolf is between G# and Eb, you use F#, C#, G#, Eb, Bb. This is also easier to understand because A# is an unfamiliar note whereas Bb is very common. Then the flats and sharps form a mnemonic for which intervals are 'bad': the fifth G#-Eb is bad as are all thirds where the root is a sharp or where the third is a flat. --Tdent 21:53, 14 Apr 2005 (UTC)

Definition and history
The definition of meantone should not imply that the major 3rd is 'just', since in 1/5 comma, 1/6 comma, etc. etc. (all of which were historically referred to as mean-tone) it is not. The correct definition is that each perfect fifth is an equal interval apart from the wolf, with a corollary that the fifths are narrower than equal-tempered. (Otherwise Pythagorean tuning and equal temperament would also qualify.)

It would be useful to indicate the historical uses of (the various types of) mean-tone, which was widespread in the Renaissance, Baroque and early Classical eras. Even though keyboards began to be tuned to other types of temperament in the 18th century, vocal and wind/string instrument intonation was still taught according to a 1/6-comma meantone scheme in the time of Mozart. It appears likely that keyboard instruments were tuned to a slightly altered form of meantone through the 19th century, for example temperaments based on 1/8 comma meantone. --Tdent 22:24, 14 Apr 2005 (UTC)

Unequal Temperaments book and website
Dear friends,

The Unequal Temperaments book of 1978 was described-in writing-as the definitive reference on the matter by authorities such as John Barnes, Hubert Bédard, Kenneth Gilbert, Igor Kipnis, Rudolf Rasch and others. In the 1990's I also developed the first professional-grade temperament spreadsheets.

Eventually I setup the "Unequal Temperaments" website, where I uploaded the spreadsheets which, kept permanently updated, are available for FREE. I also uploaded years ago a provisional "Update" to the book of 1978. The website lately gives information on the recently released new version of Unequal Temperaments 2008, which includes a detailed treatment of MEANTONE TEMPERAMENT. (Please note: the website does NOT sell the book)

I would find it useful to Wikipedia readers if my website was included among External Links:


 * Unequal Temperaments website, with Preview of the new 2008 book by Claudio Di Veroli

Kind regards

Claudio

Dr. Claudio Di Veroli

Bray, Ireland 86.42.128.58 (talk) 17:19, 26 February 2009 (UTC)

Suggestion for improvement
As a musician with a strong math and physics background, I can still say that the introduction of the concepts of "5-limit musical", "7-limit musical", etc. early on without clarification makes the article difficult to understand at the outset. One would hope that at least some cursory explanation can be offered for these terms. Thanks KarlRKaiser (talk) 03:33, 21 September 2009 (UTC)


 * Thirteen years later, I agree strongly. I am a once-competent cellist and have a math degree and am sitting here listening to the Well-Tempered Clavier and wondered "What was Bach reacting to?" so I pulled up this article, and it might as well be written in Sumerian. Might make sense if you already have a Ph.D. in music theory? Tim Bray (talk) 06:01, 11 January 2022 (UTC)


 * @Tim Bray, it seems to me that thirteen years later, the article changed. It seems among others that the mentions of "5-limit" and "7-limit" luckilly disappeared. However, if you read the commentaries below (which often are as complex as the article itself), you'll see that the discussion has not been simple and that we are many who would agree with you.
 * The question "What was Bach reacting to?" is an interesting one. Bach was in good terms with some of the theorists of his time who discussed temperaments, particularly the "good temperaments" (i.e. those allowing to play in all major and minor keys) and probably had some knowledge about this – which does not really concern meantone temperaments.
 * I slightly modified the lead, in the hope to make it more understandable. Make your criticisms more specific, stating what sounds Sumerian to you: this would help us improve the article. — Hucbald.SaintAmand (talk) 13:38, 11 January 2022 (UTC)
 * One specific thing that might be useful would be a quick introduction to the problem that meantone exists to solve - that while just tuning is the simplest approach mathematically and arguably the "truest" musically, it is not possible to construct keyed or fretted instruments which can play such a temperament in multiple keys. Or perhaps I'm wrong about this? In any case, I do think it would be useful to have some words about the motivation. Anyhow, thanks for the update, I do find the opening of the article a little smoother now.  I will try to dig in a little deeper and highlight the parts that still seem a bit obscure.  Related: I obviously also looked at Just intonation and despite all the badges at the top, I found the path through the material well-thought-out. It seems that it would be good for this article and that to be somewhat symmetric, as they are really two halves of the same problem. Tim Bray (talk) 04:18, 14 January 2022 (UTC)
 * Oh dear, now I have looked at Equal temperament and Musical tuning and the *huge* "Well-tempered tuning" section in The Well-Tempered Clavier, and I am feeling terribly ignorant. My initial impression is that a bit more harmony [hah hah] between these discussion might be achieved. In any case, thanks to the editors who have gathered all this rich material.Tim Bray (talk) 04:29, 14 January 2022 (UTC)

Clarification
What the hell does the shape of the keyboard have to do with anything? —Preceding unsigned comment added by 184.189.220.19 (talk) 00:28, 21 May 2011 (UTC)

Difficulties following argument; missing thummer.jpg
1> It seems that the .jpg of the "Thummer" keyboard is missing.

2a> As a physicist / singer / early music listener I find this article excellent, but have trouble following the para with Easley Lockwood's "R" and its relation to various mean tones. Accepting that the stated relationship is exact: log2 [interval of fifth] = (3R+1)/(5R+2) then it seems to me that R cannot be the ratio of (frequencies) for wholetone to semitone, but rather its logarithm (log2). Am I missing something?


 * It's not even the logarithm of the ratio, but the ratio of the logarithms. —Tamfang (talk) 00:11, 23 October 2011 (UTC)

2b> The table "Meantone Tunings" is very useful. The relationship of column 1 to column 2 is clear (once I interpret R as a logarithm). But I have trouble relating (approximately) column 3 (the syntonic adjustment) to column 1 (R). I would be helpful to me to see explicit constructions from stacked tempered fifths of the wholetone and the diatonic semitone whose ratio defines "R". Is "R" the (log) ratio of pitch [F] to pitch [E] after [F]:[C], [C]:[D] and [D]:[E] have been constructed from stacked tempered fifths? And why is the relation of column 3 to column 1 only approximate -- are not the meantone fractions (1/4 or 1/6 or 1/11) also logarithmic?

68.239.154.181 (talk) 14:32, 19 September 2011 (UTC)

But what is it?
I came to this article because I read of an organ being a mean-tone organ, and I wanted to find out what that meant. I read the introduction and attempted the rest of the article, and I still don't know what it means. Is there anyone who could rewrite the introduction so that it's understandable by non-experts? Thanks. G (talk) 09:19, 14 July 2012 (UTC) -- Yes, but, if they do, it will be deleted. (...by the non-experts who wrote the over-pseudo-technical article.) — Preceding unsigned comment added by 97.82.109.213 (talk) 02:38, 15 June 2021 (UTC)

You won't find that in this article, which is only intended to show off the pretentious & pseudoscientic use of some people's undefined overtechnical jargon, illustrated by a graph that doesn't tell what quantity is being graphed with respect to what other quantity.

However, my (deleted) brief & clear explanation can be found in an article-history search.

...and at the end of this talk-page, at least until it's deleted from there too. --


 * Do you understand what musical temperament is for? (That's the first link in the intro.) —Tamfang (talk) 15:05, 14 July 2012 (UTC)

Tempering the syntonic comma to unison???
Early in the article, one can find the following utterly puzzling (IMO) statement:
 * [Meantone temperament] by choosing an appropriate size for major and minor thirds, tempers the syntonic comma to unison.

This not only is not true, but it also is meaningless. I vainly tried, since quite a long time, in many discussions either on WP or elsewhere, to eradicate this strange idea. I won't correct the article because I know that well-intentioned contributors would soon revert my correction, but let me at least explain here why such an idea is incorrect and silly.

All temperaments ever documented result from a tempering of the fifths. Meantone temperaments temper the fifths to less than pure (i.e. less than the 3:2 ratio of Pythagorean tuning) or, if one wants to exclude equal temperament from the set of meantone temperaments (I can see no reason to do this, but ...), they temper the fifth to less than the ET-tempered fifth.

Doing so has the explicit purpose of producing better major or minor thirds, or both. The syntonic comma is, by definition, the difference between the pure major third and the Pythagorean one, or the pure minor third and the Pythagorean one – the syntonic comma corresponds to 81:80 in both cases. Meantone temperaments always result in lessening this difference. None, however, achieves this by tempering the syntonic comma itself, once again only the fiths are tempered.

1/4-comma meantone is that which makes the major third pure (ratio 5:4): the syntonic comma that separates the major third from the Pythagorean one vanishes in this case. The minor third of 1/4 comma-meantone, on the other hand, is only 3/4 of a syntonic comma larger than in Pythagorean tuning. In other words, the 1/4-comma meantone makes the syntonic comma vanish for the major third, but retains 1/4 of the one of the minor third. 1/3-comma meantone, similarly, makes the syntonic comma of the minor third vanish but, as a result, produces a major third 1/4-comma lower than pure: obviously, the syntonic comma of the major third does not vanish.

Other meantone temperaments produce various results for the intervals between the major and minor thirds and their Pythagorean equivalent. None of them, however, is able to make any of these two intervals vanish – it would be utterly impossible to make both vanish in any regular temperament. (Irregular temperaments might make the interval vanish for some major thirds and some minor ones, but never for all of them.)

Tempering an interval means reducing (or widening) it to obtain a desired effect. Once again, all meantone temperaments reduce (temper) the fifths in order to obtain better thirds: pure fifths (3:2 ratios) are lessened to an interval which cannot be expressed in the form of a ratio. The syntonic comma, by definition, corresponds to the ratio 81:80. Diminishing this ratio transforms it in another interval that can no more be said a syntonic comma (no more than a tempered fifth can be said to be a pure one). Therefore, to claim that "the syntonic comma is tempered" is meaningless: first of all, which syntonic comma? and, in addition, if the interval is tempered, it is no more a syntonic comma...

What the statement quoted above means is this:
 * [Meantone temperament] by choosing an appropriate size for major and minor thirds, creates an interval between them and their Pythagorean equivalent; 1/4-comma meantone produces major thirds a syntonic comma away from their Pythagorean equivalent, 1/3-comma does the same for the minor thirds.

Meantone temperaments, in brief, create an interval between their thirds and the Pythagorean ones, an interval which is some cases may reach a full syntonic comma (or more) — is that "tempering to unison"? Am I making all this more or less clear? — Hucbald.SaintAmand (talk) 21:20, 8 August 2016 (UTC)

Yes, it does make sense, perfectly. It's just a pity that the "talk" page is easier to understand than the "real" page! I think this is an example of what happens when several people make a very honest attempt at explaining a complex concept. The result is a bit like listening to two people talking at the same time - a bit confusing; the foundations built by one author weren't intended for (and don't quite fit) the superstructure erected by the next. The meat of the article is really the section

"The family of meantone temperaments share the common characteristic that they form a stack of identical fifths, the tone being the result of two fifths minus one octave, the major third of four fifths minus two octaves. Such temperaments are also called "regular" or "syntonic". Meantone temperaments are often described by the fraction of the syntonic comma by which the fifths are tempered: quarter-comma meantone, the most common type, tempers the fifths by 1/4 syntonic comma, with the result that four fifths produce a just major third, a syntonic comma lower than a Pythagorean major third; third-comma meantone tempers by 1/3 syntonic comma, three fifths producing a just major sixth, a syntonic comma lower than a Pythagorean one.

This, together with the first line of the "Wolf" section, is really all that needs to be said on the physics of the situation (the little historical/current usage section is also valuable, and could be extended without losing relevance or interest, in my view).

I would completely support the removal of the "tempered to the unison" comment as it honestly makes no sense, and is only saying that the fifths are narrowed in 1/4-comma to create a just third, which the article says much more clearly later anyway.

I would also suggest that the comment about wolf intervals being an artifact of keyboard design be removed too, because it's focusing on the wrong thing. The design feature of the keyboard that dictates the existence of a wolf (in the sense of having at least one fifth worse than an ET fifth) is that it has 12 keys for each octave. And it has 12 keys because someone decided to have 12 divisions per octave. It would be better, therefore, to say that a wolf interval is an inevitable feature of having 12 notes per octave. The isomorphic keyboard avoids a wolf not because it's a clever shape, but because it has more notes per octave (in the sense that if you actually build it, you need more than 12 organ pipes per octave, because you need to provide separate sharps and flats). It is a clever shape too, of course! --149.155.219.44 (talk) 16:34, 20 October 2016 (UTC)

The caption to the figure reads "The Pythagorean A♭ (at the left) is at 792 cents, G♯ (at the right) at 816 cents; the difference is the Pythagorean comma." Pythagorean comma says the Pythagorean comma is approximately 23.46 cents while syntonic comma says the syntonic comma is "around 21.51 cents". I'm changing "Pythagorean" to "syntonic". Lewis Goudy (talk) 21:00, 30 May 2017 (UTC)
 * 816-792 = 24 cents! The difference between the Pythagorean A♭ and the Pythagorean G♯ by definition is the Pythagorean comma. Values in the image are rounded up to integral figures; because of the rounding up of the values 816 and 792, the Pythagorean comma (23.4600) appears here as 24 cents, while the syntonic comma (21.5062) is rounderd up as 22 cents (316-294 or 408-386). There is no doubt that the comma that is labelled "Pythagorean" in the image is the Pythagorean comma. Hucbald.SaintAmand (talk) 13:47, 31 May 2017 (UTC)
 * I couldn't stand it any longer and I removed the "tempering of the comma to unison". The reason why should be clear from the comments above. We'll see how long the change lasts before it is once again modified... — Hucbald.SaintAmand (talk) 14:03, 31 May 2017 (UTC)


 * Well done! It hasn't come back yet, thank goodness. :-)  yoyo (talk) 19:54, 21 March 2018 (UTC)


 * "I would also suggest that the comment about wolf intervals being an artifact of keyboard design be removed too..." I second this, as I believe it's just plain wrong. The Wolf also doesn't depend on the number of notes in the scale.  Any finite scale will have tempering issues in reconciling P5s and P8s.Unhandyandy (talk) 15:52, 5 September 2018 (UTC)
 * As the person who added the section of the article that discusses the "wolf being an artifact of keyboard design" (in 2009) I vote against removing it. Having fewer note-controlling elements per octave makes the wolf more salient; having more, makes the wolf less salient. The number of notes per octave is a design choice. Hence, the wolf is an artifact of keyboard design. --JimPlamondon (talk) 10:04, 7 May 2020 (UTC)

Keyboard poorly suited to meantone???
The article writes:
 * "On the other hand, the piano keyboard has only twelve physical note-controlling devices per octave, making it poorly suited to any tunings other than 12-ET."

The keyboard in question was invented centuries before the piano and therefore hardly can be described as the "piano keyboard".

Meantone temperaments were devised for keyboard instruments. I cannot figure how keyboards were "poorly suited" to meantone temperaments, but these (the temperaments) obviously were suited to them (the keyboard instruments).

As a historian of music theory, I find quite irritating the way in which some today rewrite the history. Hucbald.SaintAmand (talk) 08:01, 5 December 2016 (UTC)

Beside, the Musical temperament article states:
 * "Temperament is especially important for keyboard instruments"

Hucbald.SaintAmand (talk) 08:16, 5 December 2016 (UTC)

Keyboard shape and the Wolf
I believe this is flat out wrong. The tuning system has nothing to do with the keyboard layout. If an isomorphic keyboard is tuned in 1/4 comma meantone then it will have a wolf fifth.

Perhaps the writer meant that use of an isomorphic keyboard would have made the development of unequal temperaments less likely? Unhandyandy (talk) 16:00, 5 September 2018 (UTC)
 * You are perfectly right, and actually the article itself says so:
 * "The problem is at the edge, on the note E♯. The note that's a perfect fifth higher than E♯ is B♯, which is not included on the keyboard shown (although it could be included in a larger keyboard, placed just to the right of A♯, hence maintaining the keyboard's consistent note-pattern)"
 * It goes without saying that adding a note for B♯ merely would shift the problem to between B♯ and F. What produces the wolf interval is the (unescapable) limitation in the number of notes available. As soon as this number is limited, there will arise somewhere the necessity to take one degree for its enharmonic equivalent. The only thing that the isomorphic keyboard achieves, is that it cancels the difference between black and white keys of the ordinary keyboard, so that each interval, when playable on the keyboard, is played everywhere with the same fingering. The keyboard illustrated produces 19 different pitch classes (instead of 12): the wolf intervals appear whenever a 20th pitch class is needed...
 * The trouble is that people working with these modern devices are so convinced that they realized the squaring of the circle that they become unable to see clearly. — Hucbald.SaintAmand (talk) 17:58, 5 September 2018 (UTC)
 * Dear Hucbald.SaintAmand: I clearly see that I wrote, in 2009, the content that you are citing in explanation of Unhandyandy's question. I am not sure what it is that you think I am not seeing clearly. I would, again, note that using electronic transposition to keep the current key's diatonic notes centered on a 19-note-per-octave keyboard would tend to make the playing of wolf intervals extremely rare. If this is incorrect, I would welcome your correction. :-)  --JimPlamondon (talk) 10:16, 7 May 2020 (UTC)
 * Dear JimPlamondon, I only mean that what the article says about the wolf is unclear, probably because it is unclear in the mind of who wrote these passages.
 * If "there are no wolf intervals within the note-span of the isomorphic keyboard", then there are none either in the span of a 12-note common keyboard. Wolf notes always are at the edge of the span or, more precisely, outside the edges.
 * There are wolf intervals only if the music exceeds the span of the tuning (keyboards may have more keys than the tuning has notes). Many historical compositions can be played on a 12-note keyboard in any tuning merely because they keep within the limits of the span E♭–G♯ (or possibly A♭–C♯, with a retuning of G♯ to A♭).
 * Extended keyboards have been described since the early 15th century (Prosdocimus de Beldemandis, 1413). They extend the span, but that becomes useful only if there is a specific music requesting that span.
 * There is another problem related to that of the wolf intervals, that of enharmonic modulation. See the "Enharmonic keyboard" article in The New Grove, particularly what it says about Bull's chromatic fantasy on Ut re mi fa sol la.
 * Isomorphic keyboards as such do not afford a solution that mechanical keyboards with as many keys would not afford. They may facilitate fingerings, but that is another matter.
 * Electronic instruments may make transpositions possible, displacing the span and the problem of the wolf, but they cannot solve it. They reduce it to the same extent (or only slightly more, if they adjust the span in real time), as any other keyboard with more than 12 notes in the octave.
 * Most of these points, in addition, do not specifically concern meantone temperament. They would exist in most tunings other than ET. Most of what is said in this article is also said in Wolf interval, which is an unnecessary and potentially confusing redundancy. In other words, the passages concering wolf intervals could be much improved. I won't do it myself. — Hucbald.SaintAmand (talk) 17:58, 7 May 2020 (UTC)


 * Well, put, esteemed Hucbald.SaintAmand. I doff my hat to you. [Doffs hat.] :-)  --JimPlamondon (talk) 13:11, 9 May 2020 (UTC)

"R" ?
What follows is an attempt to make sense of this alinea found in the article:
 * Meantone temperaments can be specified in various ways: by what fraction (logarithmically) of a syntonic comma the fifth is being flattened (as above), what equal temperament has the meantone fifth in question, the width of the tempered perfect fifth in cents, or the ratio of the whole tone to the diatonic semitone. This last ratio was termed "R" by American composer, pianist and theoretician Easley Blackwood, but in effect has been in use for much longer than that. It is useful because it gives us an idea of the melodic qualities of the tuning, and because if R is a rational number N/D, so is 3R + 1/5R + 2 or 3N + D/5N + 2D, which is the size of fifth in terms of logarithms base 2, and which immediately tells us what division of the octave we will have. If we multiply by 1200, we have the size of fifth in cents.

• "by what fraction (logarithmically) of a syntonic comma" probably means "by what fraction of a syntonic comma (expressed logarithmically)". I wonder however whether the expression "1/4-comma meantone" does not predate the invention of logarithms in the early 17th century. But this merely would prove that the musicians had an unconscious notion of logarithms before their invention.

• "what equal temperament has the meantone fifth in question" would deserve a definition of both "equal temperament" and of "fifth". For most people, "equal temperament" is the temperament that divides the octave in 12 semitones and the idea of other equal temperaments may be somewhat puzzling. The idea that an "equal temperament" has a "meantone fifth" also would require explanation. The whole idea is more complex than it seems. A regular temperament is one where all the fifths are equal, being equally tempered; a meantone temperament is a regular temperament in which after a number of tempered fifths (and fourths), one comes back to the starting point (or its octave). This, however, in turn raises the question of the definition of the "fifth": to what extent can one temper a fifth and still consider it a fifth? Figure 1 in the article shows fifths from 686 to 720 cents, covering a span of 34 cents, 1/3 of a semitone, from the narrowest to the widest; but arrows on both sides indicate that the "generator" could be flattened or sharpened more than that... Is one still speaking of fifths, of meantone temperament, or even of temperament?

• Figure 1 apparently describes a continuous tempering of the fifth, continuously from 686 to 720 cents; the mention of equal temperaments, from 7-ET at 686 cents to 5-ET at 720 cents, however shows that only a few ETs appear in this continuum: this seems to indicate the difference between regular temperaments (any equally tempered fifths) and meantone ones (equally tempered fifths that produce an octave, and therefore an ET). The ETs indicated divide the octave in 5, 7, 12, 17, 19, 22, 26, 31, 43, 50 or 53 equal intervals. Many intermediate ETs are missing in this list: they would require a more extreme tempering of the fifth, once again raising the question of what is a fifth. Also, 5-ET is described as "Indonesian slendro" and 7-ET as "Thai traditional". But does this imply that these people consider their scales as formed of a cycle of 5 or 7 equal "fifths"? I very strongly doubt that they do.

• But let's turn to R. It is defined by the ratio of the whole tone to the diatonic semitone; in addition, R is said to be a rational number, which implies that the whole tone and the diatonic semitone are commensurable. One may suppose that in the ratio N/D, N is the value of the whole tone and D that of the diatonic semitone. "Whole tone" probably means the result of two fifths (as C-D = C-G-D) and "diatonic semitone" the result of five fifths (as C-B = C-G-D-A-E-B). Should one understand that R = N/D = 2/5? Does that explain why "3R + 1/5R + 2 or 3N + D/5N + 2D" are rational? 3(2/5) + 1/5(2/5) + 2 = 3, unless I am mistaken, and 3*2 + 5/(5*2) + 2*5 = 17 ... No, this makes little sense. Let's suppose that N and D are the values in cents of the tone and the semitone respectively. Then R = 200/100 = 100 in equal temperament, or roughly 204/90 in Pythagorean tuning. But 204 and 90 are roundings for irrational values, and values for the tone and the semitone would be irrational, I think, in any meantone temperament other than 12-ET — for which therefore R also would be irrational (this being linked to the definition of the cent itself, and of the tone, and of the semitone). To sum up: I utterly fail to understand anything of this.

Any explanation that could be given would be welcome. — Hucbald.SaintAmand (talk) 11:25, 24 January 2018 (UTC)


 * I further tried to make sense of this "R" and I realize that it is linked to the following statement, removed from the article itself but quoted on top of this talk page:
 * "If the semitone is a rational fraction of a tone logarithmically, one gets a finite number of notes in an octave."
 * Easley Blackwood's "R" ratio is this ratio between tone and semitone. Whoever introduced this in the article (it apparently has been there since quite some time) must not have realized that in any temperament that is not an equal temperament, the ratio between tone and semitone never is a rational fraction. To which may be added that 12-ET is the only ET that include tones and semitones properly speaking.
 * That is to say that this mention of "R" and the table labeled "Meantone tunings" that goes with it might have their place in an article about ET, but not here. And the tunings represented in the table are but poor approximations of meantone tunings: the logarithmic values of the meantone tones and semitones are rounded so as to produce rational fractions. In other words, this table forces meantone tunings to correspond to a definition that does not concern them.
 * This section also mixes considerations of n-limit systems, which are just intonation systems, with matter of temperaments with which they have little in common. If whoever wrote this section is still among us, I will gladly hear her or his arguments. — Hucbald.SaintAmand (talk) 15:37, 26 January 2018 (UTC)
 * Dear Hucbald, please see the video below, on the mapping of partials to notes in the syntonic temperament. It directly addresses this issue.JimPlamondon (talk) 10:21, 8 June 2020 (UTC)


 * Something else: The caption to Figure 1, in the same section, claims that the names of the meantone tunings illustrated are «of the form "n/d-comma."» This merely is not true. I presume that n and d, in this, have the same meaning as in the definition of "R" above; but the names of the meantone tunings merely indicate by what fraction of the syntonic comma the fifths are tempered. In addition, this figure is not the one given by Milne 2007, mentioned as its source. Milne 2007 mention 5-limit,but in other contexts, and does not mention 7- or 11-limit. This section as a whole appears to have been written by someone who did not really understand the matter. — Hucbald.SaintAmand (talk) 15:51, 26 January 2018 (UTC)
 * [I am sorry (or am I?) to occupy this talk page, but I stumbled on something more with respect to the above. I more closely read Milne 2007 (actually, Milne, Sethares and Plamodon 2007) and discovered that these authors may be responsable for many inconsistencies in this (and other) articles. I don't want to invoke the argument of authority (that is, I will not disclose my own authority in this domain), but I must forcefully stress that I am in no way impressed by the fact that these authors published in Computer Music Journal or elsewhere: some of what they publish is mere nonsense. They seem responsible for this silly idea of tempering an interval to unison (or to vanishing) − they actually refer about this to "Smith 2006", a websource that seems no more available. Never mind, this idea is unacceptable on account of both the definition of "tempering" and of that of "interval", as I already explained above. They tend to use strange terms that they may not fully understand, for instance when they state about tempered intervals that "using semiotic terminology, the sounded interval is an indexical signifier of the just ratio it approximates". Semiotics has nothing to do here; even just intervals do not "signify" the just ratio to which they correspond. This is a misuse of language. They mix acoustical perfection with musical in-tuneness, for instance when they consider that a musical minor third is musically in tune when it approximates 6:5, which every sensible musician knows is not true in all contexts. Although they explain that tempering an interval may lead to a point where the perceived interval no longer can be perceived as corresponding to the untempered one, they nevertheless call "perfect [!] fifth" intervals that may vary over more than 35 cents – or, as I said above, they can still call "syntonic comma" an interval which they consider "tempered to unison". More generally, they project on music (in which they may not be fully competent) a geometric or algebraic view in which they apparently are competent. Etc.
 * This raises a major problem of Wikipedia itself, the idea that anything published is a valid source... My own opinon, to turn back to meantone, is that history is the best source for us: a meantone tuning is a tuning that historically has been termed "meantone". We may extend this to more recent historical developments in the terminology of people advocating microtonal music (among whose, apparently, the authors mentioned here). In my opinion, however, it would be silly to try to explain the historical usages of the term on the basis a questionable modern terminology – especially at the level of popularization that we should tend to reach. — Hucbald.SaintAmand (talk) 21:27, 26 January 2018 (UTC)
 * Hello, my dear Hucbald! I apologize for my late reply; I only just today stumbled on your criticism of the work of Milne, Sethares, myself, and our collaborators.


 * In this response, I choose not to follow the ad hominem tone of your criticism of our work. Let's get past that, shall we? 😊


 * You seem to be particularly upset by our work's redefining common terms and introducing new ones -- yet this is a normal and inevitable aspect of paradigm shifts. Let us consider one specific case that seems to get particularly under your skin: the definition of the syntonic comma, and the idea of its being "tempered to zero." This boils down to one thing: the definition of the syntonic comma. You define it to be the ratio 81/80. You are right that such a ratio cannot be "tempered to zero." I am happy -- delighted! -- to yield that a ratio cannot be tempered to zero.


 * We then take one step back, and ask, "why does the syntonic comma have that value?" To which our answer is, that ratio is the difference between "four JUST perfect fifths minus two JUST octaves" and "two JUST octaves plus one JUST major third." In Dynamic Tonality, we generalize that definition to be the difference between "four TEMPERED perfect fifths minus two TEMPERED octaves" and "two TEMPERED perfect octaves plus one TEMPERED major third." That is, we see commas not as fixed ratios, but as fixed relationships among tempered intervals. If the amount of tempering is zero, then one gets the ratio 81/80 from this latter definition, as a special case of the generalized definition. (It is necessary for a new paradigm's generalizations to embrace the previous paradigm's assumed realities as special cases.)


 * In Dynamic Tonality, we define a temperament using a period, generator, and comma sequence. The comma sequence is a list of commas that are to be tempered to zero. Obviously, this is going to rile you up, because you are defining all commas as being fixed ratios, not as being fixed relationships among tempered intervals.


 * In the syntonic temperament, the width of the tempered major third is defined, by the appearance of the syntonic comma as the first comma in its comma sequence, as being equal to the width of "four tempered perfect fifths minus two tempered octaves." Hence, it is also defined to be exactly equal to the width of "two tempered octaves plus one tempered major third." That is: in the syntonic temperament, the syntonic comma is tempered to zero. That's what the comma sequence is FOR: to define a list of fixed relationships among intervals, by defining a sequence of commas that are tempered to zero. Importantly, this also informs the mapping of partials to notes (thereby ensuring that the tuning and timbre are "related" as we define that term, ie., that the tuning's notes align with the timbre's partials). You can see, about 40 seconds into the video below, a visual representation of the syntonic comma being tempered to zero in the syntonic temperament.Mapping Partials 1080.webm


 * I understand that this is mind-boggling to someone who sees the musical universe in terms of the fixed ratios of Just Intonation and the Harmonic Series. That's NORMAL in a paradigm shift. You shouldn't feel bad about it. Dynamic Tonality is allowing entities that you have always thought were as fixed as the stars in the heavens to wander about like planets. It's discombobulating. I understand that. I don't hold it against you. I experienced the same thing when Plate Tectonics challenged the textbook geology that I was learning back in the 1970s, and when dynamic programming languages challenged the statically-typed programming languages I had used for decades. It was VERY disconcerting for me to see these fixed things become dynamic, and it is obviously disconcerting to you, too. We are but human, you and I. 😊


 * Even when you launch ad hominem attacks, cite our updating of Wikipedia articles to reflect our published research as being a demonstration of "a major problem of Wikipedia itself," and describe our ideas as being "unacceptable" -- all of which you did in your comments, above, on our work -- I see this merely as a manifestation of the Semmelweis Reflex and confirmation bias, to which we are all subject, more or less. The deeper our knowledge of a given domain, the stronger our Semmelweis Reflex. You know a lot about the Static Timbre Paradigm (even if that name is unfamiliar to you), so your Semmelweis Reflex in defense of it is correspondingly strong. Furthermore, the Semmelweis Reflex is always strongest when those who back a novel paradigm come from outside the domain being challenged, which is surprisingly common (Pasteur was a chemist, so why would any doctor believe his germ theory? Wegner was a meteorologist, so why would any geologist believe his plate tectonics theory? Etc.). I'm a computer scientist and marketer; Sethares is a professor of electrical engineering; and Milne has a BS in music, a MA in "Music, Mind and Technology," and a PhD in "Computational Musicology" -- but you still might consider him to be an "outsider" from your perspective. This is normal in paradigm shifts.


 * I encourage you to take the following steps to minimize your Semmelweis Reflex and confirmation bias in this matter:
 * Stop thinking of commas as being fixed ratios, and start thinking of them as defining fixed relationships among intervals. These relationships are where the fixed ratios come from, after all.
 * Stop thinking of consonance as arising from "fixed ratios of small whole numbers" (which is only true in the special case of Harmonic timbres played in Just Intonation) and start thinking of consonance as arising from the the alignment of a tuning's notes['s fundamentals] with a timbres' partials (which is true for every real-world combination of tuning and timbre).
 * Play our synths, using your computer keyboard as a controller. You will find that you can do some amazing things with Dynamic Tonality -- polyphonic tuning bends, tuning progressions, and novel timbre effects. Compose a piece of music that relies on Dynamic Tonality to enhance tension and release. That is: Pause your arguing about the theory, just for a moment, and compose music with the tool. I have every confidence that you could make music that sounds terrible. Everyone can, using any tool. Instead, make an honest attempt to make music that sounds good, using this new tool. Set up a steelman, not a strawman.


 * Taking the above-listed steps also inevitably leads to viewing music, the history of music, and the history of music theory, etc., through a new lens. How could it be otherwise? A new paradigm gives us new tools with which to examine the past. This is not "re-writing history" in a pejorative sense; it is bringing new knowledge to bear in understanding the past. We've been doing that since Herodotus. That is a good thing! 😊


 * In closing, dear Hucbald, I and my co-authors wish you well in your journey into the new paradigm of Dynamic Tonality. Should you choose to intransigently oppose Dynamic Tonality... well, we expected that some people would do so; it always happens in paradigm shifts. We ask only that you do not abuse your power on Wikipedia to suppress ideas that have ample support in respected journals.


 * Respectfully,
 * JimPlamondon (talk) 08:37, 8 June 2020 (UTC)
 * @JimPlamondon, a short answer which I'll try to keep ad rem. I don't think that the meaning of words can be changed, nor that redefining common terms is a "normal" procedure. You may create new concepts (signifieds), but you would do better to give them new names (signifiers), otherwise you destroy the language itself.
 * To me, a "syntonic comma" remains what it has been said to be from the Renaissance up to recently (I don't think that the expression existed in Antiquity, even if it has been attributed to Ptolemy): the difference between a just major third and four just fifths minus two octaves (and not, as you write, "between 'four JUST perfect fifths minus two JUST octaves' and 'two JUST octaves plus one JUST major third'," which would be between a ditonic third and a just 17th), i.e. 80:81. This entails definitions of all the terms used, among others of "third", "fifth", "octave", and "just" – which I also take as defined by the Renaissance authors (Glareanus, Salinas, Lippius, and others) who spoke of the syntonic comma.
 * You want to "generalize that definition to be the difference between 'four TEMPERED perfect fifths minus two TEMPERED octaves' and 'two TEMPERED perfect octaves plus one TEMPERED major third'," (with the same error as above: the difference is not between a third and a 17th, but between two versions of the third), as if "just" was but a special case of "tempered". But this once again forces the terms to meanings that they do not have. If the fifths are properly tempered (i.e. in 1/4-comma meantone), the major thirds are not tempered (they are just) and there is no comma anymore – which is not at all the same as saying that the comma itself (or the major third!) has been tempered. This is not the meaning of "tempered", neither in the Renaissance nor today. And how (i.e., in what language) could 1/4-comma meantone have made the comma vanish and still be a 1/4-comma meantone?
 * When I read in the Dynamic tonality article that "a vibrating string, a column or air, and the human voice all emit a specific pattern of partials called the Harmonic Series" (I don't mind who wrote that, I mind what I read), I wonder who "sees the musical universe in terms of [...] the Harmonic Series." This has nothing to do with the "universe", there is nothing universal in this. Producing harmonic partials with a vibrating string, a column of air or the human voice is quite a tricky Western achievement, involving not only aspects of playing or singing, but also of instrument making. In does not seem to be an important concern in many non Western musical cultures.
 * Yours friendly, Hucbald.SaintAmand (talk) 08:14, 9 June 2020 (UTC)
 * My dear Hucbald, You wrote above that "I don't think that the meaning of words can be changed, nor that redefining common terms is a 'normal' procedure." Please allow me to draw your attention to Wikipedia's excellent article on Semantic change, which discusses the changes in the meaning of words over time. From my perspective, the gist of the article is this sentence: "Every word has a variety of senses and connotations, which can be added, removed, or altered over time, often to the extent that cognates across space and time have very different meanings." The creators of Dynamic Tonality are merely doing to, and with, musical terminology what has always been done: evolving it. So long as our use is well-defined and internally consistent (given that our own understanding and use of these terms has inevitably evolved over time), then we meet the same standard as everyone else, in every other discipline. We can (and may) do so. It's normal.


 * You wrote above that the syntonic comma should be defined as "the difference between a just major third and four just fifths minus two octaves (and not, as [Jim wrote], "between 'four JUST perfect fifths minus two JUST octaves' and 'two JUST octaves plus one JUST major third'," which would be between a ditonic third and a just 17th), i.e. 80:81." I took the definition that I used from Wikipedia's article on the Syntonic comma, specifically the third sentence in the first bulleted paragraph of the section of the article labelled Relationships, which reads: "The difference between four justly tuned perfect fifths, and two octaves plus a justly tuned major third." If you disagree with that definition, I encourage you to edit the article to correct it.


 * You wrote above that "This entails definitions of all the terms used, among others of 'third', 'fifth', 'octave', and 'just' – which I also take as defined by the Renaissance authors (Glareanus, Salinas, Lippius, and others) who spoke of the syntonic comma." I would argue that they -- Heinrich Glarean (1488-1563), Francisco de Salinas (1513-1590), and Johannes Lippius (1585-1612) -- were all dead and buried before:


 * Joseph Sauveur presented in 1701 his more findings re strings vibrating the notes of the Harmonic Series all at the same time;
 * Hermann Ludwig Ferdinand von Helmholtz published in 1863 On the Sensations of Tone (Ellis' English translation following in 1875), describing his experiments that identified the coincidence of partials as being the source of consonance;
 * Plomp and Levelt described in 1965 their experiments expanding Helmholtz's findings to consider the critical bandwidth;
 * Sethares published in 1992 his experiments expanding the foregoing to encompass arbitrary patterns of vibration.
 * We consider the findings of these later researchers to be quite sufficient to justify an evolution of the nomenclature of music and music theory beyond that of the Renaissance, just as the definition of "atom" has evolved as scientific findings piled up after the Ancient Greeks first gave that new meaning to their word for "indivisible" (and noting that atoms are now seen as being assembled from sub-atomic particles, thus dividing the "indivisible").


 * You wrote previously that our definitions and reasons were "mere nonsense." We, of course, beg to differ.


 * I appreciate the time that you have taken, in this thread, to develop your arguments. Thank you.


 * Respectfully.
 * JimPlamondon (talk) 09:13, 10 June 2020 (UTC)


 * Where to from here? Hi again!  You make some good points - particularly about rewriting history.  But let's not throw out anything that is both accurate and useful.  With your background in the history of music theory, we should be able to produce a page that is
 * accessible to most readers,
 * historically accurate,
 * well-organised, and
 * conducive to further reading.


 * The use of the ratio R by Blackwood is also already history! However, it certainly wasn't the way the various meantone temperaments came about.  Yet those who created and used such temperaments would have been well aware of the relative sizes of the major and minor tones and semitones involved, even when their ratios were irrational numbers.  What does history tell us about that?


 * I believe we should focus on getting a few simple and relevant facts right, clearly stated and supported by reliable sources. Why not let's start with your comment above?:

A regular temperament is one where all the fifths are equal, being equally tempered; a meantone temperament is a regular temperament in which after a number of tempered fifths (and fourths), one comes back to the starting point (or its octave).
 * Surely we can find reliable sources for these definitions? They are both relevant and necessary in this article - as so much is not.


 * PS - by "alinea" above, do you perhaps mean "paragraph", as in the Dutch? yoyo (talk) 20:25, 21 March 2018 (UTC)

@yoyo, three points:

1) I wonder to what extent the use of the ratio R by Blackwood "is already history". Meantone temperament, in its present form, provides no reference. I presume that the idea comes from Blackwood's The Structure of Recognizable Diatonic Tunings, 1985, which would make hardly more than 30 years of history. The statement, in the article, that "Meantone temperaments can be specified [...] by the ratio of the whole tone to the diatonic semitone" is puzzling because that ratio could specify any temperament, meantone or not (i.e. including irregular temperaments). In most temperaments, if not all of them, the ratio would be quite difficult to calculate, because it would be a ratio of irrational values. In ordinary equal temperament, for instance, the ratio is 2 (a tone equals two semitones), but it is the ratio between $\sqrt{2|6}$:1 and $\sqrt{2|12}$:1, and that could hardly have made sense in the 16th century, when meantone first appeared. The tone of 1/4 comma meantone is 1,118033989 (roughly, it is an irrational number corresponding to about 193,16 cents), and its diatonic semitone is roughly 1,069984488 (about 117,11 cents); the ratio is about 1,044906727 – my calculations are made using logarithms; a calculation starting from ratios would have to involve complex roots. So what? Certainly, the users of meantone temperaments were aware that the meantone diatonic semitone was larger than the chromatic semitone, but I strongly doubt that they were able to quantify the difference. And to turn back to Blackwood, his book received mixed reviews in its own time and has not been much quoted since. And it remains that R is not rational, unless in the case of equal temperaments: it should be mentioned in the Equal temperament article, not here. (With the additional problem, though, that the definition of "tone" and "semitone" is unclear in any other ET than 12-ET.)

2) My definitions of regular and meantone temperaments that you quote merely are wrong – my apologies for that. According to J. Murray Barbour (Tuning and Temperament), the meantone temperament properly speaking is 1/4 comma meantone (I don't see why, though), and for the rest the class of meantone temperaments is identical with that of regular temperaments. Of these, the ones that close on the octave are equal temperaments. These definitions can easily be documented, in J. M. Barbour's book and in the references that he quotes.

3) The Greek word παραγραφη ("paragraph") refers to a marginal annotation defining a subsection of a work; more generally, it denotes a subsection with a heading title, often written in red in old books (hence the name "rubric"); an "alinea", as the name indicates, is a short section delimited by line jumps and often an identation. I did not enough realize that this usage is so to say lost in English. In French, we consider that the usage of paragraphe for alinea results from a careless French translation by Microsoft of the terms in Word (the software), and I am of those who try to fight this usage. I checked that you are right about Dutch, thanks.

Hucbald.SaintAmand (talk) 13:31, 22 March 2018 (UTC)


 * @User talk:Hucbald.SaintAmand, thanks for the detailed reply. Briefly:

paragraph n., & v.t. 1. distinct passage or section in book etc., marked by indentation of first line; symbol (usu. ¶) formerly used to mark new ~, now as REFERENCE mark ; detached item of news etc. in newspaper, freq. without heading, whence ~ER, ~IST, ~Y, nn. 2. v.t. Write ~ about (person, thing), arrange (article etc.) in ~s. Hence paragraphIC a., paragraphICALLY adv. [f. F paragraphe f. med. L f. Gk PARA- (graphos f. graphō write) short stroke marking break in sense ]
 * 1) Blackwood's book has certainly been influential on a growing number of microtonalists, especially in his native USA.  It's a major (early modern) historic source for many discussions on  the very active xenharmonic and other microtonal groups on social media.  So please don't discount it altogether.  My only concern with quoting Blackwood or his concepts here is that they're relevant to the topic of meantone temperament.  As a modern way of interpreting such temperaments, they make sense.  In particular, the ratio R provides another view of the effect such temperaments have on the quality of the resulting tunings; one of more immediate perceptual relevance than what fraction of a comma was used to construct the tuning.
 * 2) I know that Barbour's book is often cited, but he's neither the first nor the last theorist to classify or define temperaments.  My understanding of the topic is that if one speaks of "the meantone temperament" without qualification, one usually means quarter-comma, but that the expression "meantone temperaments" means the whole class of temperaments in which the tone is some one of the various mathematical kinds of mean of the two tones provided by [usually 5-limit] just intonation: the 9:8 major tone and the 10:9 minor tone.  (And for those interested in 7-limit tunings and beyond, even the 8:7 septimal tone may enter the equation.)  So I don't think we have to take Barbour as prescriptive of the "proper" usage of the term in English.  Even authoritative sources only gained their authority by their accurate description of a state of affairs, rather than by fiat.  What we need the article to do is to reflect the actual usage in reliable sources; I'm going to look for others to see what they have to say.
 * 3) So French also uses "alinea"?  I didn't know that; my French teachers in the 1960s, including the erudite polyglot Dr. Kowalski and the charming Francophiles Mr. & Mrs.  Ryder, used "paragraphe" in usages corresponding to the English "paragraph".  And its use in English absolutely predates Microsoft!  For example, it's found in my Concise Oxford Dictionary, Fifth Edition (1964) thus:
 * Nor have I ever seen "alinea" used in English (except here by yourself); it's certainly not in the dictionary I just quoted from. So my guess is that English probably hasn't lost it; it may never have had it.  Perhaps the Normans of William the Conqueror's court would have used it?   But in the end, the English language even conquered their descendants ;-).  By all means, keep French pure – if you can! (L'Académie Française hasn't kept the people from eating "le bifteck" on "le weekend".)  But this bit of linguistics, diverting tho' it's been, is a digression from our topic …


 * So, as I've said, I'm going to seek other reliable sources on the usage of "meantone temperament"; also on "regular temperament" and "equal temperament" (even tho' your view on the latter seems the only reasonable one) – perhaps you have some better ones to hand? But what do you propose we do with the section on "R" - does it need a rewrite?  yoyo (talk) 04:05, 25 March 2018 (UTC)


 * @yoyo, let me begin with a short answer about Murray Barbour. Of course he was neither the first nor the last, but his approach (I think it was his PhD thesis) is extremely schorlarly and his book is excellently documented. It remains in my opinion one of the most serious among accessible sources about temperaments. His usage of "meantone" is exactly that which you mention and about which I agree: without qualification, it refers to 1/4-comma meantone, and otherwise to regular temperaments at large. We may find confirmation in other sources if necessary. If it appeared that the microtonalists developed another meaning of "meantone", I'd argue that this is a modern usage, and that it should be presented as such. Barbour's description corresponds to the historical usage at least up to the publication of his book.
 * About the section on "R", I have no proposition to make, because my own inclination would be to remove it entirely. I reckon that this would have been rather extreme and this is why I opened the discussion in this page. And I am very thankful for your participating in the discussion.
 * You write that "the ratio R provides another view of the effect such temperaments have on the quality of the resulting tunings; one of more immediate perceptual relevance than what fraction of a comma was used to construct the tuning". I am not the slightest convinced by this. If R is calculated on logarithmic values of the intervals (but our article fails to say so!), for 1/4-comma meantone R is about 1,65; for 1/3-comma meantone it is about 1,50. So what? What is the meaning of this difference? What does it say of the "quality" of these tunings? (And, anyway, what is the quality of a tuning?).
 * 1/4-comma meantone tempers its fifths by 1/4 comma, i.e. by about 2,25% of a semitone: this is a value that I can figure out, all the more so that I know how a meantone fifth sounds; I know in addition that this value of the fifth results in perfect major thirds, and therefore that the minor thirds become 1/3 of a comma too narrow. This is the particular quality of 1/4-comma meantone: perfect major thirds and acceptable fifths; I don't know what kind of quality could result from the fact that the tone is about 1,65 semitone. In the case of 1/3-comma meantone, the fifths are slightly more tempered (closer to 3,5% of a semitone), resulting in perfect minor thirds and with major thirds a 1/3 of a comma too wide narrow. I know that this tuning in most cases will be less satisfying because (a) the fifths are less pure; and (b) the minor thirds so tuned are theoretically pure (ratio 6/5), but musicians know that minor thirds often do not correspond to that theory. Now I admit that I am not the ordinary Wikipedia reader in all this. I have some (limited) experience of tuning meantone temperaments and above all I have a strong knowledge of the theoretical aspects of tuning.
 * Our aim, when writing a WP article, should be to explain complex things in simple terms for the ordinary reader. I think I can explain why a tuning with better fifths and better major thirds often is more satisfying than one with slightly worse fifths and better theoretical minor thirds. I might even be able to explain why it is less important to have theoretically pure minor thirds, because the actual music does not often make use of them. But I feel unable to explain (because I do not understand it myself) why a ratio of 1,65 between the tone and the semitone is better or worse than one of 1,50. I know that the ratio is 2 for 12-TET, but that does not help me, unless somebody can explain to me why a regular temperament would be better (or worse) if it is closer to 12-TET. (Barbour somehow appears to think so, but this is not really true. He merely thinks that if one must measure the deviation of a tuning, one must measure against a yardstick with equal graduations – much as if one wants to measure a length, one does best to use centimetres that are all the same.)
 * As for the other tunings advocated by the microtonalists, especially for the equal temperaments other than 12-TET, the meaning of R puzzles me all the more than I don't understand what the terms "tone" and "semitone" mean in these cases.
 * To makes thinks short, I think that this section on R utterly fails to explain things in simple terms. I think in addition that this was not its aim and that whoever added it only tried to show themselves more intelligent that any of us. — Hucbald.SaintAmand (talk) 22:12, 25 March 2018 (UTC)

I wonder whether anyone ever tried to understand what the table at the end of the section "Meantone temperaments" really means – what it really says about meantone temperaments. Let's consider the line marked "1/4", i.e. 1/4-comma meantone.

The second column, "Approximate size of the fifth in octaves", reads "18/31". It took me some time to figure out what "the fifth in octaves" could mean (as a matter of fact, I still don't know), but I soon understood what "18/31" meant: if you divide the octave in 31 equal parts, then 18 parts gives you an approximation of the 1/4-comma meantone fifth. From there one begins to understand that the table is not about "Meantone temperaments", despite what it claims, but about "How to approximate meantone temperaments with equal divisions of the octave".

The third column gives the "Error (in cents)", that is, how much the approximation deviates from the meantone fifth it is meant to represent. In the case considered, one reads "+1.95765×10−1". I dont know what this apparent precision means, but would it not be simpler to write "0,195765082 cents" or more simply (since we are dealing with approximations), "about 0,2 cents"? It must be realized that, in order to appreciate the approximation for the other degrees of the tuning, one must multiply this value by the number of fifths necessary to reach them. For the 8th degree of the chromatic scale (say, G♯, reached after 8 fifths, C–G–D–A–E–B–F♯–C♯–G♯), the error reaches about 1,7 cents. This is the maximum deviation between 1/4 comma meantone and 31-ET, and it begins to appear that 31-ET indeed produces a good approximation of 1/4 comma meantone.

The last column gives the "Ratio R", which still puzzles me. One reads "5/3", which gives "the ratio of the whole tone to the diatonic semitone", as explained higher in the article. But it must be realized that this is the ratio in 31-ET, not in 1/4 comma meantone (even if, once again, the first is a good approximation of the second). What this means (and note how my explanation here is simpler than the one given in the article) is that the whole tone in 31-ET takes 5 units (5/31 of the octave) while the diatonic semitone takes 3 units; one easily deduces that the chromatic semitone takes the difference, 2 units. The ratio between the two semitones is 1,5 in 31-ET; it is 1,54 in 1/4-comma meantone. Or the ratio from whole tone to diatonic semitone is 1,667 in 31-ET, while it is 1,65 in 1/4 comma meantone. So what? This ratio may give us an idea of the quality of the approximation, but certainly not "of the melodic qualities of the tuning" (which tuning, meantone or its approximation in ET?).

I repeat: this table and all the commentary that accompanies it concern how to approximate meantone tunings with equal temperaments; in addition, it succeeds in making complex what otherwise would be relatively simple. I think that a mere comparison of the values in cents of the degrees of each meantone tuning and the corresponding values in the ET approximations would say more.

Let me add, to conclude, that some of the "historically notable meantone tunings" in the left column of the table are not at all historically notable, that I know; the only thing notable is that they can be approximated by ET. I defy anyone to produce any historical source for 1/315-comma meantone, and any practical source for 1/2-comma meantone. — Hucbald.SaintAmand (talk) 09:50, 14 July 2019 (UTC)

Dynamic Tonality paradigm
Having recently completed a major revision to Wikipedia's Dynamic Tonality article -- which I encourage the followers of this page to review -- I have recently started updating other articles to refer to it and/or incorporate some of its content (e.g., its explanatory videos). The meantone article is an early beneficiary/victim of this process. I look forward to the inevitable debate over its beneficiary/victim-hood. Such debates are inevitable whenever a new paradigm arises, and serve to improve the content, explanation, and rate of adoption of the new paradigm. 😊 JimPlamondon (talk) 07:02, 8 June 2020 (UTC) - An encyclopedia isn't the place to promote or introduce a "new paradigm". — Preceding unsigned comment added by 97.82.109.213 (talk) 19:39, 15 June 2021 (UTC)

...therefore, the "New Paradigm" material needs deletion. If it isn't deleted within a calendar-month from today (June 16th, 2021), I'll delete it myself.

The article should be about Meantone Temperament, not about a proposal for a "New Paradigm". Visitors to the article are only looking for an explanation of what Meantone Temperament is, why it was used, and how it's generated. They don't look the subject up because they want new paridigms to be promoted to them.

Discussion between JimPlamondon and Hucbald.SaintAmand
I create this new section and move to here the discussion between Jim Plamodon and me. I am not sure this is a normal procedure in WP, but I thougth that this would make things clearer. The discussion results from the section on "R"? above, but does not directly concern it. I thought this would be clearer if it appeared in a separate section. Hucbald.SaintAmand (talk) 09:47, 11 June 2020 (UTC)


 * Hello, my dear Hucbald! I apologize for my late reply; I only just today stumbled on your criticism of the work of Milne, Sethares, myself, and our collaborators.


 * In this response, I choose not to follow the ad hominem tone of your criticism of our work. Let's get past that, shall we? 😊


 * You seem to be particularly upset by our work's redefining common terms and introducing new ones -- yet this is a normal and inevitable aspect of paradigm shifts. Let us consider one specific case that seems to get particularly under your skin: the definition of the syntonic comma, and the idea of its being "tempered to zero." This boils down to one thing: the definition of the syntonic comma. You define it to be the ratio 81/80. You are right that such a ratio cannot be "tempered to zero." I am happy -- delighted! -- to yield that a ratio cannot be tempered to zero.


 * We then take one step back, and ask, "why does the syntonic comma have that value?" To which our answer is, that ratio is the difference between "four JUST perfect fifths minus two JUST octaves" and "two JUST octaves plus one JUST major third." In Dynamic Tonality, we generalize that definition to be the difference between "four TEMPERED perfect fifths minus two TEMPERED octaves" and "two TEMPERED perfect octaves plus one TEMPERED major third." That is, we see commas not as fixed ratios, but as fixed relationships among tempered intervals. If the amount of tempering is zero, then one gets the ratio 81/80 from this latter definition, as a special case of the generalized definition. (It is necessary for a new paradigm's generalizations to embrace the previous paradigm's assumed realities as special cases.)


 * In Dynamic Tonality, we define a temperament using a period, generator, and comma sequence. The comma sequence is a list of commas that are to be tempered to zero. Obviously, this is going to rile you up, because you are defining all commas as being fixed ratios, not as being fixed relationships among tempered intervals.


 * In the syntonic temperament, the width of the tempered major third is defined, by the appearance of the syntonic comma as the first comma in its comma sequence, as being equal to the width of "four tempered perfect fifths minus two tempered octaves." Hence, it is also defined to be exactly equal to the width of "two tempered octaves plus one tempered major third." That is: in the syntonic temperament, the syntonic comma is tempered to zero. That's what the comma sequence is FOR: to define a list of fixed relationships among intervals, by defining a sequence of commas that are tempered to zero. Importantly, this also informs the mapping of partials to notes (thereby ensuring that the tuning and timbre are "related" as we define that term, ie., that the tuning's notes align with the timbre's partials). You can see, about 40 seconds into the video below, a visual representation of the syntonic comma being tempered to zero in the syntonic temperament.Mapping Partials 1080.webm


 * I understand that this is mind-boggling to someone who sees the musical universe in terms of the fixed ratios of Just Intonation and the Harmonic Series. That's NORMAL in a paradigm shift. You shouldn't feel bad about it. Dynamic Tonality is allowing entities that you have always thought were as fixed as the stars in the heavens to wander about like planets. It's discombobulating. I understand that. I don't hold it against you. I experienced the same thing when Plate Tectonics challenged the textbook geology that I was learning back in the 1970s, and when dynamic programming languages challenged the statically-typed programming languages I had used for decades. It was VERY disconcerting for me to see these fixed things become dynamic, and it is obviously disconcerting to you, too. We are but human, you and I. 😊


 * Even when you launch ad hominem attacks, cite our updating of Wikipedia articles to reflect our published research as being a demonstration of "a major problem of Wikipedia itself," and describe our ideas as being "unacceptable" -- all of which you did in your comments, above, on our work -- I see this merely as a manifestation of the Semmelweis Reflex and confirmation bias, to which we are all subject, more or less. The deeper our knowledge of a given domain, the stronger our Semmelweis Reflex. You know a lot about the Static Timbre Paradigm (even if that name is unfamiliar to you), so your Semmelweis Reflex in defense of it is correspondingly strong. Furthermore, the Semmelweis Reflex is always strongest when those who back a novel paradigm come from outside the domain being challenged, which is surprisingly common (Pasteur was a chemist, so why would any doctor believe his germ theory? Wegner was a meteorologist, so why would any geologist believe his plate tectonics theory? Etc.). I'm a computer scientist and marketer; Sethares is a professor of electrical engineering; and Milne has a BS in music, a MA in "Music, Mind and Technology," and a PhD in "Computational Musicology" -- but you still might consider him to be an "outsider" from your perspective. This is normal in paradigm shifts.


 * I encourage you to take the following steps to minimize your Semmelweis Reflex and confirmation bias in this matter:
 * Stop thinking of commas as being fixed ratios, and start thinking of them as defining fixed relationships among intervals. These relationships are where the fixed ratios come from, after all.
 * Stop thinking of consonance as arising from "fixed ratios of small whole numbers" (which is only true in the special case of Harmonic timbres played in Just Intonation) and start thinking of consonance as arising from the the alignment of a tuning's notes['s fundamentals] with a timbres' partials (which is true for every real-world combination of tuning and timbre).
 * Play our synths, using your computer keyboard as a controller. You will find that you can do some amazing things with Dynamic Tonality -- polyphonic tuning bends, tuning progressions, and novel timbre effects. Compose a piece of music that relies on Dynamic Tonality to enhance tension and release. That is: Pause your arguing about the theory, just for a moment, and compose music with the tool. I have every confidence that you could make music that sounds terrible. Everyone can, using any tool. Instead, make an honest attempt to make music that sounds good, using this new tool. Set up a steelman, not a strawman.


 * Taking the above-listed steps also inevitably leads to viewing music, the history of music, and the history of music theory, etc., through a new lens. How could it be otherwise? A new paradigm gives us new tools with which to examine the past. This is not "re-writing history" in a pejorative sense; it is bringing new knowledge to bear in understanding the past. We've been doing that since Herodotus. That is a good thing! 😊


 * In closing, dear Hucbald, I and my co-authors wish you well in your journey into the new paradigm of Dynamic Tonality. Should you choose to intransigently oppose Dynamic Tonality... well, we expected that some people would do so; it always happens in paradigm shifts. We ask only that you do not abuse your power on Wikipedia to suppress ideas that have ample support in respected journals.


 * Respectfully,
 * JimPlamondon (talk) 08:37, 8 June 2020 (UTC)
 * @JimPlamondon, a short answer which I'll try to keep ad rem. I don't think that the meaning of words can be changed, nor that redefining common terms is a "normal" procedure. You may create new concepts (signifieds), but you would do better to give them new names (signifiers), otherwise you destroy the language itself.
 * To me, a "syntonic comma" remains what it has been said to be from the Renaissance up to recently (I don't think that the expression existed in Antiquity, even if it has been attributed to Ptolemy): the difference between a just major third and four just fifths minus two octaves (and not, as you write, "between 'four JUST perfect fifths minus two JUST octaves' and 'two JUST octaves plus one JUST major third'," which would be between a ditonic third and a just 17th), i.e. 80:81. This entails definitions of all the terms used, among others of "third", "fifth", "octave", and "just" – which I also take as defined by the Renaissance authors (Glareanus, Salinas, Lippius, and others) who spoke of the syntonic comma.
 * You want to "generalize that definition to be the difference between 'four TEMPERED perfect fifths minus two TEMPERED octaves' and 'two TEMPERED perfect octaves plus one TEMPERED major third'," (with the same error as above: the difference is not between a third and a 17th, but between two versions of the third), as if "just" was but a special case of "tempered". But this once again forces the terms to meanings that they do not have. If the fifths are properly tempered (i.e. in 1/4-comma meantone), the major thirds are not tempered (they are just) and there is no comma anymore – which is not at all the same as saying that the comma itself (or the major third!) has been tempered. This is not the meaning of "tempered", neither in the Renaissance nor today. And how (i.e., in what language) could 1/4-comma meantone have made the comma vanish and still be a 1/4-comma meantone?
 * When I read in the Dynamic tonality article that "a vibrating string, a column or air, and the human voice all emit a specific pattern of partials called the Harmonic Series" (I don't mind who wrote that, I mind what I read), I wonder who "sees the musical universe in terms of [...] the Harmonic Series." This has nothing to do with the "universe", there is nothing universal in this. Producing harmonic partials with a vibrating string, a column of air or the human voice is quite a tricky Western achievement, involving not only aspects of playing or singing, but also of instrument making. In does not seem to be an important concern in many non Western musical cultures.
 * Yours friendly, Hucbald.SaintAmand (talk) 08:14, 9 June 2020 (UTC)
 * My dear Hucbald, You wrote above that "I don't think that the meaning of words can be changed, nor that redefining common terms is a 'normal' procedure." Please allow me to draw your attention to Wikipedia's excellent article on Semantic change, which discusses the changes in the meaning of words over time. From my perspective, the gist of the article is this sentence: "Every word has a variety of senses and connotations, which can be added, removed, or altered over time, often to the extent that cognates across space and time have very different meanings." The creators of Dynamic Tonality are merely doing to, and with, musical terminology what has always been done: evolving it. So long as our use is well-defined and internally consistent (given that our own understanding and use of these terms has inevitably evolved over time), then we meet the same standard as everyone else, in every other discipline. We can (and may) do so. It's normal.


 * You wrote above that the syntonic comma should be defined as "the difference between a just major third and four just fifths minus two octaves (and not, as [Jim wrote], "between 'four JUST perfect fifths minus two JUST octaves' and 'two JUST octaves plus one JUST major third'," which would be between a ditonic third and a just 17th), i.e. 80:81." I took the definition that I used from Wikipedia's article on the Syntonic comma, specifically the third sentence in the first bulleted paragraph of the section of the article labelled Relationships, which reads: "The difference between four justly tuned perfect fifths, and two octaves plus a justly tuned major third." If you disagree with that definition, I encourage you to edit the article to correct it.


 * You wrote above that "This entails definitions of all the terms used, among others of 'third', 'fifth', 'octave', and 'just' – which I also take as defined by the Renaissance authors (Glareanus, Salinas, Lippius, and others) who spoke of the syntonic comma." I would argue that they -- Heinrich Glarean (1488-1563), Francisco de Salinas (1513-1590), and Johannes Lippius (1585-1612) -- were all dead and buried before:


 * Joseph Sauveur presented in 1701 his more findings re strings vibrating the notes of the Harmonic Series all at the same time;
 * Hermann Ludwig Ferdinand von Helmholtz published in 1863 On the Sensations of Tone (Ellis' English translation following in 1875), describing his experiments that identified the coincidence of partials as being the source of consonance;
 * Plomp and Levelt described in 1965 their experiments expanding Helmholtz's findings to consider the critical bandwidth;
 * Sethares published in 1992 his experiments expanding the foregoing to encompass arbitrary patterns of vibration.
 * We consider the findings of these later researchers to be quite sufficient to justify an evolution of the nomenclature of music and music theory beyond that of the Renaissance, just as the definition of "atom" has evolved as scientific findings piled up after the Ancient Greeks first gave that new meaning to their word for "indivisible" (and noting that atoms are now seen as being assembled from sub-atomic particles, thus dividing the "indivisible").


 * You wrote previously that our definitions and reasons were "mere nonsense." We, of course, beg to differ.


 * I appreciate the time that you have taken, in this thread, to develop your arguments. Thank you.


 * Respectfully.
 * JimPlamondon (talk) 09:13, 10 June 2020 (UTC)
 * @JimPlamondon, a few points:
 * 1) Semantic changes, like all linguistic changes, are the result of an evolution. Benveniste writes that "the system of language changes only very slowly and on the pressure of internal necessities, so that – this is a condition that must be stressed – in normal conditions of life, speaking humans never are witnesses of the linguistic change" ("Structures et analyses", in Problèmes de linguistique générale II, p. 96, my translation). Linguistic changes cannot be decided.
 * 2) Please compare what you wrote in the Syntonic comma article:
 * – "The difference between four justly tuned perfect fifths, and two octaves plus a justly tuned major third."
 * and what you wrote here above:
 * – "The difference between four just perfect fifths minus two just octaves and two just octaves plus one just major third."
 * As you can see, the second statement includes a "minus two just octaves" that is unneeded.
 * 3) Neither Sauveur, nor Helmholtz, nor Plomp and Levelt modified the meaning of "third", "fifth", "octave" and "just". You may note in particular that "fifth" may appear to make sense only in a diatonic context (where the interval of a fifth indeed is the distance between the first and the fifth degrees). Passing to a chromatic context means that a fifth is the interval between the first and the seventh degree! But the size of the interval is not really modified, it remains somewhere between, say, 695 and 702 cents. I am not sure that your "generator" interval between 686 and 720 cents (i.e. with a possible variation of more than 80 cents) always qualifies as a (tempered) fifth.
 * 4) The change of the understanding of "atom" is the result of a gradual change of conception since Antiquity: that is how concepts normally change. You want the names of concepts to change according to your own theories (and you apparently also consider that we should abandon music produced by acoustic instruments). I don't think that possible. See that as "a manifestation of [my] Semmelweis Reflex" if you want, I don't mind.
 * Hucbald.SaintAmand (talk) 09:47, 11 June 2020 (UTC)
 * I must apologize for my mistake above. A "fifth" in a span between 686 and 720 cents obviously varies only by at most 34 cents. I for a moment mistakenly considered that 686 and 720 were frequencies, which would mean a span of more than 80 cents. But this does not change much to the matter. A variation of 34 cents, quite larger than any type of (historically recognized) comma, also is wider than any variation of the fifth documented in historical temperaments – which is about 7 cents. — Hucbald.SaintAmand (talk) 21:37, 13 June 2020 (UTC)

To add to article
To add to this article: mention of sixth-comma meantone. Why isn't it already included here? 173.88.246.138 (talk) 14:09, 24 January 2021 (UTC)
 * Should the article mention all sorts of meantone temperament? There are many, among which 1/6-comma is not so notable. Sorge mentioned it in 1748 as one of Silbermann's temperaments, and it has been described in France in 1758 as an approximation of the division of the octave in 55 intervals. That's all, so far as I know, and the practical purpose of this temperament remains unclear. — Hucbald.SaintAmand (talk) 17:55, 25 January 2021 (UTC)

1/6 Comma Meantone compromises between the minor 3rd & the 5th. It sounds (from its definition) the most appealing to me, though the only Meantone Temperament that I've heard was 1/4 Comma Meantone, and it sounded incomparably better than Equal-Temperament, in harpsichord music with chords. The harmonic content was much, much more present and clear, in comparison to Equal-Temperament.

To add to this article: mention of sixth-comma meantone. Why isn't it already included here? 173.88.246.138 (talk) 20:22, 31 December 2021 (UTC)
 * 1/6-comma meantone is mentioned in the table in the Meantone temperaments subsection. What do you mean when you say that it "compromises" between the minor 3d and the 5th? The 5th in 1/6-comma meantone by definition is narrowed by 1/6 of a comma, i.e. about 4 cents, and the minor 3d is widened by 3/6 = 1/2 of a comma, about 12 cents. In what sense does that "compromise"? And how do you tune it? It would involve tuning the tritone a comma narrower than the Pythagorean one, but how does one do that? — Hucbald.SaintAmand (talk) 10:30, 1 January 2022 (UTC)

Proposed, very brief, improvement & replacement for the article
With its uniform sequence of half-steps, an equal-tempered instrument can start anywhere in the instrument’s range, and make, starting from there, any scale that uses its half-steps. Likewise with a scale gotten from the Pythagorean sequence of 5ths (product of factors of 3/2) and frequencies gotten by multiplying or dividing those by two. …except that somewhere there must be a discontinuity because the 3/2 factor of those 5ths doesn’t multiply to something evenly divisible by two. But, other than, aside from, that one discontinuity, a Pythagorean instrument, like an equal-tempered instrument, can start anywhere in the instrument’s range and make the same Pythagorean scale. Like an equal-tempered instrument, a Pythagorean-tuned instrument doesn’t achieve the Just scale’s harmonious intervals. Equal Temperament & Meantone Temperament improve the Pythagorean’s 3rds, by compromising the 5ths. Meantone temperament merely differs by adjusting the 5th’s 3/2 by a factor that fixes or improves the 3rds. The ¼ comma is the 4th root of the factor by which the Pythagorean major 3rd differs from the true (Just) major 3rd. The 1/3 comma is the 3rd root of the factor by which the Pythagorean major 6th differs from the true (Just) major 6th. The 1/6 comma is the square-root of the 1/3 coma. In the sequence of 5ths, adjusting each 5th’s 3/2 (in the appropriate direction) by one of the above factors will make the following improvements in the 3rds: ¼ comma: Will make the major 3rds true. (because the major 3rd is gotten from a product of four factors of 3/2) 1/3 comma: Will make the major 6ths (& therefore the minor 3rds) true. (because the major 6th is gotten by a product of 3 factors of 3/2) 1/6 comma: Will compromise between the minor 3rd & the 5th. --- I'll comment inline on what you say below:


 * The above has shortly been added on top of the article itself. This addition was removed by Jonesey95, and rightly so, I think. The above appears to attempt a description of what a "regular" temperament is, i.e. a temperament with all fifths equal, unless for the wolf fifth (if any).

- It was more than attempt. It was a description of the generation of Meantone Temperament...three of them in particular. Of course I didn't mention the much less familiar term "regular temperaments", and there was no need to.


 * But the wolf fifth does more than merely add a mere "discontinuity". It results in that any music played, because it should not involve the wolf fifth, should not exceed the range of (usually) 12 notes, say from E♭ to G♯, in the order of the cycle of fifths. A regular temperament allows pieces to "start anywhere in the instrument's range" only if they do not exceed that range. But that means that pieces transposable by one fifth must not exceed 11 notes of the cycle, pieces transposable by two fifths must not exceed 10 notes, etc. This is a very strong restriction. It applies in all regular temperaments, including meantone temperaments; ET is the only one to escape it.

- Yes, I intended to tell only a brief motivation & procedure for generating Meantone Temperaments. I didn't want to lengthen the section by talking about the problems. But yes, you're right--I shouldn't say that a scale can be started anywhere in the instrument's range. I should have qualified that by mentioning that the choice of where a scale or piece can be played is limited by the need to not use the wolf-note resulting from the discontinuity. ...and by the need for more accidental-keys than might be feasible on the keyboard. So yes it isn't literally true that a piece can be played starting anywhere in the instrument's range, and something about that should probably be said in that brief introductory description (which the article needs).

But Meantone Temperament does achieve something in that direction, by making a variety of keys available, and sounding a lot better than Equal Temperament in some keys. - Discussing all this may find its place in the Regular temperament article better than here. - No. It belongs in the Meantone Temperament article. Meantone Temperament is the term that people look-up. ...and it belongs at the top of the article. -
 * As to comparing meantone temperaments in terms of "factors," I am not sure to understand what that means.

- A factor is one of at least two numbers to be multiplied together. That's what that word means in mathematics. --- To say, for instance, that "The ¼ coma is the 4th root of the factor by which the Pythagorean major 3rd differs from the true (Just) major 3rd" seems to me unduly complex. -- But without saying that, it's quite impossible to say what Meantone Temperament is.

It isn't possible to avoid some mathematics when explaining how Meantone Temperament is generated. But anyone who wants to find out what Meantone is, and how it was made, will expect to, & be willing to, read a little mathematics. ...because it's unavoidably a mathematical matter.

You want unduly complex??? :D Just look at the current article's pretentious, over- technical, pseudoscientific language, full of undefined terms & symbols. -

The syntonic comma (with two m's!)

The section was deleted after I fixed the spelling. --- You said:

The syntonic comma is the interval between a pure third and the third obtained by four pure fifths (or fourths). Therefore, diminishing each of these fifths by ¼ comma will yield a diminution by a full comma after four fifths, so reaching a pure third. In this, "¼ comma" merely means the 4th part (¼) of a comma, and I don't see that "the 4th root of the factor [etc.]" in any way simplifies this description

- You're speaking in terms of logarithmic measure. But when we're talking about ratios (like 3/2), it's simpler to not bring in logarithms. A sequence of four 5ths means that four 3/2 factors are multiplied together.

Meantone Temperament is about ratios, and, instead of obscuring that by logarithmic-measures, the subject is clearly & simply explained by ratios & factors of 3/2. Don't lose the clarity by muddying it with logarithmic-measures. — Preceding unsigned comment added by 97.82.109.213 (talk) 01:03, 15 June 2021 (UTC)

, which I think is also relatively clearly illustrated in the figure at the outset of the article.
 * It might be that the description above is more rigourous (it really depends in what terms the comma is defined, in frequency ratio, 81:80, or in logarithms, ~22 cents), but it seems to me certainly less understandable.

So you think logarithms are more understandable to most people? Of course it's universal & unavoidable to speak of logarithmic measures like cents, and ET half & full steps. But to tell the mechanics, it's clearer, simpler & better to speak of a 5th as a factor equal to 3/2, to tell what's really going on. -- The fact is that ¼ comma meantone was described under this name at a time when neither the idea of "4th root" on the one hand, nor that of logarithms on the other hand, already were in existence. That it was described in terms of "¼" merely evidences an intuition of what logarithms would be – the same intuition makes this description more understandable today for people who really figure out neither "4th roots", nor "logarithms." - But if you speak 1/4 of the syntonic coma, without clarifying that you're talking about logarithmic-measure, then you're deceiving the reader. You're not doing her/him any favor. And if you bring up logarithmic-measure, you're complicating the explanation. --- And, once again, this discussion should better appear in the Regular temperament article -- When people want to find out where Meantone Temperament comes from, they look up Meantone Temperament, not Regular Temperament.

(which I also find hardly understandable, but that's another problem). — Hucbald.SaintAmand (talk) 19:36, 14 June 2021 (UTC)
 * I removed the text because it was malformatted, unsourced, and in the wrong place in the article, not for any content-related reasons. – Jonesey95 (talk) 21:34, 14 June 2021 (UTC)

--- 1. Malformatted, compared to what? The rest of the article? :D

In the section that I added, I explained the generation & construction of Meantone Temperament briefly, clearly, in ordinary language, & without the pretentious pseudoscientific undefined terms & symbols and unlabeled graphs that your article is full of.

2. Unsourced: By Wikipedia rules, sources aren't needed for uncontroversial and obvious arithmetic.

3. Wrong place: No. People look up "Meantone Temperament" to find out what it is, and how it's generated or constructed. That should be first in the article. What would you put first? Something like "Expressed in terms of the Dynamic Tonality paradigm" :D

4. Try to remember that discussion should precede deletion.

Rewriting the article
This message is addressed first of all to user 97.82.109.213, but also to everyone interested in this article.

@97.82.109.213, you are welcome to participate in the reorganization of this article, but I think that you should first consider the following: This being said (and don't take it negatively), I agree with you that the article became sort of a mess and that browsing through its revisions shows that it had been much better some years ago, see for instance its state early in 2009. However, if you reread the messages above, you'll realize that this whole affair may soon become a war and that we probably would do better to carefully plan the modifications (which probably also involves dispatching some of the information in other articles).
 * 1. You should sign your contributions, and not let robots do it in your place.
 * 2. In order to best be able to do that, you should create your own WP page, possibly using a surname for the purpose (as recommended by WP itself).
 * 3. You should better not add your comments inline within text already written, as it makes it very difficult to follow who said what. This is one of the main reasons why I create this new section, where your comments will be welcome below what I write here.
 * 4. You write that "discussion should precede deletion", but I'd say even more: discussion should precede modification. Reread the discussions above, and you'll see that some of what you suggest already had been discussed, in one way or another.

Concerning what we already discussed above, about 1/4-comma meantone involving a quarter, or a fourth root, of a syntonic comma, it strikes me that meantone tunings were conceived at a time when neither roots nor logarithms made any sense. On the other hand, logarithmic thinking seems quite natural to human thinking at large (everyone would easily agree that an octave is a fifth plus a fourth and few would understand that it may be a fifth times a fourth). Leave me some time to find out when 1/4-comma meantone was first named that way.

An evident problem of the article is that it lacks references. There already is a template to the effect that it needs reorganization, but it would deserve another template calling for references.

Anyway, let's begin with a discussion of all this here. — Hucbald.SaintAmand (talk) 10:26, 17 June 2021 (UTC) --- If the article has been modified any since the most recent discussion, the modification isn't enough. e.g., this following paragraph, whose meaning, if any, is entirely unclear:

[quote]

"Meantone" can receive the following equivalent definitions:

The meantone is the geometric mean between the major whole tone (9:8 in just intonation) and the minor whole tone (10:9 in just intonation). The meantone is the mean of its major third (for instance the square root of 5:4 in quarter-comma meantone).

[/quote]

Are you sure that powers & roots, including 4th roots, were unknown in the days of Meantone temperament, and that it wasn't known that the ratios for the notes in the Pythagorean & Meantone scales were gotten by raising a (true or temperament-adjusted) 3/2 to a power?

The ancient greeks used cube-roots, and invented a mechanical device to generate them, for the purpose of catapult-design. You can't say that powers & roots were unknown during the days when Meantone Temperament was devised & used.

When speaking of a quarter of the syntonic comma, what is the nature of the quantity of which that's a 4th? Some unspecified goo? It certainly doesn't constitute an explanation.

In explaining any of this to readers interested in an explanation, you can't leave-out ratios. Ratios of small whole numbers are the basis of consonance...due to the harmonics of musical tones.

Therefore, since the reader must know that, then you can either explain Meantone temperament in terms of ratios, or else introduce logarithms.

I've tried the explanations out with a few people, and found that multiplictive-powers, and roots (easily and briefly explained as the number that, when raised to the corresponding power, would result in some specified number), are a lot more immediately understood, and more welcome, than bringing logarithms into it.

Therefore, since ratios can't be ignored in any explanation of consonance, scales & temperament, and since powers & roots are a lot better-understood and welcomed than logarithms, then Meantone Temperament is best explained in terms of powers & roots.

I notice that the "New Paradigm" is still present in the article, and, in fact, still predominates in the article. Does it occur to anyone that there's no need to complicate an otherwise simple & brief topic with all that unnecessary terminology & unnecessary talk??? Your article has been raided & vandalized to promote someone's pet theories & elaborate generalizations. It's astounding that the article still remains in that state.

It's a simple & brief matter to tell what the situation was, and how it was remedied.

-- p.s. The graph now at the beginning of the article remains unlabeled. When graphing a quantity, one really does need to specify what quantity is being graphed with respect to what other quantity. That's just one example of the article's sloppiness and lack of definitions of its terms and specification of its meanings. — Preceding unsigned comment added by 97.82.109.213 (talk) 21:22, 18 June 2021 (UTC)

[quote] You should sign your contributions, and not let robots do it in your place. [/quote]

I thought that's what labor-saving robots were for.

I have or had an editing-account, but it was a long time ago, and I'd have to start all over finding out how to use the account.
 * @97.82.109.213,
 * 1. If you have had an editing account, it must still exist and you should find it if you remember its name.
 * 2. The article has not been modified since the last discussion. As I said, discussion should precede modification.
 * 2bis. However, following your remark that the graph in the beginning of the article remained "unlabelled", I added a few words better explaining what the figures at the left represent. I thought that it was obvious, but I obviously was mistaken.
 * 3. The definitions of meantone that you quote above as "unclear" more specifically concern the sense in which the tone in this family of tunings can be defined as a "mean": either as the mean between major and minor tones (of just intonation), or as the mean of the major third (which is formed of two tones). What do you find unclear in this, and how would you reformulate it?
 * 4. I am aware that roots and powers have been calculated or approximated by mathematicians in Antiquity, already in Babylonian tablets, say c1500 BC. In Latin Middle Ages, Leonardo Fibonacci, in his Liber abaci (early 13th c.) computed roots by something resembling continous fractions. This is linked with the introduction of so-called "Arabic numbers", but these remained largely unknown, unless in specialized circles, at least until the 15th century.
 * 5. Even in the early 16th century, musical ratios were ratios of string lengths, and the idea of elevating such ratios to powers or of taking their roots is somewhat problematic.
 * 5bis. Computing roots or powers of a syntonic comma (80:81) may be relatively easy; but for the Pythagorean comma (524288/531441) the situation becomes more complex.
 * 6. When speaking of a quarter of the syntonic comma, the implied quantity obviously is logarithmic – what type of logarithm does not matter. The interesting point is that it is intuitively understood even by people who do not know what logarithms are. There are studies in cognitive science that explain that (see for instance Weber%E2%80%93Fechner_law).
 * 7. About the "New Paradigm", see in the discussion above. I would agree with you that it may have no place in our article, but that remains a controversial issue.
 * But anyway it is much too soon to decide what we could make of all this in the article. We should perhaps begin by drawing a possible table of contents. The important point to me is that, when I say that discussion should precede modification, I mean it. Let's first discuss – and I hope that others will join in the discussion. — Hucbald.SaintAmand (talk) 10:13, 19 June 2021 (UTC)

-- You wrote:

[quote] But anyway it is much too soon to decide what we could make of all this in the article. [/quote]

...and, in the meantime, everyone who looks at the article is going to think that either the article is bull****, which it is, or else that the subject is much too complicated for them.

I recognize two issues:

1. Logarithmic measure vs powers & roots. . 2. The New Paradigm.

Number 1 is almost a non-issue, because it's obvious that both could be mentioned in the article. It's just a question of which should be in the initial easy explanation...a matter of which is easier. . (Sometimes what seems easier only seems easier because it's simplistic, and evades, dodges, complete explanation. Powers & roots are the genuine underlying basis, when the matter is expressed in terms of ratios--the form of expression relevant to what makes consonance. That can't (or shouldn't) be swept under the rug.) . Number 2 is too obvious to be an issue. You know it doesn't belong in the article. I know it doesn't belong in the article. A visitor to the article asked us if anyone here could define Meantone Temperament in a way that's understandable. Yes we could, if we start by deleting "The New Paradigm".

How is that question controversial?? Because a promoter of the New Paradigm doesn't agree?

For as long as we postpone deleting the New Paradigm material (and all its accompanying pseudo-technical gibberish)the article will make every visitor believe that the subject is too complicated. I suggest that deletion of the New Paradigm should be immediate.

Reminiscent of the story of King Solomon & the disputed baby, I say let's immediately replace the New Paradigm gibberish with a brief expression of the logarithmic wording that you prefer,and then discuss whether the initial easy explantion should use logarithms or powers-&-roots. - How much discussion was there before Plammandon vandalized, garbaged, the article? :D — Preceding unsigned comment added by 97.82.109.213 (talk) 23:06, 19 June 2021 (UTC)

First suggestions for a revised article
@97.82.109.213, I for one see more numerous issues in the article than just two. We have to reorganize it as a whole. Here follow some of my suggestions: As you see, none of this is simple. I trust that, before correcting the article here or there (and possibly introducing new problems) we might do better to further discuss, to carefully plan the modifications (possibly on a draft page, a "sandbox"). What do you think? — Hucbald.SaintAmand (talk) 13:23, 20 June 2020 (UTC) -- I think that's fine. A sandbox would be a good place for proposed articles or sections, if that would be more convenient than writing them at this talk-page. If you know how to set it up, just tell me how to reach it.
 * Lead ("lede") of the article: "Meantone temperament is a musical temperament, that is a tuning system, obtained by slightly compromising the fifths in order to improve the thirds. Meantone temperaments are constructed the same way as Pythagorean tuning, as a stack of equal fifths, but in meantone each fifth is narrow compared to the perfect fifth of ratio 3:2."This seems to me reasonable enough. It does give an idea of what a meantone temperament is without raising any of the questions that will be treated further in the article. The only point that might deserve additional explanation is that the main difference between Pythagorean tuning and meantone temperaments is that the first is not a temperament.
 * Figure illustrating the lead: I find it satisfying (all the more so that I drew it). If you think it (or its caption) should be modified, say so.
 * Dynamic tonality paradigm: even if we may agree that this does not belong to this article, one of its promotors indeed does not agree, and this makes the whole affair controversial. We may discuss the fact that the statement that "Dynamic Tonality can apply only to sounds that are digitally processed" found in Talk:Dynamic tonality somehow disqualifies it from figuring here. (Note that the Dynamic tonality page has mainly one author, who also is the sole contributor to its talk page, and that seems to me problematic in itself, as it confines to auto-promotion. But it is not our problem here.) Most of what is said in this section already appears in the Dynamic tonality page, so that a link may suffice.
 * Notable meantone temperaments. This section needs expanding.
 * The tone as a mean. This seems acceptable.
 * Meantone temperaments. This section appears largely redundant with things already said, and includes additions that I think do not belong to this article. Much of it has the same sources as what concerns "Dynamic tonality". In includes a highly confusing discussion of Easley Blackwood's "R" (already discussed above) and tables labelled "Meantone tunings" and "Comparison between 1/4-comma meantone and 31-ET" which apparently describe meantone tunings that could be approximated by equal division (also discussed above). Let's agree that this all needs cleaning.
 * Wolf intervals. Same source in "Dynamic tonality", similar problems.
 * Extended meantones. This should concern instruments with more than 12 notes in the octave, which may try either extended just intonation or extended meantone. There is a historical part in this, mixed with considerations of the "isomorphic keayboard", once again from the same source. This section also needs cleaning.
 * Use of meantone temperament. The section seems reasonably acceptable, but I'd have some doubts about its title.
 * We should also add, probably very soon after the lead, a section discussing descriptions of meantones in terms of parts or of roots of the syntonic comma. Zarlino and Salinas, in the second half of the 16th century, certainly described meantone temperaments in terms of parts (fractions) of the syntonic comma, more specifically in "equally proportional parts".

This is just a brief reply. I'm going to post a more complete reply to the general matters in your most recent post, such as:

The real main issue is: "Overhaul piece-by-piece" vs "Junk & Replace"

(I strongly recommend the latter. Much easier & less resulting junk.)

I'll comment on a few of the current sections that you mention, to support the above recommendation.

I'll briefly tell of a reason why "4th-root of the comma-ratio" is better than "1/4 of [something]".

But you didn't answer this:

Ratios are necessary in the explanation because they're what consonance is about. They're what we're trying to improve. Now, when you talk about 1/4 of the Syntonic Comma, a ratio, people will think that you're talking about 1/4 of the difference between two ratios, or a ratio divided by 4. ...unless you explain that it's logarithms. The former way, letting people misunderstand it like that, isn't helping anyone. The latter way, making it a discussion of logarithms, is as roundabout explanatory approach, unnecessarily wordy & complicated.

Do you have an answer to that argument?

But I re-emphasize that immediately deleting the New Paradigm nonsense is top priority.

I think your Lead-In is fine. Plainly both approaches to the construction can be included in the construction/generation section to immediately follow it, and it's just a matter of which one (root or fraction)should be the easy first explanation. That, as you suggested, can be discussed and sandboxed.

Anyway, this has been just a brief reply before I write my reply.

But I can't help asking you to have some concern for the people who are visiting the article every day and being discouraged, & convinced that Meantone Temperament is much too complicated...because of all the New Paradigm garbage dumped when the article was raided & vandalized.

And you didn't answer my question:

How much discussion was there before Plammandon raided, vandalized, rewrote & ruined the article???!!

Then I'll reply to comments in your previous post.
 * @97.82.109.213, you write
 * "Ratios are necessary in the explanation because they're what consonance is about. They're what we're trying to improve. Now, when you talk about 1/4 of the Syntonic Comma, a ratio, people will think that you're talking about 1/4 of the difference between two ratios, or a ratio divided by 4. ...unless you explain that it's logarithms. The former way, letting people misunderstand it like that, isn't helping anyone. The latter way, making it a discussion of logarithms, is as roundabout explanatory approach, unnecessarily wordy & complicated."
 * First, I don't think that ratios are what consonance is about – consonance is about the concordance between partials, which may or may not be harmonic and which, therefore, may or may not be described by (simple) ratios. Second, I don't understand what you mean by "what we're trying to improve" – improve ratios??? Third, the syntonic comma is not in itself a ratio, it merely is best represented in the form of a ratio, but it obviously also can be represented as a logarithm.
 * The reason why I think we should (also) explain meantone temperament by fractions of the syntonic comma (implicitly expressed as a logarithm, in this case) is that otherwise we would be at loss to explain why well-known meantones are named "1/4-comma" or "1/3-comma" meantone. I did not invent these names, they are there since the late 16th century, about half a century before the invention of logarithms.
 * You also ask
 * "How much discussion was there before Plammandon raided, vandalized, rewrote & ruined the article???!!"
 * Well, see by yourself. Each modification and each discussion are dated – unless yours, as long as you don't sign them. As you can see above, the article had been created in 2002, discussions began in 2005; Jim Plamondon began modifying the article in 2008. The concepts of "5-limit" and "7-limit" were questioned in 2009, the relevance of the keyboard shape in 2011, Lockwood's "R" also in 2011, I began modifying the article in 2015 and discussing it in 2016, Plamondon joined the discussion in 2020, etc.
 * Hucbald.SaintAmand (talk) 07:04, 21 June 2021 (UTC)

You said:

[quote] First, I don't think that ratios are what consonance is about – consonance is about the concordance between partials [/quote]

Yes, concordance between partials, or lack of it, is what makes Meantone harpsichord chords sound a lot better than ET. The harmonics of the ET chord-notes interfere, beat & clash, making the harpsichord sound muted, without the rich harmonic content heard in Meantone Temperament.

As for melodies with notes played one-at-a-time, the notes sounding like eachother's overtones likely has something to do with the pleasingness of the relation among scale-notes.

When only one note is played at a time, obviously more liberty can be taken with the scale, and ET doesn't sound bad (just different, when it exaggerates important features of the Just scale)

So yes, of course harmonics are the reason for why ratios of small whole numbers are desirable as frequency-ratios among certain notes in scales. But that doesn't change the fact that ratios among small whole numbers are desirable for frequency-relations of certain notes in scales.

So yes, it's about frequencies related by ratios of small whole numbers.

[quote] , which [partials] may or may not be harmonic and which, therefore, may or may not be described by (simple) ratios. [/quote]

No, we're talking about musical scales, not drums, cymbals or blocks.

[quote] Second, I don't understand what you mean by "what we're trying to improve" – improve ratios??? [/quote]

Yes. The thirds of the Pythagorean & ET tunings are off. They're off in the sense that they differ by too much from ratios of small whole numbers...5/4 & 6/5 specifically, in the case of 3rds.

[quote] Third, the syntonic comma is not in itself a ratio, it merely is best represented in the form of a ratio, but it obviously also can be represented as a logarithm. [/quote]

Since harmonious relation in the triads in the diatonic scale are a matter of frequencies in ratios of small whole numbers, then I'd say that it's reasonable to speak of the Syntonic comma as the factor by which the Pythagorean major 3rd & major 6th differ from 5/4 & 5/3 respectively. Because it's fundamentally, ultimately, about ratios, then there's no point in wandering into other measures in an explanation.

[quote] The reason why I think we should (also) explain meantone temperament by fractions of the syntonic comma (implicitly expressed as a logarithm, in this case) is that otherwise we would be at loss to explain why well-known meantones are named "1/4-comma" or "1/3-comma" meantone. I did not invent these names, they are there since the late 16th century, about half a century before the invention of logarithms. [/quote]

Throwing good money after bad. A roundabout, overcomplicated explanation, just to justify a semi-medieval name?? I'll explain the justification for "1/4 comma" & "1/3 comma": They're brief and convenient. ...as names,not as explanation. — Preceding unsigned comment added by 97.82.109.213 (talk) 14:38, 21 June 2021 (UTC)

Hucbald: I’d like to reply to all your comments, and the easiest and most efficient way to do that is by replying inline. It seems more efficient to use the same writing-space, demarcated by horizontal-lines, labeled by author. But, if it has to be in separate sections, then I’ll do quotes & inline replies to them. I’ll do that in a subsequent post here. Here’s a summary of things that I want to say: 1. My brief proposal for an **immediate** big provisional change in the article. 2. Inline replies to your comments in your most recent two posts. 3. The wording of my suggested explanation-section for the generation/construction of four meantone-temperaments. …consisting of a subsection for the problem, and one for the meantone solution. It could be at a sandbox, or it could be here at this talk-page. Let me start, right now, with my brief proposal for an immediate big change in the article. Leave your Lead-Piece in. In fact, because you wrote your initial graph-figure, leave that in too. Delete the rest of the article. There’s nothing worth keeping. It’s pointless to look at each section and decide whether to keep it, or how to modify it. The new article will be a LOT briefer and more to the point.. It will be much briefer, neater, & cleaner to just delete the article & write a new one. Replace the deleted article with your initial (at least partial) proposal for it. Then we can improve it & discuss replacing the logarithmic-measure with powers & roots. Maybe much of that writing can be done at the sandbox. I have a number of criticisms of your graph-figure, including suggested improvements, and reasons why it isn’t as helpful or clear as well-chosen words. But keep it in for the time-being, because you wrote it…but move it to the bottom of your proposed article, because something that technical and confusing to most people definitely doesn’t belong at the beginning of the article. We can discuss its merits & improvements later. One criticism that I’ll say now is that its information doesn’t help with telling the problem that motivated Meantone Temperament, or the rationale, principle & procedure for generation of Meantone-Temperament. It’s a digression on a different matter, separate from those matters. And, take my word for it, it doesn’t look inviting or easy to most people. So anyway, replace the current New Paradigm article with something brief and clear that you prefer (but keep your graph & your Lead-In). Do it within one week from today, or else tell me a good reason why the “New Paradigm” belongs in the article. Do that within one week, or I’ll do it myself. …and you know that you won’t be able to justify reverting my deletion of that crap article. As you said, the article is inextricably, un-dis-entagle-ably, riddled with New Paradagm, and by far the best solution is to delete the whole thing and replace it with something brief, clear & to-the-point. This is one of those instances where replacement is a lot better & more feasible than overhaul. When that is done (by whoever does it first), then it will be time to resolve the matter of logarithmic-measure vs powers & roots. As you said, the current article is a mess. That’s an understatement and a euphemism for what the article is. It has to go, **immediately**. As you yourself said, no one likes or wants it, other than the person who raided and ruined the article with it. (He evidently isn’t having anything to say about why his material should stay in. Doesn’t that say something about how much support there is for it? Even its perpetrator won’t try to defend it.) Next will be my inline replies to your individual comments. Then will be my proposed wording for the explanation for the generation & construction of four Meantone-Temperaments.

--- One thing that I can’t resist saying now: You said that you didn’t know what purpose the 1/6 comma Meantone serves. It compromises between the minor 3rd and the 5th. The only reason why you didn’t notice that is that you were looking at the matter via the logarithmic approach instead of the powers & roots approach. Likewise, the 1/8 comma Meantone compromises between the major 3rd & the 5th. --- Didn’t you say that the Meantone 3rd is the mean of the Just major & minor 3rds? No, the arithmetic mean of the ratios for the Just major & minor 3rds is 49/40. Their geometric mean is the square-root of 3/2. The ¼ comma Meantone major 3rd is identical to the Just major 3rd. The 1/3 comma minor 3rd is identical to the Just minor 3rd. But the 1/6 comma Meantone minor 3rd is the geometric mean of the Just minor 3rd & the Pythagorean minor 3rd. …and the 1/8 comma Meantone major 3rd is the geometric mean of the Just major 3rd & the Pythagorean major 3rd. But those facts aren’t necessary to the article, and only add unnecessary length & clutter. The 1/6 comma & 1/8 comma Meantone Temperaments are relevant as compromises between the 5th, and the minor or major 3rd, respectively. All that part about what meantone is the mean of, is part of what serves no purpose and should be discarded.
 * @97.82.109.213, it seems now obvious to me that you don't want to discuss, nor to conform to the usages of WP. I'll therefore stop here (I have other more interesting things to do) and let you do as you want. We'll see what the results may be. — Hucbald.SaintAmand (talk) 17:55, 21 June 2021 (UTC)

_____________________________ First, this isn’t addressed to Hucbald, because he isn’t speaking to me  :-) But it has been claimed that I refuse to discuss. That’s a serious accusation at Wikipedia, where it implies that I want to change the article in disregard of others’ opinions that I don’t want to hear. I’m surely entitled to answer such an accusation. So we’re told that I “don’t want to discuss”, though I’d been answering everything that Hucbold said  :D Additionally, I invited Hucbald to put up his own preferred wording of the explanation (as opposed to mine), because that matter could later be discussed unhurriedly. So then what’s this unwillingness to discuss?  The matter of the deletion of “The New Paradigm”?  No, I’d just finished asking Hucbold for a good reason why the humungously-lengthy & unnecessarily-elaborate “New Paradigm” material belonogs in the artice.   i.e. I was calling for discussion regarding a change (“New Paradigm” deletion) that I proposed. I asked for discussion about that matter, and it isn’t my fault if Hucbold refuses to discuss it, or give some reason in opposition to the deletion of “New Paradigm”. Maybe Hucbold didn’t like my announcement that, if no one is willing to support “New Paradigm” for the article, then I’d delete it after one week. So that could be taken to meant that I was unwilling to discuss it for more than a week. Well, how long would it take Hucbold to share with us why he wants to keep “New Paradigm” in the article? That material has been in the article for a long time. Presumably Hucbold has been around this article for a while, and has been discussing the matter for a while. So yes, I must admit that I’m unwilling to have the “discussion” about that go on for more years. …while, daily, people are arriving at the article to find out what Meantone Temperament is and how it’s generated, and discouraged by an impression that it’s far too complicated for them. Take another look at what that visitor said, above at this talk-page. She or he asked us if there’s anyone here who can explain it in an understandable way. So I’ll repeat my invitation: I invite anyone to tell why they think that the over-lengthy & unnecessarily-elaborate “New Paradigm” material belongs in the article. Anyone who wants to keep “New Paradigm” in the article should say so now, or at some time before Monday, June 28th. …the date on which I’ve scheduled my deletion of the “New Paradigm” material from the article if no one objects with reasons. Though I’d said that the article is such a mess that it would be better to delete the whole thing and start over with something brief and to-the-point--and I stand by that statement—I also realize that, as a practical-matter, deletion of the whole thing could be construed as justifying more objections to the change. Minimal change minimizes objections, especially when someone’s own text is involved. And so here’s what I intend to do on June 28th: I’ll delete every section containing New Paradigm and nothing else substantial. I’ll delete New Paradigm from whatever other sections it’s in. I’ll leave Hucbold’s intro, maybe with a few wording-improvements. I’ll (for the time-being) leave his graph, though it still doesn’t specify what its graphed-quantity is being graphed with respect to. …a necessary requirement for any genuine graph. For that reason, and because it doesn’t contribute to an explanation of the need, principle & generation of Meantone Temperament, I’ll move that graph to the bottom of the article. Directly after Hucbold’s intro, I’ll write my easy explanation regarding the need for, and the generation/construction of, Meantone-Temperament, in four versions (1/4, 1/3, 1/6 & 1/8 comma). Other than the obfuscatory “New Paradigm”, I don’t really care what remains in the article, as long as it’s below the easy explanation. Though I’m leaving everything other than “New Paradigm” in the article, I’ll later check some of it for accuracy & coherence of meaning & expression. But, before deleting any of it, I’ll state what’s wrong -with it, and invite anyone to either defend it or fix it.

I repeat that I’d invited Hucbold to (when deleting “New Paradigm) put up his own construction/generation explanation wording, to be unhurriedly-discussed later. Now that Hucbold refuses to discuss, then I’ll just put up my own explanation. ..certainly at least by June 28th, but maybe before. My argument for the need for explanation in terms of ratio and multiplication has several justifications: 1. People are familiar with ratios and multiplication. 2. Ratio’s are what consonance of scale-notes is about. 3. Given the above, any explanation regarding scale-note-consonance must start with ratios. Talking about additive pitch-intervals in an explanation starting with ratios is inconsistent & misleading, implying that ratios are being added, or added-to, or divided by a number. ...unless the logarithms of ratios are explicitly brought-in…making the explanation unnecessarily-roundabout. But here’s the basis of Hucbold’s disagreement about that: Hucbold says: “First, I don’t think that ratios are what consonance is about” And “I don’t understand what you mean by ‘what we’re trying to improve’. Improve ratios???” There can be no resolution of the matter, with someone who’d say those things. Are those statements characteristic of the “New Paradigm”? So yes, I guess that maybe it’s just as well that Hucbold isn’t speaking to me, because, on matters of consonance of musical-tones, there can be no meaningful or worthwhile discussion with someone who would say the above-quoted things. So, in closing, I invite anyone to tell why they think that the unnecessary and humungously over-lengthy and elaborate “New Paradigm” should remain in the article. …but please do so before June 28th. I won’t do the deletion without discussion…unless no one is willing to tell why they think the “New Paradigm” belongs in the article. I emphasize that I invite discussion first. -

June 25th, 2021

Though New Paradigm needs deletion, the more important thing is to add, at the top of the article (right after the existing intro) an easy explanation of why Meantone Temperament was needed, and how it’s generated/constructed. That’s the important thing, and it really doesn’t matter how much garbage, of whatever kind, is below it in the “article”. Ideally it wouldn’t be a matter of “either/or”, but Hucbald seems pretty sure that he’ll get his way here (until the question is taken to Wikipedia administration). In these circumstances, where the matter will probably have to be taken to Wikipedia management/administration, it’s best to be asking for as little as possible, and not let a less important issue distract from a more important one.. Therefore, I’m not going to bother trying to take away Hucbald’s New Paradigm. As I said, it’s irrelevant how big is the garbage-pile below the easy explanation. But an easy explanation is needed immediately. And yes, Hucbald is right about one thing: I reject and don’t recognize his rule that no changes be made in the article until there’s agreement on every aspect of a complete re-organization & restructuring of the article, after a “much longer discussion” (in Hucbald’s words). No, sorry, but that won’t do. Contrary to what he might believe, Hucbold isn’t Supreme Rule-Maker, Honcho, or Majordomo here. And there’s no Wikipeda rule that says that no change can be made without complete re-organization and rewriting of an article. As in the current case, with the current completely, bizarrely, inadequate state of this article, there are times when an article is so badly-lacking that, no, there’s a change that is needed sooner. Though my refusal to recognize Hucbold’s rule-making authority, and my rejection of his rule about there being no changes without complete rewriting, has upset Hucbald, and made him pout, I stand by what I say here. Anyway, I’m withdrawing my announcement to delete “New Paradigm”, or anything else. No deletion. I’m just going to add an easy explanation, immediately after the existing intro. Because of the extreme inadequacy of the article in its current form, I’m going to do that one week after posting, to this talk-page, a proposed wording of that easy explanation, unless there are objections to address, fix, or answer. Sorry, but it isn’t something that should be delayed more than that. The problem that required the use of Meantone Temperament was that the Pythagorean thirds (major & minor) were felt to be too far-off. Specifically, their frequency ratios with respect to the scale’s bottom-note differed from (respectively) 5/4 and 6/5 by too great a factor, causing noticeably poor musical sound, especially in chords. Note that the problem was about ratios that were too wrong. Speaking about the especially-popular ¼ comma Meantone: An accurate major 3rd is a frequency-ratio of 5/4 The Pythagorean scale gets its major 3rd by multiplying together four factors of (3/2), resulting in 81/16. …which, when adjusted by octaves (factors of 2), to a value between 1 & 2, gives 81/64 5/4 divided by 81/64 = 5/4 X 64/81 …which, divided above & below by 4, equals: (5X16)/(1 X 81) = 80/81. Therefore the Pythagorean major 3rd ‘s frequency must be multiplied by 80/81…or divided by 81/80, in order to get 5/4. 81/80 is called the Syntonic comma. Because four factors each equal to 3/2 are multiplied together to get the frequency of the Pythagorean major 3rd (…after then being repeatedly divided by 2 to get a number between 1 & 2), then, if the 3/2 ratio of the perfect 5th is divided by the 4th root of 81/80  (that’s the number that, when raised to the 4th power (that is, multiplied by itself in 4 factors) equals 4.), then the resulting product of  factors of the adjusted 5th will equal 5/4, as desired. (There will be a short paragraph assuring the reader that the article doesn’t require any special math background, and it will, imitially, very briefly define the few mathematical terms needed (powers & roots) ). Because the problem was a ratio that was too far off, and needed to be adjusted, by adjusting the size of a 5th, ratios and multiplications are the only basis for a first explanation of the need for, and generation/construction of, Meantone Temperament. To explain the procedure in terms of cents or semitones would first require defining them logarithmically. …and talking about the 12th or 1200th root of 2. …therefore still needing roots, but now logarithms too. Obviously the above explanation is is the same for 1/3 comma, where each 5th must, for the dame reason, be divided by the 3rd root (“cube root”) of the Syntonic Comma…and likewise for the 1/6 & 1/6 comma Meantone Temperaments, which compromise between (respectively) the minor or major 3rd and the 5th. That’s only a summary of the central material of my easy explanation. Of course, for the 1/3 comma, it will be necessary to clarify that it’s the major 6th’ ratio that is being sought, because ratios for the Just major 6th and minor 3rd multiply to 2, so if one is accurate, the other will be as well. This is only a summary of the central material, not the full detailed explanation-wording, which will speak in some detail about the problem, and what’s wrong with the Pythagorean & Equal-Tempered tunings, with some necessary brief explanation about the Just Scale and consonance …which will unavoidably be about ratios of small whole numbers. Anyway, soon I’ll post my proposed easy explanation, because advance notification should be given, so that any objections can be made and addressed or answered. Then, a week after posting that, if (after I’ve agreed to any changes that are consistent with the article’s purpose) there are no objections (or at least none that are consistent with the article’s purpose and that I believe will stand up to administrative judgment), then I’ll add my easy explanation to the article, directly below the existing intro. — Preceding unsigned comment added by 97.82.109.213 (talk) 06:02, 25 June 2021 (UTC)
 * @ 97.82.109.213. Just a word: you write that "Therefore, I’m not going to bother trying to take away Hucbald’s New Paradigm." If you reread the discussion above, particularly this section, you'd see that I have always been much against this "New Paradigm", which in no way could be considered "mine." — Hucbald.SaintAmand (talk) 17:38, 25 June 2021 (UTC)

Remove everything related to Syntonic Temperament and Dynamic tonality
Maybe about 40% of this article seems to correspond to a fringe theory by user JimPlamondon. This is against wikipedia's philosophy, as you can see here. In the fringe theories page. This has nothing to do with whether I agree or disagree with this theory, it is an objective fact that this is not broadly supported and as such should be deleted. Wikipedia is not a place for promoting your own theories, it should reflect what is broadly acceped in a field, there are many other websites that can be used to promote or support this research.

In a nutshell, as my reference states: To maintain a neutral point of view, an idea that is not broadly supported by scholarship in its field must not be given undue weight. I don't think this deserves extra discussion, and JimPlamondon's own comments on the matter make this very clear. I will procede to delete everything associated with Dynamic Tonality IgnacioPickering (talk) 23:52, 17 July 2021 (UTC) IgnacioPickering (talk) 00:39, 18 July 2021 (UTC)
 * Let me stress, in addition, that there exists a Dynamic tonality article that says all what WP readers might want to know (and even more). A link to that article might prevent further discussions. There also is a Quarter comma meantone article, with which this one at times is redundant. I wonder whether the present article could not be renamed "Meantone temperament s " (plural), leaving details about specific meantones to specific articles. — Hucbald.SaintAmand (talk) 09:17, 18 July 2021 (UTC)

The definition of ‘Meantone’
It has become customary to refer to any regular temperament with better thirds than equal temperament as meantone. However, as far as I can tell, this is modern terminology. Contrary to what is often claimed (without clear citations from sources), I am not aware of any historical sources that use this term other than for the 1/4 (syntonic) comma tuning. The historical concept seems to be clearly based on the fact that the whole tone (logarithmically) is the average of the major and minor whole tone from a pure tuning. To base the definition on the fact that a whole tone is half of a major third seems rather pointless to me for two reasons.

First, this terminology does not imply a delineation between temperaments with better thirds than equal temperament (as is meant in common usage). What then is this limitation based on, and why would just this term have come into being for this subset of regular temperaments? Especially as a delineation from the traditional pythagorean tuning, this definition of "meantone" is inadequate!

And second, what does "mean" mean if we don't assume well-defined different whole tones with simple string length ratios? However, these exist only in simple mathematical systems, such as pure tunings (which does not exclude that such intervals can also occur in some temperaments, but there they do not constitute a theoretical basis for the definition of a certain kind of whole tone). Therefore, also linguistically, the term "meantone" does not seem to be the most obvious designation for the notion of a "semi-(major)third".

Since I don't pretend to have read all possibly relevant historical sources, my first question to the other participants in this talk page is therefore: can you provide me with historical sources for the use of the term "meantone" in the currently common sense of a general third-oriented regular temperament?

And if not, shouldn't we make it clear in this article that the definition that in fact includes a whole range of regular temperaments is a modern definition that differs from historical usage?

I have no fundamental problem with unhistorical definitions if they describe a meaningful modern concept. However, we must be aware that the use of a historical term that does not (or worse, only partially) correspond to the historically associated concept has the serious disadvantage of causing confusion. Therefore it should then be clearly communicated. There are enough shaky theories circulating in the early music world that have more to do with the interpretation of some performers than with the sources. If these lead to a good musical result in practice, I have no problem with that. However, we should not ascribe to them a historical authority that they cannot pretend to have.

Koos van de Linde — Preceding unsigned comment added by Niehoff54 (talk • contribs) 11:29, 26 January 2022 (UTC)
 * Meantone temperaments other than the quarter-comma meantone have been documented since the 16th century, even if they may not have been called "meantone". Zarlino mentioned tempering the fifths by 2/7 comma in his Istitutioni harmoniche (1558), chapter 42, pp. 126-127. Salinas describes three methods of obtaining equal tones (vt Toni æquales fiant), namely 1/3 comma, 1/4 comma and 2/7 comma, in De musica, Book III, chapter XV (pp. 143-145). Both call the major third ditonus, which stresses its being made of two [equal] tones. (Ditonus comes from the Greek ditonos, of course, and Pythagoreans in Greek antiquity already were aware that the Pythagorean third also was a ditone.) Several other divisions of the syntonic comma were discussed in the 17th and 18th centuries: 1/5, 2/9, 3/10, 5/18, 1/6, 1/7, 1/8, 1/9 and 1/10, always coming closer to 1/11 (i.e. 1/12 Pythagorean comma) that would produce equal temperament. The authors of these descriptions were aware that these divisions produced major thirds formed of two equal tones. These temperaments may not have been described as "meantone" – determining this would involve a lot of checking. Ellis makes an abundant use of the term in his translation of Helmholtz' Tonempfindungen, but Helmholtz himself never names that temperament. I am not even sure that 1/4 comma meantone ever was so called before the 19th century. What other name would you suggest? — Hucbald.SaintAmand (talk) 14:25, 26 January 2022 (UTC)
 * @Koos van de Linde, on second reading of your message above, I realize that you worte that you are "not aware of any historical sources that use this term [meantone] other than for the 1/4 (syntonic) comma tuning." If you are aware of historical sources that do use it for 1/4 comma temperament, I'd be genuinely interested. I think to have read that Sauveur did, but was unable to find in Sauveur nor it in his writings (I suppose that the French term would have been ton moyen, nor in any of the French treatises that I have). In German, Mittelton has been used in discussions of enharmonies, or also to denote the third (Mediante) as dividing the fifth in a triad, etc., but not, that I am aware of, for temperaments – neither Helmholtz, nor Riemann, nor Brokhaus, use it in that meaning. — Hucbald.SaintAmand (talk) 15:04, 26 January 2022 (UTC)

Diagram caption
The caption on the diagram comparing the various temperaments is way too long. If the diagram can't be explained in a few sentences, it's probably not of any value to our readers. Half of the caption is just pointing out various spots on the diagram, which, if they are notable, should be expressed in the diagram itself. I'm going to boldly chop off a huge chunk of the caption and see how that works. Orange Suede Sofa (talk) 00:10, 18 August 2023 (UTC)

Major tone and minor tone
The term "major tone" is wikilinked to major second, which does NOT explain what a (just) major tone is: 9/8. In the given context, which explains the term "meantone" as a mean between a major tone and a minor tone (i.e. 10/9), we had better not confuse the readers with the major and minor seconds of scale construction. If there's a Wikipedia entry that explains "major tone" accurately, let's wikilink to that instead. yoyo (talk) 22:01, 15 October 2023 (UTC)