Talk:Equal temperament/Archive 3

Inaccurate description of historical temperament
I believe that the "History" section requires a major revising, if not a complete rewriting.

The section on history has almost no discrimination between "equal temperament" and "well temperament." Although equal temperament have existed in theory since late Renaissance, it was 1)deemed impossible to achieve due to technical restriction, and 2)often deemed unusable and offensive, for having neither purity of consonant intervals nor tonal color. All of the historical "circular" temperament described in the section, meaning those temperaments that could play all 24 major/minor keys, were in fact "unequal" (which is "well temperament," a concept completely different from equal temperament with different aim--well temperament was for equal playability of intervals while equal temperament was for complete equality of intervals). Weickmeister was the pioneer of well temperament, or "wohl temperiert," as Weickmeister himself put forward and coined, and although he indeed discussed and popularized Mersenne's equal temperament theory in his posthumous work(which, in turn, is completely different from modern equal temperament of twelveth-root-of-two), the wording misleadingly represents him as THE advocate of equal temperament.

Until the twentieth century, there was absolutely no method of achieving equal temperament, even with the "a tuning fork tonometer in 1834" described in the text, since the beating between keys were to be counted manually--a method impossible to specify irrational number of beats, which is what exactly modern equal temperament is. Description of 18th century is thus completely inaccurate: at the time, the tuning system that dominated the scene was never "equal temperament" but "well temperament." Everything from Baroque to late Romantic period was either non-12TET meantone temperament(since 12tone equal temperament is meantone temperaments with 1/11 syntonic comma) or variations of well temperament.

Here are the sources. http://www.millersrus.com/dissertation/ http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html http://www.kylegann.com/histune.html http://en.wikipedia.org/wiki/Andreas_Werckmeister http://www.radfordpiano.com/historical.html http://www.1911encyclopedia.org/Sound#Scheibler.27s_Tonometer

Mondschatten (talk) 04:10, 15 February 2009 (UTC)

I came here to mention the exact same thing. The reference to the "Sabbatini method" says that it made a precisely equal temperament possible. That's wrong. The Sabbatini method gave a quarter comma mean tone temperament. The Thirds and Minor Sixths were "perfect" (just); the fifths were "narrow". Here's a link to a series of articles that appeared in the Piano Technicians Journal in 2010-2011, discussing this. That description of the Thirds as "just" or "perfect" is a dead giveaway: in Equal Temperament, NONE of the intervals are just or perfect. They can't be. They're all (except for the octave) a little bit off, with the discrepancy spread equally among the 12 tones. The article should be updated to correct the inaccuracy. — Jim Hardy (talk) 07:23, 1 June 2018 (UTC)

"Until the twentieth century, there was absolutely no method of achieving equal temperament" Is this really correct? If you look at a 12TET stringed instrument such as a Spanish guitar , the fretting positions up the neck to achieve 12TET are quite simple to calculate and the maths existed. AnotherPath (talk) 19:58, 25 October 2013 (UTC)


 * Yeah, it is correct. I'm not sure you're right that the irrational numbers required by ET were simple to MEASURE in manufacturing.  But the larger point is, "temperament" really only applies to fixed-pitch instruments: keyboard, woodwind, something like that.  With a guitar, a player can "bend" a note a little bit by changing their action on the fret hand.  The pitch isn't purely a factor of the instrument maker's fretting.  So any "small" inconsistency in tuning can be easily covered/adjusted by the player.  Another factor is the range of the instrument.  A piano has seven full octaves of playing range: enough to go halfway around the Circle of Fifths.  Temperament choices start to matter.  Most other instruments have much smaller ranges, so not as important.  Also a piano can play in every key signature.  Not every instrument can.  So the full comment is, "Until the twentieth century, there was absolutely no method of achieving equal temperament on a keyboard instrument (like the piano)."  And that's true  Jim Hardy (talk) 07:36, 1 June 2018 (UTC)
 * Funny. One wonders how Hummel and probably Werckmeister could have supported equal temperament (source) if they could not achieve it. Indeed, the book I linked (Temperament: How Music Became a Battleground for the Great Minds of Western Civilization) contains quite a good debunking of the idea that equal temperament was unattainable before the late 19th century. Double sharp (talk) 12:51, 1 June 2018 (UTC)
 * I read the book in March. Enjoyed it: a fun, gossipy little history.  It doesn't contain ANY debunking of the idea that ET was unattainable; it contains no practical discussion of HOW TO TUNE fixed-pitch instruments (like keyboards).  Almost all of the controversies Isacoff reports are theoretical.  Werckmeister's temperament was a "well" temperament, not the equal temperament: see http://kylegann.com/histune.html#hist3  Hummel's was a "quasi-equal" temperament: see here for the offsets from ET http://rollingball.com/images/Hummel.gif
 * Werckmeister & Hummel may have believed in the aptness of ET; and they may have strived for it in the tunings they did. But the problem they would have run into is that there there was no way to accurately measure the frequency of a sound until about the early 20th C.  You could KNOW that the correct frequency ratio between successive notes a half-step apart is the 5th root of 12: the math was doable a long time ago.  But when tuning an instrument, how do you measure that the interval between (say) F# and G is exactly 100 cents, and not 90 cents or 110?  Sure they could tune one or two notes to a reference pitch off a tuning fork (pretty accurately, actually); but that gets them started on the tuning, not finished with it.  What they would have done then is tune the intervals by ear, judging (probably by the beating) whether they sounded right or not.  I'm not trying to dog those guys: they were excellent practical tuners, as well as being very fine musicians (H a genius composer).  But their ear didn't "know" what the 5th root of 12 sounded like.  Their piano tunings would have involved the same sort of compromises and approximations that every other piano tuning does.
 * I have another argument with Isacoff. Somewhere in the book (near the passage you linked to) he writes "It is abundantly clear to me -- on the basis of the music itself -- that Beethoven, Chopin and others had equal temperament in mind."  This is not "abundantly clear" at all: if fact it is controversial.  It proceeds from the assumption that "well" temperaments are unplayable because of Wolf Intervals, and that assumption is untrue.  What you are supposed to get in a "well" temperament is that the keys close to C on the Circle of Fifths are consonant, gentle; and the keys on the other side of the Circle close to F# are more dissonant, harsher.  Not UNPLAYABLE: they just have a different, less comforting sound.  Like the diff between scary/tension music and lullaby music.  The key sigs that people played in the most sounded the most harmonious; the more remote key sigs sounded a little wild.  That's what makes the temperament "Well", as opposed to other harsher tunings.  This seems to me a fundamental misunderstanding on Isacoff's part — maybe more likely, a deliberate simplification to keep his book to a manageable, readable size.  But it sets up this binary "ET Listenable / Unequal Sound Bad!" construct, that doesn't match the real world.
 * Isacoff devotes some time in his Afterward to his discussions with Jurgenson. Here's a few posts on the PianoWorld forums from a Registered Piano Tuner, and teacher & examiner with the Piano Technicians Guild, who's also an ardent advocate for a Well Temperament of his own design.  Like Isicoff, this guy worked with Jurgenson (I think mostly to get J's opinion of the temperament).
 * (In the posts linked below, "EBVT" is the name of the temperament this guy created: you can read that as a synonym for "a good well temperament.)
 * http://forum.pianoworld.com/ubbthreads.php/topics/2448779/re-which-one-sounds-better-to-you.html#Post2448779
 * http://forum.pianoworld.com/ubbthreads.php/topics/2507488/re-my-piano-in-ebvt-iii.html#Post2507488
 * http://forum.pianoworld.com/ubbthreads.php/topics/2700946/re-ebvt-iii-and-wolf-tones.html#Post2700946
 * http://forum.pianoworld.com/ubbthreads.php/topics/2701085/re-ebvt-iii-and-wolf-tones.html#Post2701085
 * As I understand his argument: he's saying that if Beethoven/Chopin et al were playing & writing in Equal Temperament, then that must mean that their choices of key signatures were basically random, because under ET all key sigs are supposed to sound the same. All half-step intervals are the same.  But if you assume a Well temperament, then their sig choices make obvious sense, because of the "character" they give each piece; and also within pieces, the way they move thru different keys, increasing in tension by moving into more remote and "harsh" keys, then releasing the tension and resolving back into a more gentle home key.  He's saying that you can hear all that in a Well temperament; but not in equal temperament, because in ETs all the key signatures sound the same, just as gentle/harsh as each other.  ET renders the harmonic wandering "aimless"; whereas in a Well temperament the harmonic changes are purposeful and convey a charge.  Mind you, this line of argument is a little beyond me musically; also historically and technically.  But I find it persuasive.
 * To me, these items all trend together: (1) no way to accurately measure frequency intervals until the 20th C; (2) the known historical tunings by advocates of ET like Werck & Hummel are clearly & measurably UNequal temperaments (Well and Quasi-Equal respectively); (3) the descriptions by modern piano tuners of the methods of aural tuning (as opposed to using an Electronic Tuning Device) and what they achieve; (4) the presumption that Mozart's & Beethoven's & Chopin's choices of key signatures could not possibly have been random, the keys must have meant something to them; and (5) that guy's posts about how particular pieces sound in Well temperament vs ET, the way they progress harmonically toward dissonance and then resolve back toward peace. Those points all buttress the idea that the composers of the Common Practice Period were usually working with (and presuming) a piano tuned in a Well temperament.  Not a pure ET; and further that pianos tuned to a true pure ET didn't appear at all until around the middle of the 20th C.
 * To me, any assertion that Bach & Mozart & Beethoven & Chopin &c intended and assumed and worked with Equal Temperament pianos, has to explicitly address and explain away those 5 items. I can buy that they were working toward Equal Temperament.  Well temperaments are a step toward ET, from Quarter Comma Meantone and the like.  The "well" temperaments were more equal than what had gone before.  But they weren't what we could consider a true modern ET.
 * I'm not a scholar in this area. But this is what makes sense to me.  Jim Hardy (talk) 06:38, 29 June 2018 (UTC)
 * I think I wrote "5th root of 12" a couple times. That can't be right.  I think I meant "12th root of 2"?  Something like that.  Please overlook.  Jim Hardy (talk) 06:41, 29 June 2018 (UTC)
 * To Double sharp – I just read more closely the series of articles I linked above, originally published in the Piano Technicians Journal in 2010-2011: "A Clear and Practical Introduction to Temperament History" The articles are fascinating, I highly recommend.  Anyway: it's causing me to think that I was wrong when I argued with you above.  My idea that true Equal Temperament tuning was mostly unknown until the early 20th C, seems mostly to have originated with Jurgensen; and Jurgensen focused on England.  He seems to have been out of touch with the history of tuning in Continental Europe.  The author, a member of the Piano Technician's Guild, argues convincingly that Equal Temperament was probably the most widely-used tuning in Germany/Austria from about 1750 on.  He wraps up with a paragraph to the effect that, while we can't know how precisely 18th & 19th century tuners were able to achieve their ET, in practice there are variances in any tuning; the range of tunings that would be accepted by musicians as "equal temperament" is larger than modern tuners (with their electronic tuning devices) tend to think.  A decent practical "ET" tuning done in the  18th & 19th century, probably sounded pretty much like our current ET does.  And with no accepted pitch standard, two tunings could be wildly different from one another, without it having anything to do with "temperament".  All this is to say, I'm leaning toward thinking that you were right and I was wrong.  Jim Hardy (talk) 07:11, 20 September 2018 (UTC)
 * Thanks for the links! Truly, they are a fascinating read, and it's nice to see that they corroborate the point I was making. It's also nice to see that we're now in agreement. ^_^ Indeed, expressive intonation will absolutely be used for all instruments that do not have every note pre-tuned like the piano does, so that a whole tone could indeed be realised as a 110-cent or a 90-cent interval instead of a 100-cent interval. However, it remains a realisation of the theoretical 100-cent interval of 12-tone equal temperament, as it must to ensure that a whole tone scale starts and ends on the pure octave. (We can consider the intervals of 12-TET to be abstract equivalence classes with a variety of realisations whose variety itself depends on the musical context, which is why it is not nonsense to say that one can distinguish enharmonic intervals even in 12-TET.) We do not hear this and similar expressive deviations as deviations at all, and since studies have revealed differences in intonation of minor seconds ranging from 62-cent intervals to 118-cent intervals (a spread of 56 cents!), it seems clear that our aural ability to detect equal temperament is fuzzy enough that the 18th-century method of tuning three major thirds to the octave and interpolating fifths between them would have been good enough to sound just as good as the more accurate 12-TET that we can achieve today to almost everyone.
 * Since these articles mainly address the points of practice (your 1, 2, and 3), I'll briefly address your items 4 and 5 by summarising a paragraph of page 28 of Charles Rosen's The Classical Style (a book that addresses so many points with such wonderful logic that I often find myself unable to do better than refer to it, although I will add a point of my own at the end). It is true that many composers have said that keys have characters. However, it should give one pause to note that Beethoven once specifically said that C♯ major and D♭ major had different characters, even though on a piano they would have sounded exactly the same at the time even in a non-equal temperament (because you cannot have C♯ and D♭ on the same keyboard unless the keys are split). This should tell us that the distinction is at least partially theoretical, and probably based on what one sees on the page (sharps imply a dominant direction while flats imply a subdominant direction), with some additions from traditional key–instrument associations (e.g. E♭ major for horns). In the classical style, harmonies are more consonant or more dissonant according to their distance from the tonic, and this is a cogent relationship that can be heard even if one uses equal temperament because one can always keep the original tonic in mind just as one can keep it in mind during a chord progression (a modulation to the dominant is simply an expansion of the dominant chord, and the same for any other key). In fact, equal temperament can realise this better than well temperament if the original key has many accidentals. The slow movement of the Emperor Concerto is in B major (five sharps), and the section in the middle in D major (two sharps) is in the flattened mediant and should be heard as more dissonant and requiring resolution (which indeed occurs). But since well temperaments make keys with less sharps or flats purer, they produce the opposite result here, and would hence give a worse realisation of the theoretical structure of this composition than equal temperament would. Double sharp (talk) 16:01, 20 September 2018 (UTC)

"...and for pipe length by $\sqrt{2|24}$"
Why this value of $\sqrt{2|24}$ ? What does it means in a twelve tempered system ?

--AXRL (talk) 20:35, 9 September 2017 (UTC)

Appearance of equal temperament in Europe
The source given for the claim that equal temperament was brought to Europe from China relies on Robert K. G. Temple, who is known for pseudoarcheology, a la Ancient Aliens. This claim is a big deal, so we should be able to back it up by reliable sources.

Here's the citation section of the self-published webpage used as a source for this fairly startling claim, which apparently comes from Robert K. G. Temple:

Temple, R., The Genius of China. New York: Touchstone Books, Simon and Schuster, 1989, pp. 209-213.

Further, our article Twelfth root of two seems to use a more rigorous source, and makes no mention of Zhu Zaiyu's accomplishment actually making it's way to Europe. Instead, Simon Stevin is credited with making a calculation in 1605.

There is a mention of a transmission from China here:

https://web.archive.org/web/20120315013436/http://en.cnki.com.cn/Article_en/CJFDTOTAL-XHYY201002002.htm

The research by Cho may very well be iron clad, but so far we have one academic source, and the source itself notes claims by other (multiple) scholars that Simon Stevin originated the mathematics of equal temperament (independently) in the West. So it seems that there is not a consensus that the system was transferred from Zhu Zaiyu to European music.

I am adding language to properly highlight the scant research and lack of consensus on this claim.

This is basically a continuation of an archived discussion here: https://en.wikipedia.org/wiki/Talk:Equal_temperament/Archive_2#Equal_temperament:_History

Thanks

BassHistory (talk) 03:35, 10 July 2019 (UTC)

The central issue of equal temperament is a compromise toward whole number ratios
Basic info was not being communicated in this article. The lede did not explain how ET came about as a compromise for the problems that happen after tuning to whole integer ratios of Just Intonation ...and then changing keys. The ratios in the new key are out of whack. Equal temperament is the tweak that was done to make any and every key sound equally close to the whole number ratios, while no single key at all would have exact ratios in their semitone intervals. All keys would be equally "off" from perfect ratios to a small extent, which was taken to be an improvement on having one key being exactly "on", with the other keys being badly off.

I had added this to the article with this edit, but my inputs promptly got clobbered within about half an hour as being Original Research. Original? This has been common knowledge for hundreds of years. As the article stands now (with the revert), it does a lame job of explaining why we have Equal Temperment, let alone why it became so popular. Just Intonation is not fully explained until more than halfway down into the body of the article. Considering how the desire for exact whole number ratios is the central issue in how equal temperament came to be, it is clear to me that this needs to be explained in the lede.

Equal temperament as a compromise toward a "close-to Just" tuning that works equally well when the key is changed is the fundamental issue of this article. And now with this revert, this central fact is never mentioned, let alone explained. -- Tdadamemd19 (talk) 20:26, 2 May 2019 (UTC) [Correction: my original edits had stated "Pythagorean Tuning", when I meant to say Just Intonation. To understand what the man Pythagoras himself valued, it is easy to conclude that he, or rather his school, were proponents of intervals based on low whole integer ratios.] -- Tdadamemd19 (talk) 21:19, 2 May 2019 (UTC)

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"Examples" deleted again
This section has multiple issues: Note that this section has had a "confusing" template since January 2017, and there was a deletion attempt last year. It is possible that there is some material in this section that is salvageable, but as currently written it is better left out of the article. Ringdown (talk) 12:19, 9 February 2018 (UTC)
 * 1) It seems to be some kind of algebra/arithmetic workbook with no clear message. Formulas are rearranged and re-expressed to define new quantities for no apparent reason. The properties of logarithms (and how to evaluate them numerically) are presented in excruciating detail.
 * 2) It is wholly uncited. It is unclear whether the terms that are introduced ("magnitude", "abstract unit", etc.) are actually established nomenclature or just idiosyncratic (i.e., straying into WP:OR territory). From some googling I suspect the latter.
 * 3) It seems to exist to point the reader to a tutorial, which is not the point of Wikipedia. Also, said tutorial is just a talk page section, which is also not kosher.
 * Mr. Ringdown, please let me say that you are right on that you deleted general formulas, because they are original work of mine. Indeed psycho-acoustical law was known even to ancient Egyptians. Also formulas with numerical values are others original work, only generalization is mine. With regards and friendship, Georges Theodosiou, The Straw Man, Georges T. (talk) 09:41, 10 February 2018 (UTC)

Address for general formulas
Ladies, Gentlemen, please let me give you address of general formulas and their background theory. With regards and friendship, Georges Theodosiou, The Straw Man, Georges T. (talk) 08:18, 2 June 2018 (UTC)

Correction
Ladies, Gentlemen, please let me correct an obvious error in Background theory > Example. Paragraph "Indeed, if we multiply ... then L = 500 cm." to be replaced by following in quotas.

"Indeed, if, in the last equation, we multiply m by d and divide log $ffreq_{C}⁄ffreq_{D}$ by d, that is


 * md $$\frac{\log\frac{\mathrm{ffreq_C}}{\mathrm{ffreq_D}}}{d}$$

then md is the magnitude and
 * $$\frac{\log\frac{\mathrm{ffreq_C}}{\mathrm{ffreq_D}}}{d}$$

is the unit, as with every quantity, e.g. length L = 5 m. If the magnitude 5 is multiplied by 100, and m (certain length, serving as unit) divided by 100 (that is, cm), then L = 500 cm. When $ffreq_{C}⁄ffreq_{D}$ = 2 and d = 1200, unit is cent (one 1200th of octave)."

Talking
Mr Just plain Bill,

Please let me express my many thanks for correcting my grammar in General formula for calculating equal temperament, General properties section.

With regards and friendship, Georges Theodosiou, chretienorthodox1@gmail.com 80.14.142.102 (talk) 11:17, 17 September 2016 (UTC)
 * Please, it is only collaborative editing. (You're welcome.) Best regards, Just plain Bill (talk) 17:13, 18 September 2016 (UTC)
 * Mr Just plain Bill, (I suppose you are the Author of this article), please let me express my sincere gratitude for the "welcome" and tell you the little story of "General formula for calculating equal temperament".
 * For first time I have posted it at Psaltologion by 30-01-09, 11:26, (2009 January 30), as an answer to Mr Zinoviev, a Byzantine Music lover, as well I am.
 * However I have to acknowledge that from when I posted it there, until I published at wiki, nobody responded, but from when I published at wiki (by 2015 December 2), I have received many thanks from music lovers.
 * I consider you music theoretician. Then please let me the question: do you consider this formula of some value for music theoreticians?
 * With regards and friendship, Georges Theodosiou, chretienorthodox1@gmail.com, Georges T. (talk) 07:50, 20 September 2016 (UTC)
 * Georges, I see you have registered an account. Congratulations!
 * I am by no means the "author" of this article; like every other wikipedia article, it is meant to be a collaboration among editors. I merely adjust format and orthography as I see a need. I would consider myself an interested amateur, not so much of a music theoretician. I did check the formula by substituting values for a perfect fifth in 12TET just intonation with a ratio of 3/2, and it gave t=701.955 or so, which comes as no surprise. I can imagine cases where the the formula may be useful. Best regards, Just plain Bill (talk) 17:00, 20 September 2016 (UTC)

Mr Just plain Bill, good morning (I live in France),

Please let me express my thanks for you replied, and for you can imagine cases where the formula may be useful.

In your computation unit is cent = $$\frac{1}{1200}$$ octave. However you can use another unit, for example, unit = $$\frac{1}{100}$$ of the tone $$\frac{9}{8}$$. Then, t = 344.247... units (hundredths of the tone). In this example, r = $$\frac{3}{2}$$, b = $$\frac{9}{8}$$, and d = 100.

With regards and friendship, Georges Theodosiou, Georges T. (talk) 10:15, 21 September 2016 (UTC)


 * The paragraph needs further work to improve the English and to make clearer what was intended. Perhaps someone else can contribute?    D b f i r s   12:27, 24 September 2016 (UTC)
 * I've done a bit more to the paragraph, but I'm not a musician, so please check that I haven't made any errors.   D b f i r s   11:48, 25 September 2016 (UTC)

Mr Dbfirs, let me express my thanks for your comments. I'm Greek living in France and my english are poor. Could you, or any other with well english, please, improve grammar, syntax, and phrasing? Merci d'avance! Now let me make some remarks:

1. Anyone familiar with unit cent, could check results in Examples. For example, ratio interval $$\tfrac{3}{2}$$ is 701.955... cents. Interval $$\tfrac{9}{8}$$ is 203.91 cents. Its hundredth is 2.0391 cents. Now, 701.955 cents divided by 2.0391 cents yields 344.247... hundredths of $$\tfrac{9}{8}$$, that is result in 2nd Example. Then,

2. Formula only directly calculates tempered interval, from ratio interval, with any unit.

3. "Pitch depends to a lesser degree on the sound pressure level (loudness, volume) of the tone, especially at frequencies below 1,000 Hz and above 2,000 Hz. The pitch of lower tones gets lower as sound pressure increases. For instance, a tone of 200 Hz that is very loud seems one semitone lower in pitch than if it is just barely audible. Above 2,000 Hz, the pitch gets higher as the sound gets louder." (Pitch and frequency: last paragraph).

With regards and friendship — Preceding unsigned comment added by Georges T. (talk • contribs) 09:41, 27 September 2016 (UTC)  Georges T. (talk) 09:49, 27 September 2016 (UTC)


 * I appreciate that it is difficult to write in encyclopaedic style in a third language. I couldn't even begin to do that in either French or Greek, so I admire your efforts.  The logarithmic law applies only approximately to sound intensity, and you make a good point about the sound pressure level changing our perception of pitch.  Personally, I think that the logarithmic treatment of pitch intervals can be derived without making an analogy with Frechner's formulation of the Weber-Frechner principle (there's really only one principle involved, since each law leads to the other).  I'm not a music theorist, so perhaps someone else can comment on this?  Best wishes from across the channel.    D b f i r s   09:55, 27 September 2016 (UTC)


 * Mr Dbfirs please accept my many thanks for appreciate that it is difficult to write in encyclopaedic style in a third language, and for the best wishes across the channel.


 * I really don't see how a ratio of frequencies can be described as a stimulus intensity. This is unscientific use of language.  I suggest that we remove the analogy with the Weber-Frechner observations, or only just mention it in passing as an analogy, since they made no claim about sound frequency.    D b f i r s   11:07, 27 September 2016 (UTC)


 * Many thanks for your comment. For example, ffreq = 200 Hz is a stimulus, ffreq = 300 Hz is another, 400 Hz another. Consider ratios $$\tfrac{300}{200}$$ and $$\tfrac{400}{200}$$. Former is 1.5 and latter 2. Clear, intensity 2 is greater than 1.5, like pressure (say intensity) 2 Pascals is greater than 1.5 Pascal. So, ratios represent frequencies intensity although in logarithmic scale. Regards. Georges T. (talk) 12:37, 27 September 2016 (UTC)
 * No, that is a very confusing analogy because the intensity of a sound is related to its volume, not its pitch.   D b f i r s   12:45, 27 September 2016 (UTC)

Mr Dbfirs, please let me express my many thanks for your comment. Clear I mean pitch intensity. You can explain it, in the article, for english is native language for you. Regards. Georges T. (talk) 13:12, 27 September 2016 (UTC)


 * I don't know what you mean by "pitch intensity" because I would interpret that as volume, and you don't mean volume. Why not just pitch or frequency?    D b f i r s   16:59, 27 September 2016 (UTC)

Mr. Dbfirs, please let me express my thanks for your reply. I just use Fechner law terminology. When law applies to pitch, intensity means pitch. I'm going to edit article. Regards. Georges T. (talk) 08:56, 28 September 2016 (UTC)


 * You are certainly persistent in your opinions. Can you link to anywhere that Fechner mentions pitch?    D b f i r s   11:49, 30 September 2016 (UTC)


 * I have been wondering about that as well. My quick reading of the Weber–Fechner law article does not show that the Fechner law applies to pitch. Just plain Bill (talk) 15:04, 30 September 2016 (UTC)

Gentlemen Dbfirs and Just plain Bill. It just is my inspiration. Regards. Georges T. (talk) 14:10, 4 October 2016 (UTC)


 * Messrs Dbfirs and Just plain Bill. Please let me explain my understanding (inspiration) about.
 * It was known to Pythagoras, and perhaps to ancient Chinese philosophers, that our subjective sensation of sound pitch is proportional to the logarithm (exponent) of the ratio of two sounds fundamental frequencies. For example, ratios 9÷8 and (9÷8)^2. Our subjective sensation of latter is double of that of former.
 * To go further. Our subjective sensation needs two sounds, for compare their relative pitch. As with every quantity. For example, for understand magnitude of any length we need another length called unit. Comparing the two, we understand magnitude of other. Similar, for understand magnitude of any pitch, we need another pith, called unit pitch, e.g. cent. Indeed, difference is that, length is proportional to unit length, while pitch is logarithmically proportional to unit pitch, that is, logarithm of any ratio interval is proportional to logarithm of unit ratio interval.
 * Fechner just expressed this law more general including sound volume. Then, citation is not needed. Understanding of relative theory is needed. Regards. Georges T. (talk) 12:56, 5 October 2016 (UTC) Georges T. (talk) 13:17, 6 October 2016 (UTC)


 * Since Fechner never considered pitch, any reference to him is superfluous and should be removed as WP:original research. I wonder if the whole section should be removed?    D b f i r s   19:04, 5 October 2016 (UTC)
 * Framing pitch perception in terms of Fechner's law seems like a stretch, or original research. In a context of imaginative brainstorming (tempête d'idées, or ιδεοκαταιγισμός?) it might be interesting to consider the metaphors of such a framing, but that would not make suitable encyclopedic text. I am tempted to put the section back to how it was in November 2015. Just plain Bill (talk) 02:24, 6 October 2016 (UTC)

Messrs Dbfirs and Just plain Bill. Please let me explain you that Fechner's law is general, including sound pitch and volume. Regards. Georges T. (talk) 12:48, 6 October 2016 (UTC)


 * No, that is just your interpretation and original research. Fechner never mentioned pitch.  I tend to agree with Just plain Bill about reverting, especially because your additions appear to me to be too verbose and confusing, but I'm reluctant to upset you as a sincere editor who persists in his belief that he is right.  If you can find any reference for your additions, then I might be persuaded to change my mind.    D b f i r s   07:53, 7 October 2016 (UTC)

Gentlemen, you can read background knowledge (relative theory), at my talk page. Regards. Georges T. (talk) 08:30, 17 October 2016 (UTC)


 * So are you confirming that this is all WP:original research?   D b f i r s   09:47, 17 October 2016 (UTC)
 * Mr Dbfris, please let me say that reliable, published sources (Sensation of tone, and Stanford Physics Dept. course online), exist in relative (background) theory. Only presentation and explanation are mine. Georges T. (talk) 05:13, 18 October 2016 (UTC)
 * If you wish to avoid having your additions deleted, then you need to provide in-line citations to the reference works that you are using. Your "Further Reading" is in the wrong place (it should go at the end of the article), and I cannot find your additions in that document, though I might have missed something.  Also, please put comments about the article on this page, not in the article itself.    D b f i r s   15:37, 19 October 2016 (UTC)

Mr Dbfris, Please let me ask: By 11:26, 25 September 2016‎, in Edit Summary you state: "... this section needs more work." Then, do you think work I have done so far is enough? Regarding threats for deleting, you could do that at any time. Regards. Georges T. (talk) 13:35, 21 October 2016 (UTC)
 * I wouldn't want to discourage you by deleting your work without a consensus here, but the problem is in "your work". If it is original research, then it doesn't belong in Wikipedia (by policy -- see No original research), so you need to provide some in-line citations if you found the details in publications.    D b f i r s   16:38, 21 October 2016 (UTC)
 * Mr Dbfris, please accept my many thanks for you respond immediately to my messages. I have copied background theory from my talk page and pasted to article. It includes many citations. I have 4 for Pythagoras research. Then, you should be satisfied, at last! In recent times Helmholtz undergone experimental research and proved psycho-acoustical law known to Pythagoras. Even formulas are widely known, but only for base interval 2 (octave). Only, background theory, general formulas (for any base interval), and their explanation are mine. Is it "new research"? To my view it is a small step ahead of an old and long research. That's all my contribution here. Regards. Georges T. (talk) 09:12, 22 October 2016 (UTC)

Mr Dbfris, please understand that we do not take the logarithm of any frequencies but the logarithm of the ratio of fundamental frequencies. As long our "subjective sensation is proportional to the logarithm of stimulus intensity", follows that, in our case, stimulus intensity, or intensity of stimulus is the ratio of fundamental frequencies. Regards. Georges T. (talk) 12:58, 22 October 2016 (UTC)
 * Please don't use other Wikipedia articles as in-line citations. That's not the way we do it.  Please see WP:inline citations and WP:reliable sources.  You may link to relevant articles in Wikipedia by using double brackets just to clarify terms used and to provide further reading from Wikipedia.  Also, please don't use the word intensity for pitch, or for a ratio of pitches.  In the context of sound it invariably refers to volume.  You might also like to learn about the use of articles in English grammar.    D b f i r s   15:48, 24 October 2016 (UTC)
 * Mr Moderator Dbfris, please let me say, I use citations from Wikipedia articles as I do from other websites. Regarding term intensity I deleted reference to Fechner's law, for it is not necessary in understanding formula, rather is encyclopedic information. Indeed, when it it applied to sound pitch, stimulus intensity or intensity of stimulus is the deference (interval) of two sounds in pitch, expressed mathematically by the ratio (not the difference) of corresponding fundamental frequencies, due to psycho-acoustical law. Georges T. (talk) 08:21, 25 October 2016 (UTC)
 * Dear Mr Georges T, I'm not a moderator, just an ordinary editor. I agree that Fechner's law is not necessary for the understanding of pitch intervals, and thank you for understanding about "intensity" of sound.  Wikipedia internal links, where appropriate, use double brackets like this.  Citations to external on-line references use a single bracket weblink between ref tags (, as you have been using), or you can use cite web (Template:Cite web) if you prefer.   Best wishes.    D b f i r s   09:17, 25 October 2016 (UTC)
 * Dear Mr Dbfirs, please accept my thanks for your instructions, and my sorrow for delay in respond. I'm homeless in France, proscribed (every where in Europe), "sans papier, marquée et compliquée" according to french hidden state, and always persecuted by it. So I'm forced to travel from town to town. I get internet access for free, usually in municipal libraries (Médiathèques), like this I'm just now, in town Niort, west France. Under these circumstances I try my best. I think have finished my contribution about equal temperament, and donate it - indeed through Wikipedia to which I'm very indebted - to music theoreticians, both professionals and amateurs. Last but not least, I have to inform you that as I have already said to Mr Just plain Bill "for first time I have posted it at Psaltologion by 30-01-09, 11:26, (2009 January 30), as an answer to Mr Zinoviev, a Byzantine Music lover, as well I am", although only the main formula, without background theory and explanation, and with only one example. So publishing in Wikipedia is the second time. I mean that Mr Coumparoulis owner of analogion.com is justified to claim copyright as the first publisher. With regards and friendship. Georges T. (talk) 12:07, 26 October 2016 (UTC)

My dear mate, please let me complain for you do not comment my recent edits. They are interesting, especially negative tempered intervals. Did you ever hear about? My greetings to your Queen. Regards. Georges T. (talk) 08:35, 3 November 2016 (UTC)


 * I'm not sure to whom your last message is addressed, but I'm not in a position to pass on your message to our own dear Queen. Are you trolling?  I've given up on trying to improve your additions to the article.    D b f i r s   09:25, 3 November 2016 (UTC)
 * My dear mate please do not worry. Word "mate" is chiefly British according to Merriam-Webster dictionary, then as long you are British, it is clear whom my message is addressed to. Really I troll your contribution at least in grammar.
 * Please let me answer your question "General formulas for the equal-tempered interval: ????"
 * Euler's formula is $$e^{ix}=\cos x+i\sin x$$ only. Regards. Georges T. (talk) 10:25, 3 November 2016 (UTC)

My dear mate, please accept my many thanks for correcting my grammar. Yes i was meaning "appendix" and "Especially".

Also let me correct that I wrote in my message by 09:12, 22 October 2016 (UTC) that "Even formulas are widely known, but only for base interval 2 (octave).". I see in the article's subsection "Equal temperaments of non-octave intervals" base intervals 3 and $$\frac{3}{2}$$. However I do not see general formula published before I publish it at psaltologion. So I stand on that "publishing in Wikipedia is the second time. I mean that Mr Coumparoulis owner of analogion.com is justified to claim copyright as the first publisher.". Regards. Georges T. (talk) 11:33, 20 December 2016 (UTC)

Messrs Dbfirs and Just plain Bill. Please accept my many thanks for your collaboration, also my wishes for the new year. Regards. Georges T. (talk) 13:08, 25 December 2016 (UTC)

Ladies, Gentlemen, please let me express my sorrow for I can not make the example more easy. One should be familiar with the notion of temperament for easy understand it. Georges T. (talk) 09:12, 21 January 2017 (UTC) With regards and friendship. Georges Theodosiou. The Straw Man. Georges T. (talk) 14:28, 18 January 2017 (UTC)

Ladies Gentlemen, please let me say that section "General formulas for the equal-tempered interval" as it is, is confusing. I suggest either, whole section be deleted or be restored.

With regards and friendship, Georges Theodosiou, The Straw Man. Georges T. (talk) 13:28, 5 July 2017 (UTC)

Tutorials on psycho-acoustical law
Ladies, Gentlemen, please let me say, regarding template that example is confusing, that, Wikipedia does not permit tutorial in article, nor link to talk page. So I post following tutorials here, with the hope that beginners in equal temperament will read it.

Informal tutorial

 * We consider 4 lengths of a string: 90 cm, 80 cm, 81 cm, and 64 cm.
 * Let P90, P80, P81, P64, be sound pitches produced by these lengths when string is plucked.
 * Now we consider two intervals: P90 - P80, and, P81 - P64. Our subjective sensation is that, latter is double the former.
 * We want an arithmetic relationship between string's lengths, agreeing with our subjective sensation.
 * We try differences in lengths: 90 cm - 80 cm = 10 cm, and, 81 cm - 64 cm = 17 cm. Latter is not double the former.
 * Now we try ratios of lengths: $90⁄80$ = 1.125, and, $81⁄64$ = 1.265625. Latter only little greater than former.
 * Eventually we try logarithms of these ratios:
 * log$90⁄80$ = 0.05..., and, log$81⁄64$ = 0.1...
 * Marvelous! Latter is double the former.
 * At last, we found arithmetic relationship between string's lengths agreeing with our subjective sensation. We say that "our subjective sensation of intervals, P90 - P80 and P81 - P64, is proportional to the logarithms, 0.05... and 0.1..., of the ratios, $90⁄80$ and $81⁄64$, of the lengths".
 * By trying many lengths ratios, we have found that

"Our subjective sensation of the interval between two sound pitches produced by two different string's lengths, is proportional to the logarithm of the ratio of the lengths".
 * It is said: "psycho-acoustical law".
 * Same law is applied regarding fundamental frequencies as follows:

"Our subjective sensation of the interval between two sound pitches is proportional to the logarithm of the ratio of corresponding fundamental frequencies". Remark: Fundamental frequency and string's length are inverse proportional.
 * Present version is mathematically correct. The before version was not, for I used term "inverse" in a liberal way. Mr. Wgrommel is right on it.
 * But present version provokes the question: "How is that ratio of fundamental frequencies and ratio of string's lengths are both logarithmically proportional to interval between pitches, while fundamental frequency and string's length are inverse proportional? I can not answer this question. Then I suggest my readers beginners in equal temperament, first understand psycho-acoustical law regarding fundamental frequencies, and then, having in mind they are inverse proportional to string's lengths, understand law regarding string's lengths. It is the easiest way for understand it, although indirect.

Formal tutorial
For math lovers I post following formal tutorial.
 * We consider 4 sound fundamental frequencies: F1, F2, F3, F4, and corresponding pitches P1, P2, P3, P4.
 * Now we compare intervals (differences): P1 - P2, and P3 - P4. Psycho-acoustical law states that: $P1 - P2⁄P3 - P4$ = $logF1⁄F2⁄logF3⁄F4$.

Any comment on these tutorials is well appreciated. Note: I regard that above tutorials are best explaining psycho-acoustical law. However, if Mr. Dbfirs threatens me by offending, I may explain it better. With regards and friendship. Georges Theodosiou. The Straw Man. Georges T. (talk) 13:51, 26 January 2017 (UTC)

I shall not comment on the twists and turns of this discussion. IMHO, "our subjective sensation of the difference (interval) in pitch (tone) is inversely proportional to the exponent of the ratio of string (chord) lengths" does not make sense. Two quantities can be inversely proportional; e.g., the gravitational attraction between two masses is inversely proportional to the distance between them. But our subjective sensation is not a quantity. It seems that something like this is meant: "If three pitches are in continued proportion c:d:e, i.e. the ratio between c and d is the same as the ratio between d and e, we hear these subjectively as two equal intervals. Since then the ratio e/c is the square of the ratio d/c, we hear the former as an interval that can be described as "double" the latter." One could then invoke logarithms. Not being well-versed in this field, I have not changed the text, but I think a clarification of this sort is called for. Wgrommel (talk) 21:57, 18 February 2017 (UTC)
 * I agree that your explanation is clearer. I've been wondering how to simplify all this without offending Georges ( to whom I was offering advice on avoiding deletion, per policy, not issuing any threat! ).    D b f i r s   22:07, 18 February 2017 (UTC)
 * My dear mate Mr. Dbfirs. Please let me say my understanding of simplify a tutorial. It means rather to add than to delete. For example "informal" tutorial is much longer than "rigorous" for it is formal simplified. Also please continue correct my grammar and phrasing as before. Regards. Georges T. (talk) 12:29, 4 March 2017 (UTC)
 * Mr Wgrommel, please accept my many thanks for you are the first one commented my tutorial. My old mate in this discussion Mr Dbfirs is now the second. Regarding "inverse proportional", I explain: "by sense, as greater the length as lower the pitch".
 * Our subjective sensation is psycho-acoustical quantity as gravitation is physical quantity. So psycho-acoustical law is similar to physical law.
 * As long I understand english (I'm Greek living in France) "difference" and "interval" are synonyms. Also chord is a string. Logarithm is exponent, "the logarithm of a number is the exponent to which another fixed number, the base, must be raised to produce that number.".
 * You confuse 22 with 2 * 2. Although are equal, in general 2a is not equal with 2 * a. Consider 3 fundamental frequencies: 100 Hz, 200 Hz and 800 Hz. Our subjective sensation is that, interval $800 Hz⁄100 Hz$ is (not 4, but) 3 times interval $200 Hz⁄100 Hz$. Latter is equal 2, former is equal 8, and 23 = 8. So we conclude that we comprehend pitch intervals with proportion (3) equal to exponent relating ratios of fundamental frequencies.
 * My dear old mate Mr. Dbfirs, please accept my many thanks for you are interested in not "offending Georges". Georges T. (talk) 09:45, 21 February 2017 (UTC)
 * Γεωργος, never fear -- I shall not confuse 2n with 2*n. If ratio b is the square of ratio a, the logarithm of b is twice that of a (log 9:4 = 2 * log 3:2); thus, as you say, the interval between C and E might be described as twice the interval between C and D. But I must disagree with the statement "Our subjective sensation is psycho-acoustical quantity." Subjective experience is generally qualitative, not quantitative. Yes, those of us with mathematical and musical training hear certain intervals as being equal to other intervals, and we can count intervals, but could the naive listener say that an octave is twelve times as big as a semitone? I respectfully doubt it. Wgrommel (talk) 21:44, 25 February 2017 (UTC)

Mr Wgrommel, many thanks for above message. I can add many things. For example that we call "ratio interval" or just "interval" is incorrect by math view. Ratio is not interval, difference, result of subtraction. Indeed we call it interval by the sense it corresponds to interval between pitches, according to our subjective sensation.

Psycho-acoustical law was known to Pythagoras and probably to ancient Egyptians and Chinese. Obviously they concluded it from many experiments. By the way let me do english humor. Mr. Dbfirs claims background theory is original work of mine! Rather it is original work of some Pharaoh. Only presentation is mine. Regards.


 * My guess would be that the basic facts were known to the Sumerians, long before the Pharaohs and Pythagoras, but I'd be surprised if they knew about logarithms.   D b f i r s   12:01, 28 February 2017 (UTC)


 * My dear old and present mate, please let me be not surprised if Sumerians knew about logarithms, given that logarithm is just exponent. I'd be surprised if they knew about psycho-acoustical law. Regards. Georges T. (talk) 13:17, 28 February 2017 (UTC)


 * They probably knew about it in music, but didn't express it in such a complicated way.   D b f i r s   09:59, 4 March 2017 (UTC)

Absolute unit
1. Main formula

We can calculate tempered interval of ratio one, by following formula:

t(r) = log(r) atu = ma atu (1)

"t" stands for "tempered", "r" for "ratio interval", "ma" for "absolute magnitude", "log" for "logarithm with base 10", and "atu" for "absolute tempered unit". It means that "log(r)" is just the magnitude of tempered interval. Unit "atu" is defined by this formula.

It is direct application of psycho-acoustical law. Indeed we can use a factor say f, as the number 1000 in 0.003 m * 1000 = 1 mm.

t(r) = log(r) * f satu = ms satu (2)

Where "satu" stands for "submultiple absolute tempered unit", and "ms" for "log(r) * f " that is magnitude with submultiple unit satu. For example f = 1000. Then, octave (ratio interval 2) is 301 satu, fourth (4/3) is 125 satu, fifth (3/2) is 176 satu, and major tone (9/8) is 51 satu.

Formal expression for the case of octave is (t = tempered): t(2) = 301 satu. Number 301 is magnitude for submultiple unit satu. 2. Derived Formula

From (2) we get: log(r) = $m_{s}⁄f$ and eventually: $$r=10^{\frac{m_s}{f}}$$

For f = 1000, and ms = 301, we get r = 2, that is octave.

3. Remark

In general formulas, if we put b = 10 (base of log), d = f and m = ms we get above formulas. Indeed formula $$m=log(r)\cdot d$$, is easier than $$m=\frac{\log(r)}{\log(b)}\cdot d$$. Georges T. (talk) 18:22, 22 April 2017 (UTC)

Any comment is well appreciated.

With regards and friendship Georges T. (talk) 14:03, 21 April 2017 (UTC)

Template
Ladies, Gentlemen, in section "2.1.1.1 Example", in the article, there is a template message for more than one year. Do you agree it is time to remove it? With regards and friendship, Georges Theodosiou, The Straw Man Georges T. (talk) 09:37, 6 February 2018 (UTC)


 * Does the template's presence for more than a year make the section less confusing?  Dbfirs  09:50, 6 February 2018 (UTC)
 * Mr. Dbfirs, please accept my many thanks for commented my message. Title "General formulas for the equal-tempered interval" without any formula is what is confusing me. With regards and friendship, Georges Theodosiou, The Straw Man Georges T. (talk) 13:26, 9 February 2018 (UTC)

Generalization and general formulas
Ladies, Gentlemen, please let me say that general formulas calculating equal temperament are generalization of formulas with numerical values posted by others. Then I have two views:

1. They are not original work of mine because are generalization of others's formulas.

2. Are original work of mine because generalization is mine.

Also I think every presentation of a subject is author's original work save when it is copied and pasted from other website, or copied from other author's book. — Preceding unsigned comment added by Georges T. (talk • contribs) 09:21, 13 February 2018 (UTC)

Apparent error in Regular Diatonic Tunings section
I know very little about this subject, but there seems to be an error HERE. A word, perhaps "steps", seems to be missing after "The diatonic tuning in twelve equal...". Lou Sander (talk) 15:00, 1 June 2018 (UTC)


 * Good catch! This is an informal usage, so I've added a note to the lead section to note the informal abbreviation twelve equal for twelve-tone equal temperament.  yoyo (talk) 02:46, 18 March 2020 (UTC)

Separate article for 12-TET
I don't see why 12-TET, which, along with various tweaks designed to make certain intervals sound better, has been used almost exclusively by the Western world over the past millennium or so, should only be covered in a section of a larger article while relatively obscure temperaments, such as 15-TET, 23-TET, and 41-TET, get their own articles. 12-TET is certainty more notable than those tunings and has more content devoted to it on this article than those articles are long. I therefore propose that this article's content about 12-TET (the third section and most of the first section) be moved to a separate article and be replaced with a short summary of 12-TET with a link to the main article. Care to differ or discuss with me? The Nth User 02:14, 13 January 2020 (UTC)