Talk:Equal temperament/Archive 1

Comparison
The end of this article is not really objective. I think discussions regarding equal temperament vs just intonation or other tuning systems don't belong on any of the individual tuning articles...Hyacinth


 * I agree, and I've taken it out of the article. Here it is for anybody interested:


 * However, equal temperament also made possible far more harmonically complex music, as it enabled musicians to move rapidly between keys without re-tuning: J. S. Bach wrote The Well-Tempered Clavier to demonstrate the musical possibilities of well temperament, a system similar in intent to just intonation, but one which did not create an octave of 12 identical semitones.


 * Apart from anything else, the Well-Tempered Clavier has nothing to do with equal temperament (it even says this, so why it was put in the article is beyond me). It is possible to compare diffrent tuning systems in an unbiased way, I think, but not like this. --Camembert

Curiously, if you either retune for each piece or transpose, you can play all of WTC Book I in meantone. Why that is I don't know, beyond the obvious point that that's the way Bach wrote it. Gene Ward Smith


 * I've re-added the above well-tempered info into the article more appropriately.Hyacinth

Good move, but where is the best place for a discussion of equal temperament vs just intonation etc? I've also placed other questionable to objectionable material here.Hyacinth

That would belong in a discussion of temperament in general, would it not? Gene Ward Smith


 * Quite possibly, music from these eras will sound subtly different if played in historical tunings, rather than modern equal temperament.


 * The composer Terry Riley has been quoted as saying "Western music is fast because it's not in tune".


 * Good question - I guess musical tuning would be the place. There's a tiny bit of comparison there now, but it could certainly take a lot of expansion. --Camembert

I've added a stubby section to musical tuning "Comparisons and controversies between tunings." I started it with a quick non-technical comparison between equal temperament and just intonation that is, hopefully, as objective as possible. Eventually I'll move most of the existing material on this from above and other articles.

There is vastly more to this topic than merely 12-tet vs just intonation, and much of it is highly mathematical, which raises the question of where and how much to Wikify. Gene Ward Smith

I moved the above Terry Riley quote to the "Comparisons and controversies between tunings" section of Musical tuning.

I also have added more to move this article away from 12-tet towards equal temperament in general, but someone should check the formatting on the equations...Hyacinth

There's still very little about anything other than 12. Gene Ward Smith


 * Someone pointed out that 12 TONE equal temperament consists of twelve SEMI-tones. I think it's confusing: the former tone being used in the English sense as Americans use the word pitch, and the latter tone is used in diatonic setting; but I don't know how to resolve it on this page.Hyacinth

Why, when talking about temperaments, do we strive to attain just intervals? For example, the octave is unusable as a just interval because of our physiology - because of the particularities of the human ear. If we make an analogy with the tempo parameters, we cannot say that we play with the same tempo all the time. Why try to play with just intervals then? We can have 12-tet equal temperament when its half-tone is greater than the one used in a just octave. Mr. Dimitrov mailto:latchezar_d@yahoo.com


 * I'm not sure what you're asking. The octave is, so to say, the most just interval, being the lowest whole numbers possible 2:1 (outside the unison 1:1). However, you may find some answers at Fred Lerdahl. Hyacinth 22:22, 13 May 2004 (UTC)

When the octave is just, the higher tone is "covered" by the lower because of the harmonic partials of the lower tone which coincide. If you want to hear the higher tone better the octave must be enlarged! The analogy with rhythm in music is the following: when we have a given tempo we don't play with the metronome, we only check our tempo with the metronome and we don't use the metronome to play for a long time in the same tempo. It is the same thing with intonation. We use only one reference tone-the note "A" (between 440-443 Hz) or the unison with this A 440 which is the only just interval in use! Mr Dimitrov


 * Actually, with practice one can hear the octave between the fundamental and next harmonic of a single pitch. You may be interested in lucy tuning. However, I don't know what you are asking, or how you are suggesting we change the article. Hyacinth 17:55, 14 May 2004 (UTC)

Well, I can resume my opinion: 1) I think that the equal temperament is the best thing we can possibly use. 2) The half-tone of this temperament can be determined by a division of one "large" (with a ratio >2/1) octave by 12.

And the result is great ! :-)

Mr.Dimitrov

See also: pseudo-octave. Technically any just octave tuned by ear on piano will be slightly greater than 2:1, due to the inharmonicity of strings. Is this based on original research by you, or do you have sources or references for this? I would be very interested in reading them. Hyacinth 22:07, 14 May 2004 (UTC)

Yes, it was 20 years since I read the book of Serge Cordier- "Le piano bien temperé et la justesse orchestrale" and started working on equal temperament. An example with numbers: 12 root of 2 = 1.059463 but 7 root of 1.5 is bigger= 1.059634. Therefore the half-tones in use are different when we use a different just reference interval. In that case the first one is too small and the second one too big. So what do we do? How big a half-tone will satisfy our ear? Those are good questions to discuss here! I can sent you example of my equal temperament if you write me to: latchezar_d@yahoo.com

Mr Dimitrov

Removed
I noticed the following text was removed and I can't find an explanation in the page history:
 * "Twelve tone equal temperament was designed to permit the playing of music in all keys with an equal amount of mis-tuning in each, while still approximating just intonation. This allows much more facile harmonic motion, while losing some subtlety of intonation. True equal temperament was not available to musicians before about 1870 because scientific tuning and measurement was not available. And in fact, from about 1450 to about 1800 musicians tolerated even less mistuning in the most common keys, like C major. Instead, they used approximations that emphasized the tuning of thirds or fifths in these keys, such as Meantone temperament. J. S. Bach wrote The Well-Tempered Clavier to demonstrate the musical possibilities of well temperament. There is some reason to believe that when composers and theoreticians of this era wrote of the "colors" of the keys, they described the subtly different dissonances of particular tuning methods, though it is difficult to determine with any exactness the actual tunings used in different places at different times by any composer." Anon.


 * Some of this text appears to have been reinstated, though somewhat modified, as follows:
 * *"J. S. Bach wrote The Well-Tempered Clavier to demonstrate the musical possibilities of well temperament, where in some keys the consonances are even more degraded than in equal temperament. It is reasonable to believe that when composers and theoreticians of earlier times wrote of the moods and "colors" of the keys, they described the subtly different dissonances of particular tuning methods, though it is difficult to determine with any exactness the actual tunings used in different places at different times by any composer."


 * I have issues with this.
 * *Doesn't the first half of the first sentence state only a probable inference?
 * *The second half of the first sentence is argumentative. It should not stand without giving at least one clear and incontrovertible example.
 * *The second sentence is too long and should be split in two at the word "though".
 * *The last third of the second sentence strikes me as altogether too vague. Surely there exists at least one composer whose tuning preferences and practice at some time is known beyond doubt?
 * *The rest of the second sentence is wrong. It is not reasonable to believe the assertion made there, since some writers were describing the difference of "mood" or "key colour" between keys in the same tuning. As a separate issue, "particular tuning methods", ie temperaments and bearing plans, create different consonances and dissonances, and those differences are not always subtle.  However, I don't know whether either of the two points made by this (second) sentence really belong in an article on equal temperament.  I suggest that they - or a more acceptable version of them - be moved to an article on "key colour" or on "tuning controversies". yoyo 07:36, 23 May 2006 (UTC)

I didn't write that paragraph, but I can comment on it:
 * 1) He called it "the well tempered claiver" for a reason. It is not really in debate whether or not he wrote in every key to demonstrate the possibilities (there are few absolutes in history though, there's always room for argument, of course) of well temperament. See Grout/Palisca A History of Western Music 6th ed (W.W. Norton), pg 392: In addition to demonstrating the possibility of using all the keys, with the then still novel equal- or nearly equal-tempered tuning, Bach had particular pedagogic intentions in Part I.
 * 2) All forms ot Temperament (music) degrade just intonation by some amount. Equal temperament spreads this out across all intervals as evenly as possible, other forms of temperament have uneven arrangments, thus some consonances are more pure, some are less (which that paragraph does fail to mention).
 * 3) If you think the sentence is too long, by all means, fix it. I think it's fine though.
 * 4) There are indeed composers who have written about tuning, but in general as we get into well temperaments, no major composer describes the process accurately enough that we can duplicate the tuning. This is not the case with the meantone temperament which was prevalent through the Renaissance and much of the Baroque. The reason is that well-temperament describes a large realm of possibility (whereas Meantone is specific), which includes even equal temperament. Descriptions of tuning drop off as well temperament is introduced, probably due to the subtlety of difference (and scientific inability to accurately tune before Helmholtz's landmark publication) between various methods of attaining well temperament.
 * 5) I think you misunderstand the assumption; if one assumes that a particular composer would use one tuning system consistently, his descriptions of "moods" and "colours" would apply to that particular tuning system, not to a difference between his tuning system and another. Of course, this "mood" and "colour" are convoluted terms and this is far from the only factor. For instance, how many wrote of E-flat's "pastoral" quality before Beethoven's 3rd, and after? (I don't have a reference on hand, but the answer is basically: few, then many.) If we're talking about orchestral music, accurate tuning is basically out the window as well, so the gravity of C-minor (staying on Btvn's 3rd for a moment) could be explained by its ability to use the stark sounding lower open strings of the string instruments.

Rainwarrior 17:14, 16 June 2006 (UTC)
 * Thanks for your responses. I've made some slight changes to the paragraph in question for greater clarity. It now reads:


 * *"J. S. Bach wrote The Well-Tempered Clavier to demonstrate the musical possibilities of well temperament, where in some keys the consonances are even more degraded than in equal temperament. It is reasonable to believe that when composers and theoreticians of earlier times wrote of the moods and "colors" of the keys, they each described the subtly different dissonances made available within a particular tuning method. However, it is difficult to determine with any exactness the actual tunings used in different places at different times by any composer. (Correspondingly, there is a great deal of variety in the particular opinions of composers about the moods and colors of particular keys.)"


 * I believe these changes improve the article. yoyo 16:32, 17 December 2006 (UTC)


 * (P.S. I left a reference below under "Precision" that is relevant to this discussion.) - Rainwarrior 17:49, 16 June 2006 (UTC)

China vs Europe

 * The first person known to introduce a mathematically accurate specification for equal temperament is Chu Tsai-Yu in the Ming Dynasty, who published a theory of the temperament in 1584. Soon after, European mathematicians Simon Stevin (1585) and Marin Mersenne (1636) accurately described equal temperament, independently from China. Equal temperament

On the other hand it is claimed on the University of Houston website at http://www.uh.edu/engines/epi380.htm that:


 * Many European inventions were made independently after the Chinese had thought of them first. Not this one. A Jesuit student of China, Matteo Ricci, attended a Chinese trade fair in Canton the same year Chu published his work. Ricci almost certainly brought Chu's work out of China and back to the West.

Does anyone have an authoritative answer, or should the claim in the article be toned down? Flapdragon 20:56, 25 July 2005 (UTC)


 * "In 1584, the great Chinese scholar of the Jesuits, Matteo Ricci, commenced his studies at Macao. From 1580, the Viceroy of the Cantonese province had established biannual 'trade fairs' lasting several weeks, at which Chinese and Westerners exchanged ideas and goods. The interchange between East and West was intense just at the moment when Chu Tsai-Yu went into print with his new theory. It is a case of perfect timing which gives one the feeling that it was 'meant' to happen. We do not know the exact mode of transmission of the idea to Europe; there can be no doubt that Western music was to be totally conquered by the Ming prince, for within fifty-two years of Chu's publication, his ideas were published by Pere Marin Mersenne. The Ming Dynasty ended eight years later, but Ming music today blares from every transistor radio in the world."

I got this from "The Genius of China, 3,000 years of science, discovery, and invention" by Robert Temple.


 * Unless Mersenne mentions Chu Tsai-Yu, then I think we're jumping to conclusions here. It is a very interesting coincidence, but I would argue that it's not strong enough to assert Chinese influence on Mersenne because: Equal temperament was very much not a new idea in Mersenne's time. Mersenne had an interest in tuning (it went well beyond equal temperament), and this was at a time when mathematics was finally developed enough to calculate 12th roots of things. It's very likely for someone to come up with an exact description of equal temperament on his own under these conditions. (It is, however, quite accurate to say that Chu Tsai-Yu came up with it first, I believe.) - Rainwarrior 17:22, 16 June 2006 (UTC)

Sound of different keys
In equal temperament, can someone without perfect pitch tell by the sound (color) which key is being played. If so, how?--Light current 14:27, 21 October 2005 (UTC)


 * No. yoyo 01:16, 7 January 2006 (UTC)


 * With specialized instrumental knowledge, it's actually possible. Certain pitches always sound a certain way on certain instruments. For instance, the open G on a violin is unmistakable. Even without perfect pitch, knowing the timbre of the instrument can clue you unto the key. (Of course, put the violin into scordatura and things are up in the air again.) Rainwarrior 22:54, 1 April 2006 (UTC)


 * I'd like to see this tested experimentally! I think that, for example, a listener without perfect pitch would be very hard put to distinguish the open G on a violin from that on a viola, by sound alone.


 * Let me expand a little on my earlier wordy reply ("No.") ;-). The reason I say so is because the phenomenon of key colour is primarily due to harmonic differences between keys; that is, to the fact that in most temperaments, each interval in a given key may be slightly different from its counterpart in another key.  However, in any equal temperament, each named interval eg fifth, minor third, augmented sixth - is exactly equal (by design) in all available keys. Any remaining component of key colour may be explained partly by timbre and partly by relative pitch. For example, one doesn't need perfect pitch to know that a piece in C, when transposed upwards by a fourth to F, is higher.  Knowing that it is higher may create a subjective difference of sensation. yoyo 07:57, 23 May 2006 (UTC)

On being understood
In the introductory paragraph, we find: "this expedites mathematical methodology when examining these temperaments". Would contributors please try harder to write so that more readers will understand them? Let's try to translate this phrase into everyday English, starting with the easy parts:
 * "methodology"? - plain "methods" would do as well.
 * "mathematical" - that's fine as it is.
 * "expedites"? - meaning "speeds up", right? Or do we really mean to say something else, like "simplifies the application of"? A more active version of the same idea is "makes it easier to apply"

Here's a suggested rewrite of the phrase: "this makes it easier to apply mathematical methods when examining these temperaments". yoyo 01:16, 7 January 2006 (UTC)


 * On 23 May 2006, I noticed the disappearance of the phrase I had complained of back in January. Good! However, I will leave this section in the discussion. I hope it may remind others that simple writing is often clearer.yoyo 14:27, 16 June 2006 (UTC)

EDO vs tet
Why does the article not mention the term EDO (Equal Division of the Octave)? The term used - tet (tone equal temperament) does not state what interval is being divided into equal parts. It simply assumes that it is the octave. But this assumption is invalid for some equal temperaments that have been intensively studied, for example the Bohlen–Pierce scale, or BP scale, which uses as its interval of equivalence the perfect twelfth rather than the octave. We might speak of "the tyranny of the octave" as someone (Stravinsky?) famously referred to "the tyranny of the bar-line"! yoyo 01:25, 7 January 2006 (UTC)


 * On 23 May 2006, I made the necessary changes, so I no longer have cause to grumble! GWS has since begun a separate article on Equal division of the octave, which someone has proposed be merged back here.  AIUI, there are two important differences of emphasis between "tet" and "EDO":
 * The interval of equivalence
 * The fact of tempering
 * An EDO is not, per se, a temperament - it is just an equal division of an octave. A tet is, by implication, a temperament that uses equal divsions of the interval of equivalence, whether or not that be an octave. The commonest meaning of 12-tet or "equal temperament" is both a temperament and an EDO. My question now is, whether this article on (equal) temperament is the right place to discuss EDOs. My feeling is that this article discusses (and confounds!) two things - 12-tet (and coincidentally other n-tets) as temperaments, and various n-EDIOE Equal Division of the Interval of Equivalence, whether or not they are temperaments. Might it not be better to build this article on a foundation of three key concepts, each separately described in an article, namely:
 * Interval of Equivalence (which of course discusses and links to Octave);
 * Equal Division of the Interval of Equivalence; and
 * Temperament?
 * yoyo 14:53, 16 June 2006 (UTC)


 * EDO is by its mathematical definition always a temperament (you can't make irrational numbers out of rational ones). I think it's fine that EDO has its own article now, but if we want to talk about EDIOE (this term I think is rather obscure), put it on the EDO page. Same with "Interval of Equivalence". We don't need 3 pages with tiny bits of related information where one would do better (why send the reader running from page to page, when you can put all of the information they need for the topic together?). An article on temperament, however, is definitely a good idea. (Call it Temperament (music) though. I might get to that in a few days. - Rainwarrior 17:28, 16 June 2006 (UTC)


 * Thanks, Rainwarrior, I couldn't have said it better. —Keenan Pepper 01:29, 17 June 2006 (UTC)


 * Which bit, Keenan? :-) No, I beg to differ with Rainwarrior on this: By definition, an ET is an Equal Temperament, and thus approximates ("tempers") a rational tuning; whilst by definition, an EDO is an Equal Division of the Octave.  Although both procedures: tempering and division - may result in exactly the same numbers (eg nth roots of an IOE), the purposes of the two procedures are very different.  One seeks to approximate a system of just intonation in a convenient fashion, possibly more simply than any other (well-)tempering; whilst the other has as its very *goal* the creation of a universe of sonic materia that rank equally with each other, equivalent atoms each separated from its neighbours by a standard distance.


 * However, I'm happy with the idea of keeping all the "tiny bits of related information" together, as long as we don't end up with unnecessary confusions caused by glossing over "tiny" difficulties because they're not the focus of the article. yoyo 17:33, 17 December 2006 (UTC)


 * I had planned to put "EDIOE" on the "EDO" page, under an even more general notion: rank one tuning. I don't know an accepted musical term for that, but it simply means a tuning where the notes are in geometric series, forming a rank one free abelian group. I'm open to suggestions as to better places to put this, but "edo" is an actual term in general use, and "rank one tuning" isn't.


 * As for Temperament (music) some generalities can be found under Regular temperament, which could be kept in mind while writing it. For irregular temperaments, if you want a general theory it becomes somewhat arbitary drawing a line between temperament and scale, and Well temperament is already an article. Gene Ward Smith 19:20, 16 June 2006 (UTC)


 * Yeah, I think Regular temperament, and Piano tuning have between them an explanation of what temperament is, but I think it deserves its own short page with a bunch of links. - Rainwarrior 22:25, 16 June 2006 (UTC)

Cent values of equal temperament
Under this heading in the article I made several very minor edits.

The C#, for instance, originally appeared on my screen as C denoting, I'm guessing, a character that could not be displayed with my set up.

I also added a 'b' to E, etc. and changed the G to Ab.

Like I said, very minor edits. Davehorne 22:33, 17 January 2006 (UTC)


 * They were real sharp and flat signs, as opposed to pound signs and lowercase 'b's. Wikipedia is not responsible for problems with your font setup. I'm reverting. —Keenan Pepper 01:22, 18 January 2006 (UTC)

Keenan, perhaps you can share with me what your browser 'view' setting is. Using Windows IE (latest version), under View \ Encoding, I have Auto Select checked and under that I have tried Western Europe (Windows), Central Europe (ISO), Unicode (UTF 8) and Western Europe (ISO). _All_ of them displayed either a  or other characters; nothing looked like ... b or #. That was the reason I changed the characters to b and #. This is the first time I even encountered not being able to read a simple flat or sharp.

My keyboard setting is United States - International, but I assume that doesn't even enter into the discussion since this is a 'viewing' problem. I initially changed the G# to Ab for general usage and familiarity; you changed it back. When discussing major keys, for instance (and not specific individual notes), I've never encountered G# major in my 45 years of playing. G# minor does exist as the relative minor of B major, but in actual usage, still rather rare. When it comes to 'individual notes' (in writing music), voice leading would make the choice for you.

This is added later - I asked a friend in the US to look at the text in question and he sees squares or rectangles. He uses an Apple computer. I suggest Keenan try viewing from another computer. Davehorne 17:38, 22 January 2006 (UTC)

I'm going to go back and edit back the changes I made. Davehorne 22:25, 23 January 2006 (UTC)


 * I see squares as well, but I know that the unicode characters: &amp;9837; ♭ | &amp;9838; ♮ | &amp;9839; ♯ all work fine for me (I think there's very widely supported), and don't seem to be the same as the squares you're using above. Perhaps you just picked characters from a different set? (There are some obscure sets for music notation but most people don't have them.) - Rainwarrior 17:33, 16 June 2006 (UTC)

Precision

 * True equal temperament was not available to musicians before about 1870 because scientific tuning and measurement was not available.

This statement is reproduced in a lot of places but maybe it is a little misleading. On one hand Scheibler invented the tonometer by 1834 (its basis, the tuning fork was invented 1711, and much developed by 1865) but on the other, if it couldn't be tuned by ear either directly (certainly unreliable, but proposed as late as 1907) or comparing intervals it seems like other tuning systems would have lacked the same kind of precision. --Mireut 15:55, 4 February 2006 (UTC)

I found a relevant passage from Ellis: "[T]he difficulty of tuning equal temperament by estimation of the ear, or even by the monochord, and of retaining the intonation of the piano or organ unchanged for even an houw, makes all temperaments in actual use really unequal. The difficulty of original tuning by estimation of the ear in the case of skhismatic temperament, where the Fifths have to be flattened by an almost inaudible skhist, is so much enhanced as to be insuperable except by Scheibler's method."(Ellis, A. On Musical Duodenes. Proc. Roy. Soc. 23, 1875) He instructs buying a set of Koenig's forks tuned flat 4 beats per second from the desired pitches. --Mireut 00:01, 11 February 2006 (UTC)


 * The best book I've read on the subject is: Jorgensen, Owen. Tuning. Michigan State University Press, 1991. ISBN 0-87013-290-3. The first chapter of this book is called "Equal Temperament Was Not Practiced on Pianoes in 1885". In it the Tonometer is discussed, 19th century tuning practices, the work of Helmholtz and Ellis. The date he gives for Equal Temperament is 1917, which is when the principal method of tuning equal temperament was finally written up, disseminated, and applied to pianos. After 1917, tempering became a skilled science based on universally accepted mathematical principles... There is little individuality, and the temperament sections of pianos tuned by different tuners match note for note when compared. Before 1917, tempering was an art based on a keen sense of color awareness for each individual interval or chord on the piano. This color sense that was developed through environmental conditioning by listening to tunings and piano music during the nineteenth century is now lost. Before this method was developed, apparently Ellis claimed that the only satisfactory way to tune equal temperament is with a fork tonometer, which while fairly accurate, was not a device being used by practicing piano tuners. - Rainwarrior 17:48, 16 June 2006 (UTC)


 * Some of the articles Jorgensen presents look different unedited (the encyclopedia one), and he interpreted one set of instructions different than other tuners (Montal, I think in one of the articles in Piano Technicians Journal at the time Jorgensen wrote that this book was obscure but Fétis says it was translated into other languages). Prony wrote in 1831, l'ensemble de mes épreuves m'a convaincu que le tempérament égal était aujourd'hui unanimement adopté pour l'accord des instruments à touches ; les très-légères anomalies de quelques comparaisons doivent être attribuées ou à des erreurs d'opérations, ou à des variations de tension. Mireut 21:04, 16 June 2006 (UTC)


 * I'll read Prony's article, though my French is a bit slower, so if you've already read it I'll ask you off the bat: does he give a practical method for tuning Equal Temperament, or is he only listing exposing the mathematics of equal temperament? You can calculate frequency ratios all you want, but in order to tune a piano you've got to have a method for obtaining these frequencies. This is what Jorgensen's argument is; that there was no widespread method for tuning it accurately until 1917, and thus it wasn't really part of "practice" until then. I'm sure that it had been accomplished several times before. - Rainwarrior 22:23, 16 June 2006 (UTC)


 * Followup: Montal gives a well temperament tuning plan, not equal temperament (though it is probably close to equal), on the page you linked. Prony discusses the physics of it, specifically the relationship of tension to the frequency of a string, but he does not outline any practical method of tuning a piano this way. Perhaps the question here is a matter of definition, which is what should be clarified in the article. Accurate, stable, standard equal temperament would not become available to the public until 1917. The idea came up long before, and various approximations had been tried all along. The question is: when do you stop calling them "well" and start calling them "equal"? - Rainwarrior 22:52, 16 June 2006 (UTC)


 * It's about the above statement, which I take meaning Ellis, or in the article currently, they might be more scientific but also don't present any more precise, accurate or stable methods for tuning or measuring. Mireut 12:48, 17 June 2006 (UTC)


 * Ahh, well, the statement you initially quoted from the article is clearly ambiguous (is that an oxy-moron?). I'll try to clean it up. - Rainwarrior 15:37, 17 June 2006 (UTC)


 * Oh, apparently I'd even tinkered with that bit before... I didn't notice the age of your original comment. I think it reflects the result of our discussion here, though I didn't add anything about Montal or Prony (I don't know much about either of them, but I'm not sure of what they added to the practice of equal temperament; both articles you linked are historical examples of advocation for equal temperament, but other than that don't seem to add much to its theory that wasn't already known in the previous century. Are they worth mentioning? I leave that up to you.) - Rainwarrior 16:09, 17 June 2006 (UTC)

removed out of context sentence
I removed:


 * However, Wachsmann (1950) used a Stroboconn to measure a Ugandan harp and women singing unaccompanied, finding variations of 15 and 5 cents respectively. ( &larr; check accuracy of fragment repair)

from the article as it doesn't make sense, and someone even noted in the article that it didn't. It may have had more context in a previous version or make sense in the book, but until someone finds what temperament it was talking about, I put around it. Rigadoun 05:37, 20 May 2006 (UTC)

inharmonicity and diversion from equal temperament
Properly tuned pianos are not tuned precisely to equal temperment. They are tuned to a "stretched" equal temperament. That is because (1) the pitch of each string is inharmonic, exhibits inharmonicity; the higher partials are not exact multiples of the fundamental, but rather each succeeding partial is a bit higher than it would be if it the strings vibrated hamonicly. The main reason for inharmonicity is that the 1/2 string, and the 1/3 string and the 1/4 string, etc, are not exactly 1/2, 1/3 or 1/4 etc. They are slightly shorter, thus they vibrate at slightly higher pitch than 2 times, 3 times, 4 times, etc. The differnece in length is due to a portion of the length of the string becoming a non-vibrating "node" equal to approx the diameter of the string. There are other factors also. (2) tuners match the upper partial of a lower note to the fundamental of a higher note, or a higher partial of the lower note to a not-so high partial of the upper note; they don't tune the fundamental of one note to any multiple of the fundamental of another. To this "stretched tuning" of just intonation, they add or subtract the frequency difference between equal temprament and just intonation. Thus each interval gets slightly "stretched" from how it would be if tuned precisely to equal temprament. This adds up so that there is quite a bit of stretch from the low end to the high end of the piano.

In actuality, the instrument sounds better this way than an instrument tuned to precise equal temprament.

Another factor that causes pianos to not be tuned precisely to equal temperametn is the fact that it difficult to control the tuning pin and string so that the pitch of the string is exactly the number of beats per second it needs to be, above or below just intonation (widened or narrowed from just intonation), for it to be precisly in a stretched equal temperament tuning. Time constraints will force to tuner to move on to tuning the next string, once he hears the beat rate falling into close to where he is aiming for. - This unsigned message was left on 03:48, 30 June 2006 by User:Nomenclator.


 * You seem to be a little bit misinformed. Piano tuning has all of the details, but what I would like to explain is that the deviation of piano tuning from equal temperament does not really have much to do with just intonation; it is about the fact that the overtones are wide, thus "stretched ocatves". (If you tuned the piano in just intonation it would still have stretched octaves.) As for claim about time constraints, I'd actually disagree in general, except on old or abused pianos where pins are loose. Excepting the treble register, it's not terribly difficult to get very accurate pitches if you check your temperament often. - Rainwarrior 04:43, 30 June 2006 (UTC)

You haven't explained why you think I am misinformed. Yes, tuning to just intonation would require stretching the intervals, just like tuning to equal temperament does. Not only are the octaves stretched, but every interval is stretched. This is due simply to the fact that the sensation of an interval is caused by hearing partials in unision, rather than by hearing the fundamental frequency of intervals in precise ratios (which is what the definiation of an interval is -- the ratio of the fundamental frequency of 2 pitches), and by the fact that piano strings (like all strings) have inharmonicity -- the partials are slightly higher than exact multiples, of 2, 3, 4 etcetera.

It makes no difference whether pins are too tight or too loose, or just right. I've done several hundred tunings on a few dozen pianos, and hearing the beating of the intervals and estimating how many beats per second I was hearing, was much easier than trying to get a pin to land exactly at that beat rate, and stay there. This is a matter that tuners put a lot of practice into, generally developing techniques that require incredible finesse, that involve pulling the string up a bit higher than the target beat rate, then nudging it down again until, if your prayers are answered, it falls exactly into the spot where the beat rate is the one they are aiming for, and doesn't jump past that spot, or land short of it. Quite often it jumps past that spot, and you have to pull it up again past the target, then nudge it down again. Estimating how far to pull each string of each pitch past its target point, then how to nudge the string down again to the target point, without going past it, takes lots and lots of practice. This is a problem throughout the whole range. Bump that tuning lever counter clockwise as gently as you can -- and often the pitch flys too far below the target.

Steel pins in laminated hardwood, pulled hard against the friction points of the bridge and especially the agraffe, never turn smoothly, even in the best hard wood that is the most cleverly laminated and the most precisely drilled, even if you have the best pins turned to the correct diameter with the most accuracy, and even if the string moves over a properly designed bridge and a properly designed agraffe made to the correct height, and even if the piano is kept at an even temperature and humidity. Gently as you can, with the highest quality tuning lever, and the best made pin-socket, you push, bump, batter, pat, wiggle, vibrate, and generally try to nudge, coax, and cajole the tuning pin counter-clockwize, and you hope the pin jumps counter clockwize just far enough, and the string skips just far enough past its friction poin on the agraffe, to get the string to beat at the rate you want it to beat -- but often it doesn't. It lands too high. Or it jumps a bit too far down. You often have to pull the string up past the target again and bump it down again, several times, before you finally get get the pitch to be where you want it to be, or at some point you compromise at a point that is a bit too high or a bit too low. And then, often, when you tune some other string, one of the strings you tuned earlier, decides to slip a bit in pitch due to the change in the total distribution of string tension across the whole set of strings.

While this is less of a problem with large pianos that are well made, it exists even in the best pianos kept at the most even temp and humidity, and is still the hardest part of piano tuning. It is due to friction of the tuning pin in the pin block and friction of the string over the bridge and the agraffe.

I still cannot get a decent tuning in less than 6 hours. A "concert" level tuning takes me 12.

Yes, my musician friends rave about my tunings. But I think that is mostly because I tune the unisons in unison, or so that they are wavering almost imperceptibly, and they stay that way for a little while. They are used to playing pianos where the unisons are beating about 1/2 a beat per second or more. They were nearly in unison when the tuner finished the job, but within a couple of days they slipped out. --Nomenclator 06:57, 30 June 2006 (UTC)


 * I said I thought you were misinformed because you wrote this, specifically:


 * To this "stretched tuning" of just intonation, they add or subtract the frequency difference between equal temprament and just intonation. Thus each interval gets slightly "stretched" from how it would be if tuned precisely to equal temprament.


 * That is a description of temperament, and not stretched octaves. As for your lengthy response here, I am well aware of the difficulties of piano tuning, I was simply referring to something more specific than you were, apparently. There is a big difference between "accuracy of equal temperament" and "accuracy of unisons". One can spend all day tuning unisons, I know, I know, that's not what I was referring to. The accuracy of the temperament, however, for the purposes of it being distinctly "equal temperament" in its aural identity need not be as precise. What I mean is, the primary quality of an equal temperament is intervals that are equivalent across all keys; if you can get your thirds in line across the keyboard, that is if they are monotonically increasing in beat frequency, you have achived equal temperament that is for all practical purposes perfect. Just intonation on the other hand, falls to the same problems as unison tuning. If pure intervals are part of your tuning system, the small differences are magnified tremendously.


 * My question for you is, to what end did you make your first lengthy post above? (Beginning: Properly tuned pianos are not tuned precisely to equal temperment.) Are you suggesting we include this information on this page? My own opinion is that it is unnecessary. The details of that are available at Piano tuning, and to some extent at Piano acoustics, and would just clutter the article here. This article isn't about piano tuning, it's about equal temperament. (And really, even with stretching, piano tuning has a rather equal temperament.) At most we might point out that piano tuning differs slightly from the mathematical ideal twelfth root of two unit of pitch some use to describe equal temperament numberically.


 * Also, if you are only continuing a previous discussion, please don't start a new heading on the talk page. It's confusing, as the intended indication is that a new discussion is beginning. - Rainwarrior 09:13, 30 June 2006 (UTC)


 * That is a description of temperament, and not stretched octaves.


 * Yes. My description of stretching was in the sentence before that one. That is: "whenever tuning intervals tuners match the upper partial of a lower note to the fundamental of a higher note, or a higher partial of the lower note to a not-so high partial of the upper note; they don't tune the fundamental of one note to the fundamental of another." (they only tune the fundamental of one note to the fundamental of another, when they are tuning unisons). It makes no difference whether they are trying to tune in just intonation, or with any temperament -- they are tuning harmonics -- rather than tuning fundamentals. This results in stretching of every interval. --Nomenclator 20:54, 1 July 2006 (UTC)


 * No no. I have just as much difficulty tuning a temperament as I do tuning unisions. Trying to get successive thirds to beat successively higher, is not easy. If I end up with a couple of successive thirds that seem to be beating at the same rate, every now and then, as opposed to have an occasional higher third actually beat slower than the one to the "left" of it, I consider myself lucky. No matter how carefully I tune 5ths and 4ths, I may not hear a continually increasing beat rate when I test the successive thirds. If, because it seems to be beating too slowly, I were the to nudge the beat rate of the second of 3 successive thirds, up the smallest amount I can, in order to make it beat just a hair faster than the first of the 3 thirds, it may end up beating faster than the third 3d, instead of beating more slowly, as theoretically it should. If I were to try and nudge it back down again, the smalest amount I can, it may beat at the same rate or slower than the first third of the sequence. Getting it to beat at the right speed may take several tries. And each time, I have to listen to the 4ths and 5ths, to make sure I haven't thrown any of those out. --Nomenclator 20:54, 1 July 2006 (UTC)


 * I seem to recall someone saying that properly tuned pianos are tuned precisely to equal temperament. I just wanted to correct the misinformation.


 * Perhaps it was this statement "precise equal temperament was not attainable until Johann Heinrich Scheibler developed a tuning fork tonometer in 1834 to accurately measure pitches. The use of this device was not widespread, and it was not until the early 20th century that a practical aural method of tuning the piano to equal temperament with precision was developed and disseminated."


 * The implication of that statement is that someone might attempt to properly tune a piano by tuning it to a set of equally tempered tuning fork tonometers.--Nomenclator 20:57, 1 July 2006 (UTC)


 * When using talk pages, it is very confusing to insert replies in the middle of someone's response. (They should be inserted at the end of the post replied to, after the signature, thus clarifying who said what.) If you need to reply to a specific part of it, please quote the original in your response. Typing '' at the beginning and end of such a quote causes it to appear in italics, which signifies that it is a quote. Furthermore, using indentation with : that has one more indent that then one your are replying to, it can be made clear which replies are a response to which. I will reply to your comments in a moment. - Rainwarrior 21:10, 1 July 2006 (UTC)


 * Now, a response. If you are not confused about which is temperament and which is stretched octaves (and in my usage, I take "stretched octaves" to refer to the stretching of all intervals in the tuning system, there is no need for you to continually point this out to me), then no more discussion need be made about that. If your original words were ambiguous, or if my reading of it was incorrect, it doesn't matter at this point because I think understanding has been reached about it. As for the mechanics of piano tuning, I don't wish to argue this further here, as I have made my argument, I have heard yours (obviously you disagree), and it has little relevance to the article anyway.


 * Someone did attempt to "properly" tune a piano by tuning it to a set of fork tonometers. Alexander Ellis was one such man. I don't really like to say proper here though, because there are many, many, many ways to tune a piano, and the preferred method has been very different over time. I don't know what his method was, he may have used the tonometer to set a temperament and then tuned out in the normal fasion, or he may have used the tonometer for the entire thing. Either way, what was produced I think can rightly be called Equal Temperament. There is this problem in definition because the history of Equal Temperament ties it directly to the piano. Maybe it can be argued that a piano does not have "true" Equal Temperament, but it was still piano tuning which brough "Equal Temperament" into the widespread use it has today. Equal Temperament under the railsback effect is still essentially Equal Temperament. - Rainwarrior 21:34, 1 July 2006 (UTC)

All I said was that pianos are not tuned precisely  to equal temperment. They are certainly tuned to equal temperament -- just not precisely. Nor does the tuner intend to tune them precisely to equal temperament. Rather, she tunes each interval a little "wide" from that (the upper note is a bit sharp, or the lower is a bit flat, depending upon which way you want to look at it). And the reason she tunes it a little wide is that the (1) the partials are tuned, not the fundamentals and (2) the partials exhibit inharmonicity -- they are not precise multiples of the fundamental; they do not precisely match the mathematical concept of a harmonic sequence, or series, whichever is the correct word. It is really very simple. By the way, neither did I say that pianos are not tuned precisely. Many tuners indeed tune pianos extraordinarily precisely. They just don't tune them precisely to equal temperament. They tune the intervals "wide" or "stretched" from equal temperment -- by a precise amount. --Nomenclator 06:59, 4 July 2006 (UTC)


 * Okay, so your complaint is about the specific passage that makes reference to the tonometer, etc., if I understand you correctly. What would you prefer it say? When I wrote that I considered it important to explain that there was really no precise way to tune something to equal temperament before this. I didn't think it important to mention right there that piano tuning is actually a slight deviation from it, because I couldn't think of a way to express this without making the language of the paragraph confusing. The idea is basically correct, and the slight inaccuracy is quite thoroughly ironed out, I think, by following the link to Piano tuning. What do you suggest it should say instead? - Rainwarrior 20:41, 4 July 2006 (UTC)

I can't find the passage that I objected to. I am, however, confused by the passage "A precise equal temperament was not attainable until Johann Heinrich Scheibler developed a tuning fork tonometer in 1834 to accurately measure pitches. The use of this device was not widespread, and it was not until the early 20th century that a practical aural method of tuning the piano to equal temperament with precision was developed and disseminated." What would prevent a tuner from attaining any kind of tuning he liked, before the development of accurate tuning forks? How would a tuning fork, or set of 52 tuning forks, make precise tuning of a temperament possible? The absence of a tuning fork would make precise tuning to an international standard for middle C, or A above middle C, or whatever, difficult. But there would be nothing to prevent the piano from being in tune with itself, or tuning it to any temperament. Are you saying that tuners did not identify, and listen to, and tune, coincident partials, until early in the 20th century? --Nomenclator 21:16, 4 July 2006 (UTC)


 * That is precisely what was meant. There is no publication on tuning any temperament using specific beat rates between coincident partials before the early 20th century, and it was this practical aural method that allowed the widespread use of equal temperament. Every treatise on tuning equal temperament before this involved some sort of undefined approximation. Scheibler's invention was the only exception to this, as it apparently allowed a great deal of accuracy in calculation. (I don't know enough about the device right now to answer your questions, but the writings of Alexander Ellis and others make claims to the effect mentioned in the article.) - Rainwarrior 22:18, 4 July 2006 (UTC)


 * There is no publication on tuning any temperament using specific beat rates between coincident partials before the early 20th century


 * Scheibler and his followers, point out that beats slower than 4 per second are difficult to count accurately. His instructions for tuning directly (at least, as recounted by Vincent in 1849) use a metronome and formulas for calculating a tempo according to the partials and whatever pitch standard you use (I think this is revisited with some modern ones, and of course in the same article inspiring another one includes some passing remarks about calculating tables of beats). Mireut 18:15, 7 July 2006 (UTC)


 * Ahh, thanks Mireut. Yeah, I did misspeak there. Umm... I haven't read any of the writings of Scheibler directly; Ellis in 1885 describes a few ways to lay bearings with various beat rates (most of them methods devised earlier by others), and talks about using the tonometer (apparently they did just use it to do a temperament octave). - Rainwarrior 02:57, 8 July 2006 (UTC)


 * I haven't either, and Fétis, Vincent and Schwiening write that Scheibler's writing was not very good, but I never read any recent reference to these interesting comments, either.


 * So urtheilen alle, welche Scheibler'sche Gabeln besitzen. So schreibt Herr M. Hauptmann unter den 28. Juni aus Cassel: ,,Ich freue mich sehr über den Besitz dieser Gabeln. Ich habe gleich am ersten Tage nach ihrem Empfang mein Instrument danach gestimmt, und es ist jetzt nach 3 Wochen noch reiner und wohlklingender als es je aus den Händen des besten Clavierstimmers gekommen ist etc." - Hr. Pape in Paris zahlt einem Stimmer 3000 Franks jährlich, um die Flügel seines Magazines nach des Scheibler'schen Methode zu stimmen, wobei er nur 6 Gabeln anwenden soll. (Schwiening, "Ueber den Unterschied der bisherigen und der Scheibler'schen Stimm-Methode und über die Wichtigkeit der letzern." Cäcilia: Eine Zeitschrift für die musikalische Welt, Heft 76, B. Schott's Söhnen, Mainz, 1837.) - Mireut 15:19, 8 July 2006 (UTC)


 * Here's another one, in a letter "Tuning and Building Organs", Mechanics' Magazine, v.6, 1827, (p.557) Arnold Merrick gives a list of beat rates (for an unequal temperament), and points out to "K., of Madras" the reason for his difficulty using Smith's numbers was he wasn't at the right pitch. ("Corio", writing to the same magazine in 1836, says equal temperament is the most common tuning but unmusical and says that "too much attention cannot be paid to the beatings, as that is by far the most accurate way of tuning by the ear" but he mistakes that equal beats means tuned the same.) Mireut 16:42, 3 August 2006 (UTC)

Verbose discussion of formulas
This is mainly in regards to Nomenclator's recent edit. I don't think a formula describing how to find frequencies in 12 tone equal temperament belongs in the lead. I removed the long discussion of it from the "12 tone equal temperament" section because I thought it was redundant, and poorly worded. This is certainly not a better place for it. We shouldn't have math in the lead.

Maybe the formula has a place in section where it was before, but it shouldn't take up a full page of writing, or have a title like "simple formula...". It should be more concise, something like:


 * The frequency of any pitch in the 12 tone equal tempered scale may be calculated in the following manner:


 * $$P_n=P_a \times 2^\frac{n-a}{12}$$


 * Where $$P_n$$ is the pitch of note number $$n$$, $$P_a$$ is the pitch of a tuning note (usually 440 Hz), and $$a$$ is the note number of the tuning note (i.e. on the piano A440 is the 49th key, hence $$a = 49$$). (See also Piano key frequencies.)

This could appear somewhere in the "12 tone equal temperament" section (which I believe is already cluttered as it is). - Rainwarrior 04:26, 5 September 2006 (UTC)


 * I have moved it down to the appropriate section, but I still think it is far too lengthy. As written, it seems to be some sort of tutorial in high school math. I don't think a Wikipedia article is really the place to be telling the reader something like: "Don't worry about going into negative numbers... the formula still works out.". I don't think someone who doesn't understand algebra is really going to be interested in this kind of formula anyway, and someone who does understand it now has to climb over your explanation to get to the useful part of it. - Rainwarrior 04:36, 5 September 2006 (UTC)

"Explanation" Section
Am I the only one who doesn't understand this section? Also, I am having trouble figuring out who wrote any particular section. So I don't know who to address my questions about that section, to.

I read "Naively, one may think that the frequency of the notes should grow linearly like the terms in the Harmonic Series."

Huh? The frequency of a justly-intoned 12-tone scale is based on the harmonic series, and such a justly intoned scale sounds more "pleasant" than an equally tempered scale -- simply because the harmonics of 2 notes in a just interval are more nearly identical the the harmonics of 2 notes in an equally tempered interval. And what is meant when it is said something is "aurally" linear, or exponential, etcetera? The terms "linear" and "exponential" only have meaning in terms of mathematics, and are meaningless in terms of how something sounds subjectively -- are they not? It just so happens that we can distinguish intervals that have harmonics that coincide, or nearly coincide, from intervals whose harmonics do not coincide. Just like we can tell if 2 notes are the same pitch, or nearly the same pitch, or not. That is the only thing I can say for sure about how an interval "sounds."


 * I don't know who wrote it, but I've moved it closer to the sections it is pertinent to, renamed it, and revised it. - Rainwarrior 17:29, 5 September 2006 (UTC)

Mathematics at the beginning
Equal temperament is a mathematical concept. It makes sense to start off a topic on the subject of equal temperament -- with a mathematical defintion of what it is. Algebraic formulas are the most consise way to show mathematic relations. You can also write mathematics out in "English." Algebra means less writing, to get the same concept across. Why not start the equal temperament section with an algebraic formula that concisely defines just exactly what equal temperament is, and further, just exactly what 12-tone equal temperament of an octave, is -- since this is generally what is meant by equal temperament and almost universally what is actually used.--Nomenclator 18:27, 5 September 2006 (UTC)


 * Equal temperament is not just a mathematical concept. And no, we shouldn't launch into a mathematical definition of 12 tone equal temperament at the start because this page isn't just about that. The kind of explanation required doesn't belong in the lead. The main question is can a reader find what he is looking for easily? If someone is interested in the math, right now it is clear from the table of contents where to find it. If someone is not interested in the math (and there are many more interesting things to equal temperaments than just their calculation; many people are distinctly not interested in math at all), putting it in the lead means that they have to read through that before they can get to what they want. - Rainwarrior 17:28, 5 September 2006 (UTC)

It is a mathematical concept. If you don't know the basic math, you don't know what equal temparment is. If you know the basic math, you know what it is. If you don't know the math but you know its history, you still don't know what it is. You know about it, but you don't know what it is.--Nomenclator

What does this mean: "the distance between each step of the scale and the next is aurally the same."

In ET, the distance between each step of the scale is the same ratio as the previous step and the next step -- 2^(1/12) for each step in a 12-TET octave. Whether any two steps sound the same is an unknown. For all I know one person may not be able to distinguish the pitch between any 2 notes, and another person may find that the interval between notes 1 and 2 sound like a way bigger hop than the interval between notes 2 and 3. All I know for sure it that they are mathematically the same ratio. I know little about how they sound to other people. Some people can't distinguish between colors as well as other people. It seems reasonable to exptrapolate that aurally, pitches may sound different to different people.

By the way, "the twelfth root of two" is just a long-winded English translation of the mathematical concept, known thruout the world as 2^(1/12) --Nomenclator 18:38, 5 September 2006 (UTC)18:27, 5 September 2006 (UTC)


 * Many, many, many musicians know equal temperament very well without having any grasp of its mathematical derivation. It is far more than just a mathematical idea, and many people are interested in knowing what it is without knowing the mathematics (and there is much to know besides the math).
 * Go to twelfth root of two to understand why I linked it. It is a particularly special number with a special history, and that's why it has its own page.
 * Interval identity is a well studied psychological phenomenon. "aurally the same" is probably a bad approximation of its description. I'll try to revise it. - Rainwarrior 20:39, 5 September 2006 (UTC)


 * Rainwarrior, perhaps we need an article on Interval identity? yoyo 18:23, 17 December 2006 (UTC)

Rainwarrior writes "Many, many, many musicians know equal temperament very well without having any grasp of its mathematical derivation. It is far more than just a mathematical idea, and many people are interested in knowing what it is without knowing the mathematics (and there is much to know besides the math)."

I don't see it. It is a mathematical concept. What else is there to know about it except that, and the story of its development and acceptance? How it sounds? It sounds a bit different than other tunings. There isn't much else you can say about how it sounds, beyond subjective descriptions that are pretty meaningless.

"Go to twelfth root of two to understand why I linked it. It is a particularly special number with a special history, and that's why it has its own page." I don't see why it deserves its own page. The only application it has is to tuning musical instruments. That is all that is mentioned on its page.

"Because the percieved identity of an interval depends on its ratio, this scale in even steps is a geometric sequence of multiplications." That edit hasn't helped much. It has made things even more obsure. The equally tempered scale would be a "geometric sequence of multiplications" (tho i'm sure what that means) no matter what it sounds like. The perceived identity of an interval depends upon the ratio of the two pitches; and the scale may consist of even steps in a geometric sequence of multiplications (tho that sounds like it contains a redundency). But the equally tempered scale is a geometric sequence of multiplications, if that is what it is -- whether you can hear it or not. No matter how it is perceived by the ear.

I'm not sure what the proper terminology is for the kind of series, or sequence, that an ET scale is, but I see that each term in the series is multiplied by the same thing, $$2^\frac{1}{12}$$ (in the language of math), the twelfth root of two, when translated into English.

The scale is a certain kind of series, and the ratio of the pitches determines how they are perceived. But the scale is not a certain kind of series because the ratio of its pitches determines how it is perceived. That just doesn't make any sense. --Nomenclator 00:44, 6 September 2006 (UTC)

Knowing what beatrates to expect, between the coincident partials of intervlas, when you are tuning to equal temperament, is not the same as "knowing equal temperament." Saying that someone who can tune a piano to equal temperament "knows equal temperament" is like saying that someone who can steer a car from place A to place B, "knows automobile steering systems." I would expect an article on automobile steering systems to contain mostly information on how steering systems are constructed, and then a little bit about how to steer. 70.23.31.48 01:28, 6 September 2006 (UTC)


 * Geometric sequence is linked, which I think is enough to explain what it means. I thought it was better to use that term (and so did a previous author) than to say "a sequence of multiplications by the same ratio", or something to that effect. This particular type of sequence has a well known name (and an article), and I think it's good to mention. If the ET scale wasn't geometric, it wouldn't sound ET... I don't know why you would suggest that it has nothing to do with its sound. If we didn't hear equal ratio intervals as equal sized intervals, we would not have created equal temperament. It would have no use. The scale is is useful because of that particular psychoacoustic phenomenon.


 * I don't really follow your "steering systems" analogy (especially as it is not borne out by our articles on steering systems). This article doesn't discuss piano tuning. This article doesn't discuss how to play in equal temperament. What we've got is its history, which can be appreciated independantly of its mathematics if the reader desires (and I would bet many do), its mathematics (which can be appreciated independantly of its history), and a list of different used ETs (which again, can be appreciated independantly of the other sections). I also think it would be good to have a section about the musical possibilities of equal temperaments, though the history section covers that a bit at least. This article is not just for mathematicians, there are many different ways in which someone may wish to study equal temperaments, and we should not favour the mathematical study simply because you think the other aspects of it are unimportant. That is quite biased. The more math you put in the lead, the more the reader who is not interested in it has to dig through to get to what he wants to read about.


 * If you don't think "twelfth root of two" deserves a page, nominate it for deletion, but as long as it has an article, it deserves a link from here in an appropriate place.


 * 70.23.31.48: if you are Nomenclator, you should identify yourself properly. If you are not, you should not be editing his talk posts.


 * - Rainwarrior 18:40, 8 September 2006 (UTC)

mathematics
"In this formula $$P^{}_a$$ refers to the pitch of a tuning note (usually 440Hz), a refers to the number of the tuning pitch, and n refers to the number of the desired pitch."

No, that is not an improvement. It is quite unclear. What is a a "tuning note," is that different from a "tuning pitch"? If not, why a create confusion by using a different word? And how is that, or those, different from a "desired pitch."

What you are talking about is a reference pitch and a pitch being tuned to the reference pitch, the pitch of each, and the consecutive integer used to name or identify each. These may not be the only choice of nomenclature that we may have, but they make things clearer than the names you used. Name things appropriately, or people won't know what you are talking about.

The term "tuning pitch" is ambiguous. We can't tell if you mean the pitch of the string you are tuning, or the pitch of the string, or tuning fork, or whatever, that you are using as a reference, the pitch you are tuning to, or tuning "from" (either preposition means the same thing, in this context). And is the "desired pitch" the pitch you are altering, or the pitch you are using as a reference? It is quite unclear. --Nomenclator 02:09, 6 September 2006 (UTC)


 * "Reference" sounds like a better name. I'm also going to change "pitch" to "frequency", as the latter is unambiguous. As for "desired", what wording would you suggest? I think there is a direct and apparent connection between "To find the frequency, $$P^{}_n$$..." and "...n refers to the number of the desired pitch." How do you propose making it clearer? - Rainwarrior 06:45, 6 September 2006 (UTC)

Instead of "desired" pitch I would say the pitch you wish to calculate. We are not necessarily tuning anything here; we are trying to determine what the pitch should be, according to the formula. That is, if one pitch is a mathematical function of the other (y=function of x), we are given the value of one and trying to calculate the value of the other. I forget what the usual correct math terminology is for the value you are trying to calculate and the value that is the given.

I removed the part, in the "twelve tone equal temperament section," saying each semitone is divided into smaller intervals called cents, because that was just mentioned in the immediately previous section entitled "general properties of equal temperament."

I removed the word "hz" from the formula and put it in the "givens" section - as this is the customary way to write a formula. --Nomenclator 13:55, 7 September 2006 (UTC)


 * Only 12-TET divides its smallest interval into 100 cents. What you have done is very inaccurate. I'm moving it back to where it was. One hundredth of a 12-TET semitone is a very useful definition of the cent, and it is where its name is derived from.


 * "2 to 1 or 1 to 2" is quite cumbersome, and looks confusing, and the ratio of the octave is mentioned in the. We don't need to provide two different notations for twelfth root of two either, the table below it already does that in spades.


 * It is inappropriate to remove Hz from the equation. Hz is a unit, like "m/s" or "mph". The only time where it is sensible to remove units from a number is when you are programming a computer to do it. Middle C is not 262. It is 262Hz, and it is not 262 of anything else. (I you have not seen this before, I suggest you consult a physics textbook that has examples of calculation in it.)


 * Something you are seeking is "desired", since it already says "are trying to find" once, I'm replacing the second one with desired to avoid the awkward redundancy. If something you are trying to find is not desired, I don't understand why you would be trying to find it in the first place.


 * - Rainwarrior 17:17, 7 September 2006 (UTC)


 * You may want to read Units of measurement. The first equation in the section is a formula for generating the number. The second equation is an example calculation. The general formula does not have unitz, because they are unnecessary, but in any actual calculation, the units are used. This is why "Hz" does not appear in the first equation (it would be inappropriate, because someone may wish to use another unit instead), but in the second equation, which is a calculation of a specific frequency, the number is meaningless without a unit qualification. Hz may be the standard unit for frequency, but that doesn't mean it should be discarded from the notation. - Rainwarrior 17:25, 7 September 2006 (UTC)

Rainwarrior writes "One hundredth of a 12-TET semitone is a very useful definition of the cent, and it is where its name is derived from."

But that is what I wrote, and what you removed. What is now there is "Scales are often measured in cents, which divide the octave into 1200 equal intervals (each called a cent)" and nothing is said about dividing the semitone into 100 equal parts!--Nomenclator 02:16, 8 September 2006 (UTC)


 * I put it back where it was under "Twelve tone equal temperament", because the 12-TET semitone is the only semitone that may be divided into 100 cents. - Rainwarrior 18:04, 8 September 2006 (UTC)

Rainwarrior: "...the 12-TET semitone is the only semitone that may be divided into 100 cents."

That doesn't make sense to me. Anything can be divided into 100 cents. "cents" is just another word for "one hundreth," 1/100, 1 part out of 100 equal parts. That is why we have dollars and cents, by the way. Anything can be divided (mathematically) into 100 equal parts.--Nomenclator 14:57, 14 September 2006 (UTC)


 * In the context of tuning there is only one definition of "cent". If you believe otherwise, you should probably be making changes to Cent (music), but before you do, do you have any published source that uses the term "cent" for tuning that is not 1/100 of an ET semitone? - Rainwarrior 16:19, 14 September 2006 (UTC)


 * (What I mean is that "cent", when it is applied to music, refers always to Alexander Ellis' cent. That definition is the one used for tuning theory, and no other.) - Rainwarrior 17:03, 14 September 2006 (UTC)

Australian aboriginal "temperament"
Is this distinct from a Just intonation that includes sucessive pitches from the harmonic series? I know that Didgeridoos are more or less harmonic-series bound, but I don't know much about Australian Aboriginal music. If it is Just, then it is not a musical temperament, and belongs in the Just Intonation article instead.

If however, somehow they have produced a scale that is in equal frequency differences without being aligned to a harmonic fundamental (this would sound very interesting), it might be considered a temperament, but it seems an unlikely construction to me. (I can think of very few physical situtaions where these kinds of harmonic relationships occur, and I wouldn't expect people to construct musical instruments this way...)

What I'm suggesting is that it be removed from here and moved to Just Intonation, or if not there, start a stub because it's different enough to deserve an article. - Rainwarrior 00:08, 10 September 2006 (UTC)


 * With no response in over a month, I have removed this bit of trivia from the article, as it seems to be entirely unrelated to equal temperament. - Rainwarrior 08:50, 20 October 2006 (UTC)

Wendy Carlos' scales
''Wendy Carlos created two equal tempered scales for the title track of her album Beauty In The Beast, the Alpha and Beta scales. Beta splits a perfect fourth into two equal parts, which creates a scale where each step is almost 64 cents. Alpha does the same to a minor third to create a scale of 78 cent steps.''

This is missing some information, or is inaccurate. The perfect fourth is 498 cents. One half of that is not 64 cents, and even if divided into 8 parts, it's still about 62 cents... where does 64 come from? The minor third of 316 cents might divide into 4 parts to become about 78. I'm not sure that the "64" and "78" figures are necessarily inaccurate, but some part of this description is. I'm not sure which part. (I think I have the album in a box somewhere... I'll check on it later.) - Rainwarrior 00:22, 10 September 2006 (UTC)


 * I have amended this according to information available on her website. The statements about which intervals these divde which are present in the liner notes for the BitB CD are approximate. These temperaments to not give rise to any rational intervals: they simply come close to them. There is also a third scale, called gammma, which I have defined here as well. - Rainwarrior 09:52, 10 September 2006 (UTC)

Merge from EDO

 * Support - I noticed a merge-to up at Equal division of the octave. Does that article actually say anything that's not mentioned here in the lead? I don't think so, really. Why don't we just change it into a redirect? (And why are there like a billion references for an article that says so little?) - Rainwarrior 15:42, 27 September 2006 (UTC)


 * Support per Rainwarrior. —Keenan Pepper 03:31, 30 September 2006 (UTC)


 * Oppose but I think there's a lot from its discussion page that would improve the article. - Mireut 13:12, 30 September 2006 (UTC)


 * Really? I looked, and almost all of it is a discussion of whether the article should exist or not. The only information I could see in there that isn't in the article is maybe the passing reffernce to Ivor Darreg? - Rainwarrior 17:29, 30 September 2006 (UTC)


 * The disagreement, too. Someone writing in a forum referred a request for layman's term explanation about what piano tuners do to here. I think if ET was described with a little reference to fifths and thirds it would answer better, as well as contrast better with EDO, and that more external articles might help make this one easier to improve. - Mireut 15:08, 21 December 2006 (UTC)


 * That is interesting that this page completely leaves out the temperamental properties of Equal Temperament. That should be included somehow; right now I'm not sure where exactly. There is now an article on musical temperament (there wasn't before), and I made some rather large changes to piano tuning a few months ago. I think splitting 12-TET from this article might well be worthwhile, clarifying at all points what scale is being discussed and making it easier to add relevant information. It might also make it easier to merge the EDO topic into the more general Equal Temperament article without the 12-TET stuff to get in the way. - Rainwarrior 20:05, 21 December 2006 (UTC)


 * Oppose - Mireut's definition is apt, and someone with a better knowledge of the term's history than I could document it there also. yoyo 18:12, 17 December 2006 (UTC)


 * I've performed the merge, since the tiny EDO article contained absolutely no information that was not present in the equal temperament article, and after several months there has been no effort whatsoever to expand it. - Rainwarrior 05:53, 3 April 2007 (UTC)

Cents
A lot of the language about cents, is not making much sense to me. I also found the chart showing how an equally tempered octave was divided into cents, to be pointless; anyone should able to understand how an equally tempered scale is divided into cents, without this kind of chart. Furthermore, while the concept of cents is frequently brought up, the concept of cents is never defined. Is the definition of a "cent" 1/1200 of an octave (2/1), 1/1200 of some other interval, or is it 1/100 of an equally tempered semitone? All 3 relations are mentioned, but which one is the defining relationship? The section on General properties of equal temperament, starting at "Scales are often measured in cents" and continuing to the end of the section, makes little sense to me. Why, or how, does dividing a scale into cents making comparison of different tuning systems easy? Why is the author bringing up the subject, if he is not going to explain how, or at least provide a link to an explanation?


 * The link to cent(music) makes it very clear. The table showing 100 cents between semitones is redundant with the other example in the article and is superfluous.     mbbradfor d  14:40, 2 July 2007 (UTC)

The assertion that integer notation simplifies and generalizes discussion of pitch material within the temperament in the same way that taking the logarithm of a multiplication reduces it to addition, begs for an example. It seems like an awfully high-falutin way to say, simply, that we often refer to any pitch within an equally tempered octave, as pitch number 1, 2, 3, 4... 12, along with the octave number, instead of calling each pitch by its frequency.

It seems like much of the talk about cents is an attempt to make something that is very simple, seem more complicated than it really is.

Does anyone ever divide an interval other than a 2/1 interval, an octave, which they have in turn divided into 12 equal parts - into 1200 equal parts? As far as I know, it is not customary to do so, no-one ever does it. What people do, to define a cent, is divide things, whatever they want, into 100 equal parts, not 1200 equal parts. As far as I know, the way the term cents is customarily used, no-one would ever divide anything into 1200 equal parts. They would multipy 100 cents, by 12, to get 1200. Yet the article says you can divide any ratio into 1200 equal parts and those parts would be cents. That is simply not true, the way the term cents is customarily used. Only a 2 to 1 octave, which in turn you have decided to divide into 12 equal parts, would be something that would be customarily divided into 1200 equal parts. A 2.3 to 1 interval, which you divided into 7 equal parts -- to find cents, you would divide that interval by 700, not 1200, as the article states. And the formula $$c = p^{}_{cents} / n$$ just is not making any sense to me either. What are "P cents." It is not stated.

What does this have to do with ethnomusicology? I think this stuff is obfuscating the subject, rather than elucidating it.

--Nomenclator 13:25, 14 October 2006 (UTC)


 * Cents are the standard way of talking about pitch. You will find them used freely in any academic writing about tuning, especially writings of ethnomusicology (which very often make comparisons between different tunings, which is best done with a logarithmic system). They only cases where cents do not appear are in tuning articles that are exclusively about just intonation and no comparison to an irrational (in the numerical sense) temperament is being made. Furthermore, there is no other logarithmic scale that is standard for tuning. There is cents (logarithmic), and there is rational notation (just intonation, which is multiplicative).


 * Cents are not defined by this article. Cents are defined clearly at Cent (music), which is linked directly from the article at the first usage of the term. The simplest mathematical definition of a cent is 1/1200th of an octave (in the logarithmic sense), but the history of the term is tied directly to equal temperament, as it is 1/100th of an equal tempered semitone (which is 1/12 of an octave). (For a person doing work with a calculator, it is usually easier to use a division by 1200 rather than first calculating and storing the 12th root of two and then subsequently dividing by 100.)


 * The 1/100 of an ET semitone definition is discussed under the "Twelve tone equal temperament" section, as it only applies to this particular equal temperament. In the general section that directly precedes it, it would not be proper to define a cent in terms of an interval that is not yet defined itself. The cent's simplest definition without presupposing 12-TET is 1/1200th of an octave.


 * As for other intervals, the only rational interval that cents divide in integers is the octave (and its compound forms). Equal divisions of other intervals are uncommon (and have nothing whatsoever to do with cents); there's a section on the rare scales that do this in the article. (The Bohlen-Pierce scale is the most well known example of such a scale.)


 * One of the most important properties of the logarithm is that it converts multiplication into addition. This simplifies many types of comparisons, and is very useful for the purposes of music theory (and other things). Integer pitch notation is a logarithmic scale that is used to describe equal temperaments in a very simple way.


 * - Rainwarrior 07:55, 15 October 2006 (UTC)


 * There's formulas with P as well as p ("period"?) - how about F for frequency instead of big P for pitch? - Mireut 22:30, 15 October 2006 (UTC)


 * In this case it's "p" for "interval"? I was using the variable names that were already on the page. Maybe "i" would be better? I dunno. P seemed as good as any to me at the time. - Rainwarrior 02:54, 16 October 2006 (UTC)


 * I've made it "w" for now. It's all arbitrary names anyway. I wish there was some standard to use. I only have "p" where it is because that was in use on the page before. - Rainwarrior 08:49, 20 October 2006 (UTC)

Article being Degraded
I'm sorry, but this whole article is becoming increasingly obtuse as the weeks go by. Anyone who looks to it for a definition of equal temperament, or to understand equal temperament, is going to be confused, rather than enlightened. Equal temperament is a mathematical concept that is applied to musical instrument manufacturing and maintence of one kind of musical instrument -- instruments with a set of fixed pitches. It doesn't even apply to music in general.

We have an unwieldy sentence in the very first paragraph. "This system is usually tuned relative to a standard pitch of 440Hz." This is not fluent English. And since A440 is part of the system of tuned pitches, this sentence is not even precisely correct.

The article needs a complete overhaul. It is floating high above the clouds in rarefied atmoshphere. For people to learn anything about equal temperment, from the article, both what it is, and the history behind its development, the article need to be brought back to earth. --Nomenclator 23:42, 19 October 2006 (UTC)


 * With regard your second paragraph here, I do not understand anything of what you are saying about that sentence. To me it looks completely fluent, and very correct. Perhaps someone else could offer their opinion on it. As for the other comments, I don't really have a response (I guess I don't really see what you see there, and you haven't been very specific). - Rainwarrior 08:48, 20 October 2006 (UTC)


 * I beg to differ with Nomenclator. I believe the article has improved out of sight in the last year.  I also think that minor criticisms do not deserve hundreds of words to express in fluent English on the Talk page.  The sentence in question is in fluent, everyday English.  This article would only be "floating high above the clouds in rarefied atmoshphere" if it were vague and airy, lacking substance and specifics - like tha criticism. yoyo 18:17, 17 December 2006 (UTC)

Terminology Clarification
I need a terminology clarification. If you have 2 pitches an octave apart, is the higher pitch the first partial or the second partial. And is the higher pitch the first harmonic or the second harmonic? According to my understanding, one of these terms, partial, and harmonic, traditionally idenfifies the unison with the label 0, and the other term gives the unison the label 1. So with one of these things, the octave is the first thing, and with the other, it is the second thing. But I can't remember which is which. And also, which designation system is traditionally used with the term "overtone."? --Nomenclator 03:53, 20 October 2006 (UTC)


 * The first harmonic is the fundamental frequency. A resonating body can be made to resonate simultaneously at several harmonics, the first one again having the lowest frequency.


 * Partial also works the same way. A tone is made up of several "partial" tones, the first one being the one with the lowest frequency, the fundamental. (If you can find Helmholtz's "On the Sensations of Tone", you'll find it spelled out there on page 22.)


 * Overtone, on the other hand, begins counting at the first tone "over" the fundamental. This term is generally not as useful as "harmonic" or "partial" (since the order of the overtone does not directly correspond to an appropriate integer multiplication), but can be very good when merely pointing out the existance of tones in addition to the fundamental. It is not usually used in a quantitative sense; one often refers to "an overtone" or "overtones", but rarely "the third overtone". There is no "zero" overtone, the fundamental is not an overtone at all.


 * There is some confusion about "partial" and "overtone" as well, and I've seen both misapplied more than a few times. (Alexander Ellis even cautions against any use of the word "overtone" in his footnotes to the Helmholtz book mentioned above, page 25.) - Rainwarrior 08:33, 20 October 2006 (UTC)

Thank you for the clarification Rainwarrior. I have Dover Publication's printout of Alexander Ellis' translation of Helmholtz' On the Sensations of Tone, which I acquired abut 30 years ago, when I first began learning how to tune a piano, but for some reason it didn't occur to me to look in it. And now I am not sure how helpful OST is for making the terminology clear. Page 22, the first page that the index refers to, under the entry partial, makes a distinction between partials and upper partials." On this page, I find the sentence "The first upper partial tone [or second partial tone] is the upper octave of the prime tone, and makes double the number of vibrations in the same time." Here a distinction is made in passing (and the distinction is enclosed in brackets, not parenthesis, which I think indicates it is a comment added by Ellis), between partials and upper partials but nothing is said about the term harmonics. I returned to the index and began looking through the subentries under under the index entries of partial and harmonics, and began going to some of the dozens of pages referenced by the index entries, but I was not soon presented with an answer to my question, and I now wonder how long it would have taken me to find a definitive answer in OST, or if I might have ever found one there. I remain thankful for the internet, and for the knowledgable people who populate it. -- --Nomenclator 15:52, 20 October 2006 (UTC)


 * Ellis I think avoids "harmonic" as well, but it is quite commonly used in English sources today (like physics textbooks). Yeah, I forgot to mention "upper partial", which is think is somewhat obscure. I can remember cases where "upper partial tones" are referred to (like I was suggesting with overtone), but not where a specific one was referred to ("second upper partial"), like with overtone. - Rainwarrior 19:39, 20 October 2006 (UTC)

Peculiar Notion
Re this sentence "Because the perceived identity of an interval depends on its ratio, this scale in even steps is a geometric sequence of multiplications."

No! The perceived identity of an interval may perhaps depend on its ratio, and an equally tempered scale, in even steps, may perhaps be a geometric sequence of multiplications, but the latter is not a consequence of the former! --Nomenclator 21:17, 25 October 2006 (UTC)


 * You've removed the sentence that begins that paragraph. The full quote is:
 * In an equal temperament, the distance between each step of the scale is the same interval. Because the perceived identity of an interval depends on its ratio, this scale in even steps is a geometric sequence of multiplications.
 * The goal of equal temperament is to compose a scale from equal intervals. Because the definition of equal intervals is that they must have equal ratios, the series of steps in equal intervals must be a series of multiplications by the same ratio. Thus the fact that it is a geometric series is a consequence of the goal of equal intervals. - Rainwarrior 23:13, 25 October 2006 (UTC)

Your explanation seems like a convoluted explantion for a simple concept, that an equally tempered scale is a geometric series. Composing a scale from equal intervals is not the goal of equal temperament, that is the definition of equal temperament. The historical goal or purpose for beginning the use of equal temperament and for why it became popular, was to find a temperament that approximates commonly used just intervals, and at the same time diverges by the same amount, from the just intervals that it approximates, and thus sounds the same, in all key signatures.

The et scale is not geometric sequence of multiplications because the preceived identity of an interval depends on its ratio; it is a geometric sequence of multiplications because -- well because that is what it is: if you multiply a by b, then multlipy the product by b, the multiply the product of that by b, you have a geometric series. It has nothing to do with anything that anyone perceives about any interval. It has to do with the mathematical relationship between the intervals. Your explanation makes sense, but I don't see how it explains the sentence that sounds erroneous to me, and it seems like an awfully long-winded way of saying that equal temperament is a geometric series. There is no "goal" of equal intervals. It just turns out, by happy accident, that certain equal intervals, that is, intervals that are elements in a specific geometric series, one where, to find each consecutive element of the series, you multiply the previous element by the same amount, in this case the 12th root of two, approximates all the just intervals that were in common use at the place and time that equal temperament began being popular. --Nomenclator 14:03, 4 November 2006 (UTC)


 * The use of the word "goal" was merely an approximation I hoped would help the explanation. It's not currently used in the article. To explain this a different way: if you construct a scale with anything but a geometric sequence (e.g. an arithmetic sequence) the steps would not be perceived to be the same. If the perception of intervals was arithmetic instead of geometric, our equal tempered scales would be arithmetic sequences instead. - Rainwarrior 05:46, 6 November 2006 (UTC)

Tuning for music students -- content dismissed as redundant?
Everywhere in Wikipedia, we seem to have the same disagreement. Should the article be written for a novice, who may be coming to Wikipedia to learn about music theory because they do not know where else to get credible academic background, or should we write the article for someone who already is expert in the material and wishes to see an examination in a thoroughly academic manner?

Western music students are now learning (or struggling to learn) to tune their instrument with an electronic tuner, which knows only equal temperament. Then we expect them to train their ear to just intervals by learning the table of "cents" to adjust the pitch. Instruments are built and tuned to equal temperament to the extent possible, so that French oboe will match an American oboe for example, and that both will match a piano from Japan. The students need to train their ear using a meter and to learn their pitch adjustments as a function of the key in which they are playing. But meters typically show cents, not percentages.

I found the existing article lacking in three respects. First there was no simple table of adjustments from equal temperament to the just intervals. I looked at several articles to consider where to add this information, and settled upon this one because it already has a comparison between equal temperament and just tuning in the 12-tone section. While the existing table shows the difference, it uses percentages rather than cents and then has them in the backward direction (it shows how to adjust a just interval to an equal temperament, which is much less useful). I suppose this is trivial to a professor of music theory, but will certainly be very confusing to a freshman music performance major who is struggling to grasp the difference. Second, students actually need to learn how to adjust the equal tempered pitch on their instrument and meter to the just pitch as a function of their key signature. In ensemble playing, the major third is corrected by holding the tonic steady to the equal tempered pitch and then lowing the pitch of the third to the just interval. The comparison section does not mention this, leaving a gap for the reader to comprehend and fill in on his own. Third, no explanation as to how the differences are derived or calculated, using actual frequencies for example as is needed in the formula for cents (music). We do not expect a student to compare the formula for equal temperament and a formula for just intervals and figure out the pitch change on their own.

I attempted to supplement the article with what I believe is a needed improvement, but it has been completely reverted and dismissed as redundant with the existing table. I considered editing the last column of the comparison table, but I think this leaves out the background for a student to see how the adjustment is calculated. Perhaps a separate article should be considered, but generally this means adding redundant content and is to be avoided. Instead I appeal to your sense of ownership of the articles content and hope you will allow it to be expanded. mbbradfor d 07:50, 25 February 2007 (UTC)


 * Sorry this reply is late; I didn't notice your comment here when you made it. I removed the table you added because it essentially duplicated the existing material (in "comparison to just intonation") in terms of content for the article. If you think a measurement in cents is necessary, feel free to change the "percent difference" column to a "cents difference" column, as cents are probably more appropriate. By the way, the formula in Cents (music) covers turning a ratio a/b to cents, just plug in the values for the just and equal for a and b to get that, e.g. for the minor third use a/b = 1.189207/1.200000.


 * However, your attempt to describe how to produce just intonation by using an electronic tuner to evaluate the differences in cents I think is for one, not relevant to the article (this is an article about equal temperament, not about how to modify ET to produce just intonation, and if it fits anywhere it would be at Just intonation, though I don't think it belongs there either). For another, cents aren't really the most useful way to describe how to do this. Just intonation can be and should be tuned by ear (which for just intonation is an extremely accurate measuring tool). The vast majority of electronic tuners are not accurate enough for the purpose you are describing, and only some brands even bother to give cents (most of the ones I've seen just give an arbitrary +/- scale). - Rainwarrior 02:54, 17 June 2007 (UTC)


 * Wikipedia is not a how to, and not the appropriate place to teach students how to tune. Hyacinth 03:11, 17 June 2007 (UTC)


 * I'm disappointed that after having your "permission" to change the percentages to cents and doing so, you have then gone back and replaced it (cents between equal tempered tones and its just tuned tone relative to the tonic) with cents relative to the tonic tone. You have replaced a fundamentally inportant APPLIED tuning lesson with academic theory.  Who really cares if the major third is 386.31 cents above the tonic?  What matters is that the minor third must be played 15 cents sharp to sound in tune in an ensemble, whereas the major third must be played 14 cents flat.  Yes, students learn just intonation "by ear" but only a fool would try to learn pitch adjustments randomly.  Music students (winds, strings, voice) are taught to learn pitch adjustments as a function of the key in which they are playing (i.e. the tonic) and the note they are playing (relative to the tonic).  They learn to train their ear, starting with a meter that shows cents and the table of adjustments between the equal tempered tone on the meter and the pitch they really want.  Why be so academic with a mathematically derived table of cents based on the tonic?     mbbradfor d  22:40, 3 July 2007 (UTC)


 * When the ET cents figure says 400 and the JI cents figure says 386, does this not clearly show that the JI case is flatter than the ET case? I think the table shows very well where the JI figures lie in relationship to the ET ones, and it merges in the material you had removed which I thought was useful (though poorly placed). When looking at what was removed and what was in the table, I figured if I put a list of 100 200 300 the cents of ET, the clearest thing to compare to it would be 111 203 315, not +11 +3 +15. These figures not only give you the relative comparison to ET, but also makes their actual sizes relative to eachother apparent as well. Also, I still think you overestimate the availability of accurate electronic tuners that read in cents. The "applied" lesson you are trying to give will not be useful to anyone who does not have one of these special instruments (and to rewrite it in such a way that it is teaching this lesson would be both off topic and unencyclopedic). - Rainwarrior 03:08, 4 July 2007 (UTC)


 * I give up. I guess when you were a freshman at university, you had a professor in the concert band tell the trombone players to play their major thirds 386.31 cents above the tonic, instead of 14 cents flat.  Don't forget to change the text in the article to match the table, as it is still the other way.      mbbradfor d  09:02, 4 July 2007 (UTC)


 * Well, I've never heard any bandleader tell any musician to adjust their tone by some number of cents or some percentage. Very often they may say "if you're playing the third in a major chord, play it just a little flat", which is perfectly sensible. "14 cents" isn't really a useful number in this situation. Few musicians really know what 15 cents means, and even if they happened to have some sort of electronic tuner with a cents measurement, band tuning is never that accurate; the adjustment must still be made aurally. Now, is 386 not obviously a little flat from 400? There is more information in this number than -14. (Thanks for the heads up about the text outside the table.) - Rainwarrior 18:30, 4 July 2007 (UTC)

2^(7/12) != 3/2
Might I suggest that the most important distinction between 12-TET and Pythagorean tuning is the 5ths. Notably: If you take 3 strings of length ratio 3:2:1, (by looking at standing waves/natural harmonic series), you get these notes: C, C+octave, C+octave and a 5th. Thus a factor of 1.5 in pitch is *exactly* a 5th. BUT, in E.T., a 5th is defined as 7/12 of an octave == 2^ (7/12) ~ 1.49. This is the key "problem" with equal temperament. To Renaissance ears, 12-TET would sound as though the 5ths were very flat; modern listners are used to hearing 5ths that aren't quite "right" - one conductor calls this "an equal-tampered" scale. I find this explanation to be the simplest; hope it's useful.--RichardNeill 02:20, 17 June 2007 (UTC)


 * In the Renaissance, the much flatter fifth of Meantone temperament was more common. The difference between an ET and pythagorean fifth is very slight. Your suggestion of 1.49 inaccurately exaggerates the difference, as the calculation gives 1.4983..., so at 3 significant digits it would still be 1.50 if you rounded. When given a thorough comparison the difference is audible to someone who knows what to listen for, but on any note that lasts less than half a second it would be very difficult to perceive. This difference is also well below the accuracy threshold for human singers or players of free-pitched instruments (violin, trombone, etc.). I think the most apparent differences between Pythagorean and ET are the wolf fifths in the Pythagorean, and if they are avoided the next most noticeable thing is probably the slightly more unjust thirds. - Rainwarrior 02:38, 17 June 2007 (UTC)

55 references

 * 55-TET, not as close as 53 to just intonation, was a bit closer to common practice. As an excellent representative of the variety of meantone temperament popular in the 18th century, 55-TET it was considered ideal by Georg Philipp Telemann and other prominent musicians[citation needed]. Wolfgang Amadeus Mozart's surviving theory lessons conform closely to such a model[citation needed]. Based on orchestral recordings, it is evident that this intonation survived as a standard practice until about 1930.

The final sentence would benefit most from a reference, I don't know if Duffin provides a non-anecdotal one (How Equal Temperament Ruined Harmony 2006), and these don't provide much further information about how widespread it was, but in articles published starting 1701 Sauveur described 55-equal temperament was what common musicans used (in Histoire de l'Académie royale des sciences avec les mémoires de mathématique et de physique tirés des registres de cette Académie, you can download from gallica; from Temperament, Chambers Encyclopedia 1753: what "Mr. Sauveur calls the Temperature of practical musicians", it was one of several counterexamples to 43-equal temperament, including 12), and Telemann presented a system with 4- and 5-comma semitones as new in 1742-3 and 1767 (Neuen Musikalischen Systems: "we should be cautious to consider Telemann's system as a rigid prescription of a 55-tone equal tempered basic scale") and apparently Mozart described a similar system (J. H. Chesnut Mozart's teaching of intonation 1977: "There is no real evidence that Mozart intended 1/6- comma meantone rather than 1/5-comma or some other meantone system." quoted at http://sonic-arts.org/monzo/55edo/55edo.htm 2001 although Monzo finds evidence supporting a definite temperament he cautions it is "for instruments other than keyboards" and unequal pedagogical "commas" were used as late as the early 19th century: "lower down, or flatten C by the smallest possible gradations, until it becomes unison with B ; with a tolerably steady hand, and a few trials, you will be enabled to enumerate forty gradations of sound, which I call commas." J. S. Broadwood, advocating 12-equal temperament, quoted in Broadwood's Tuning Edinburgh Enc. 1830). - Mireut 18:30, 30 June 2007 (UTC)