Talk:31 equal temperament

Add a sentence or two

If someone would add a sentence or two answering the following, I'd be grateful:

• Will this help me tune my guitar?
• Will this help a professional piano tuner do his/her ordinary work?
• Is this important to the understanding of psychoaccoustics?
• Is this primarily of academic or theoretical interest?
• Is this mostly of interest to listeners and composers of unconventional music?

Thanks, Barry Nostradamus Sher ( —The preceding unsigned comment was added by 151.202.47.131 (talk • contribs) 07:03, 2 January 2006 (UTC) )

I'll answer them here and you can incorporate them into the article as you see fit:

• It will not help you tune your guitar unless you rip off all the frets and put many more on.
• It will not help a piano tuner unless he/she is tuning a piano with 31 keys to the octave (which is possible but I've never seen one).
• It is not particularly important to psychoacoustics in general because it is one specific tuning system.
• For some people it is only of theoretical interest, but others actually build instruments to play it and use it as a practical system.
• Unconventional is a slightly POV word, but yes, I think music in 31-EDO would be considered "unconventional" by most standards. =P —Keenan Pepper 07:55, 2 January 2006 (UTC)

• This may be important to psychoacoustics: at least 31 EDO is easy way to listen sound properties almost just 3-rd 5-th and 7-th in real music context.
• This is of academic or theoretical interest: 31 EDO allow possibility to play 12 EDO scores of 5-limit tonal music without additional accidentals. If in the sourse 12 EDO score are notes with confused accidentals, we'll hear them as out of tuning during playing in 31 EDO. To correct this error it's enough to respell these notes with right accidentals. So 31 EDO is very important for correction 12 EDO scores.(Commator 11:55, 6 April 2007 (UTC))

It's not always enough simply to respell a 12-EDO score. It can work most of the time in well behaved music, but there are situations where things are ambiguous or unresolvable, especially where chromatic or enharmonic modulations are being used. In cases where different methods of modulation are used to go to and from the same keys, resolving this results in a choice between a sour modulation and ending the piece on a slightly different pitch (e.g. D double sharp instead of E). - Rainwarrior

Would you point me to such complex score for I could test my '31 tet Playback' plugin to Sibelius 4? (Commator 17:41, 10 April 2007 (UTC))

Sure. A while ago I was looking at John Bull's "Ut Re Mi Fantasia" precisely to see how 31-TET might apply (and came to the conclusion that it does not). It can be found in the Fitzwilliam Virginal Book. - Rainwarrior 05:32, 13 November 2007 (UTC)

At first time has read your so named conclusion 07 March 2009 only! You need Sibelius Scorch to consider my objection, which today is completely ready. Also MP3 and MIDI are available. But all explanations are placed at the score title page. Sibelius Scorch needful to consider them. By my opinion this masterpiece of Dr. John Bull sound in 31EDO much better than in rough 12-TET. Commator (talk) 19:57, 12 March 2009 (UTC)

Good 31EDO sound not by my opinion only. Today the piece already has reached 4-th place among 1938 in G7Music's top ten! Commator (talk) 12:49, 16 March 2009 (UTC)

Links are dead - please repost? I'd be interested to hear this rendition. 69.7.77.20 (talk) 15:23, 8 March 2012 (UTC)

External links to music

Please see Wikipedia talk:WikiProject Tunings, Temperaments, and Scales#External links to music. —Keenan Pepper 19:46, 3 September 2006 (UTC)

The following examples need to be replaced with links to HTML files describing them, or migrated to the wikipedia Commons. - Rainwarrior 04:51, 4 September 2006 (UTC)

Original to 31-et

• The Liberation of Gabrovo, Harold Fortuin, excerpt, mp3 file
• The beat-generation march, Joel Mandelbaum, midi file
• Clash by Night: Prelude, by Aaron Krister Johnson, ogg file
• Clash by Night: Waiting for Jerry, by Aaron Krister Johnson, ogg file
• Clash by Night: Shuffle, by Aaron Krister Johnson, ogg file

31 equal arrangements

• Impromptu in C# minor, op 28 no 3, Hugo Reinhold, midi file
• String quartet no 3, op 67, first movement, Brahms, midi file
• String quartet no 3, op 67, second movement, Brahms, midi file
• Prelude to Tristan und Isolde, Wagner, mp3 file

Clarity

What the hell does this mean? :

The single most important fact about 31-et is that it equates to the unison, or tempers out, the syntonic comma of 81/80. It is therefore a meantone temperament. It also tempers the 5-limit intervals 393216/390625, known as the Würschmidt comma after music theorist José Würschmidt, and 2109375/2097152, known as the semicomma.

More significantly, perhaps, it tempers out 126/125, the septimal semicomma or starling comma. Because it tempers out both 81/80 and 126/125, it supports septimal meantone temperament. It also tempers out 1029/1024, the gamelan residue, and 1728/1715, the Orwell comma. Consequently it supports a wide variety of linear temperaments.

—The preceding unsigned comment was added by 71.102.186.234 (talk • contribs) 08:05, 22 November 2006 (UTC)

I agree that it's pretty confusing. I don't even know what the Würschmidt comma or semicomma is, or what significance they have, or the gamelan residue and Orwell comma. However, I will attempt to explain, and we should probably also amend the article:

Some people like to classify temperaments based on the commas they "temper out"; which means that using this kind of tuning, a particular disagreement between two intervals that would be constructed with just intonation, called a comma, is eliminated by modifying each of those intervals slightly so that they line up. Interestingly, temperaments that have "tempered out" similar commas often have an easier time of converting music from one to the other, which is why it was written that this temperament might "support" other temperaments.

Now, the article doesn't do anything to explain this. Musical temperament explains it only a little. This kind of information is rather obscure (of the handfull of people that actuall play with temperaments, how many of them are interested in comma-compatibility? and how many of them would come to wikipedia for that information?). I'm not sure if it really belongs in the article. - Rainwarrior 17:04, 22 November 2006 (UTC)

I think the article as written is outright wrong. This "important fact" is an esoteric, ultratheoretical notion that does not influence anyone's perception of the music. Large ratios are important if you are generating scales from stacking intervals but they're not important if you're evaluating the validity of an equally-tempered system. The most important fact about this system is undeniably that it is one of the few equally tempered systems that offers an improved match to intervals arising in the harmonic series, relative to systems with a smaller number of divisions in the octave. This is explored in great depth in the Ph.D. thesis of Joel Mandelbaum. The key part of this is paraphrased in this article: [1]. Cazort 01:57, 6 November 2007 (UTC)

no "tone"

Why are equal temperaments called "31 equal temperament" and "53 equal temperament" on Wikipedia, without the word "tone"? CKL —Preceding signed but undated comment was added at 15:58, 25 September 2007 (UTC)

There are a lot of ways to name the scale. "31 tone equal temperament" is also valid. Some like "31 equal divisions of the octave". It's just a matter of preference which one to use. - Rainwarrior 05:21, 13 November 2007 (UTC)

The term "tone" is commonly reserved for intervals close to the 9:8 or 10:9 diatonic whole tones, whereas the steps of other equal divisions may or may not be of a comparable size. 69.7.77.20 (talk) 16:05, 8 March 2012 (UTC)

Theoretical Section

This article seems to be talking about completely esoteric theoretical stuff that has little or no bearing on how people actually perceive music. For example, ratios like 393216/390625 or 2109375/2097152...the numbers are so ridiculously large that I don't see how they are relevant to anything. I think that this article needs to justify why this stuff is interesting or relevant, and in the absence of doing so I think this stuff should be deleted. And...the way the syntonic comma is mentioned doesn't do it justice...it is not important because of its ratio (again the numbers are too high to be relevant) but rather, it is relevant because of the fact that it arises when combining intervals to build scales and tuning systems. The same is true for the other two ratios mentioned. Cazort 19:27, 5 November 2007 (UTC)

Well, there are probably consequences to tempering out 393216/390625 that aren't "esoteric", but I don't know what they are and certainly I agree that the article as it is basically does nothing to explain the significance of such an ugly number. We might as well move these statements to the talk page until someone who can justify their inclusion comes along. - Rainwarrior 05:14, 13 November 2007 (UTC)

Tempering out 393216/390625 means that a stack of eight major thirds precisely equals a perfect fifth, and tempering out 2190375/2097152 (the semicomma, or Fokker's comma) makes three augmented seconds precisely equal a minor sixth.

P.S. The syntonic comma is just 81/80 – not terribly large numbers, and smaller than what Pythagorean tuning will give you for most intervals. Double sharp (talk) 03:47, 4 June 2015 (UTC)

Which intervals to include in table?

There is unfortunately room for subjectivity in determining which intervals to include in the table on this page, as well as other pages such as 19 equal temperament, 22 equal temperament, 53 equal temperament, etc. While I admit this, I think it's hard to justify including ratios for large numbers like the just intonation for seconds. Joel Mandelbaum, who has worked with this tuning system more than most composers, once commented that a close match to just tuning was very important when it came to consonant intervals, but was of little importance in dissonant intervals. He uses this system mainly because it allows one to include the 7th and 11th partials of the harmonic series. I have never read (or heard) anything to suggest that the higher ratios play any meaningful role in the perception and use of these tuning systems. Cazort 20:15, 12 November 2007 (UTC)

scale diagram misalignment?

In the table, in the section "scale diagram", the columns are misaligned. Is each cell in the top row supposed to center across the corresponding cells in the middle and bottom rows? (In which case, what does it mean?) Or, is it just an accidental misalignment? 81.187.40.226 (talk) 19:52, 18 November 2007 (UTC)

It is the interval between two notes in the row below. - Rainwarrior (talk) 23:26, 18 November 2007 (UTC)

The article states after the scale diagram "The remaining 10 notes can be added with, for example, five 'double flat' notes and five 'double sharp' notes" but it doesn't say where. As an equal-tempered scale, obviously they go in the middle of the double-size intervals, but which go where? How do you know which should be double-flats and which double-sharps? Are they considered enharmonic equivalents in this scale? The passage implies that they are considered distinct, but I may be misunderstanding it—that "for example" makes it hard to tell. Heck, just replacing the scale diagram with a complete list of distinct pitch classes in this tuning would be an improvement (the interval sizes are redundant, since the size of all steps is stated above). — Gwalla | Talk 07:40, 2 March 2008 (UTC)

The seven flats in order of appearance are B♭, E♭, A♭, D♭, G♭, C♭ and F♭. So the next five would be double ♭'s (♭♭), each being 77.4 cents below its single ♭ counterpart, just as each single ♭ is 77.4 cents below its unflattened counterpart. These five are: B♭♭(39 cents), E♭♭(542), A♭♭(1045), D♭♭(348) and G♭♭(852).

Similarly, the seven sharps in order of appearance are F♯, C♯, G♯, D♯, A♯, E♯ and B♯. So the next five would be double ♯'s (×), each being 77.4 cents above its single ♯ counterpart, just as each single ♯ is 77.4 cents above its unsharpened counterpart. These five are: F× (968 cents), C× (465), G× (1161), D× (658) and A× (155).

There are but four enharmonic equivalents (without resorting to TRIPLE flats or sharps): D♭♭= B× (348 cents), G♭♭= E× (852), C♭♭= A× (155) and F♭♭= D× (658). — Updated by Glenn L (talk) 19:22, 30 September 2009 (UTC)

Check your "facts"

Please see for example, page 875, vol 11, of The New Grove (1995), or MGG, on "meantone" before stating the "fact" that 31-tET "is" a meantone tuning. (It can function as one, or "correspond closely" to use the term in Grove's, but to say that it "is" a meantone tuning is simply provincial.)

I agree too, I think "can function as" or "can be viewed as" would be better. 69.182.126.165 (talk) 13:25, 27 April 2008 (UTC)

In addition, 31 equal temperament is NOT "often" referred to as "31-EDO". Although I personally find the abbreviation "EDO" to be excellent, and the idea behind it (which distinguishs equal divisions by intent and usage) very good, the place where it is "often" used is within a small internet community. It is devious, or simply ignorant, to foist it upon the world as if it is a standard term, "often" used, via Wikipedia.

Frank Zamjatin (talk) 10:17, 17 April 2008 (UTC)

I agree. 69.182.126.165 (talk) 13:25, 27 April 2008 (UTC)

The whole point of 'meantone' is that a major third is equal to two 'whole tones'. That's all it means. Anyone can click on that article and read it; there's no reason to duplicate all that stuff here. —Preceding unsigned comment added by 166.84.1.1 (talk) 04:30, 24 May 2009 (UTC)

Improvement?

How is emphasizing septimal minor third and other intervals in the interval size table an improvement? Hyacinth (talk) 23:47, 7 June 2009 (UTC)

I don't fully understand the question, as the septimal minor third has not been emphasised as far as I can see. However, as to "other intervals", I agree entirely: it made the table look messy, and I cannot see how it helped, so I have reverted it. JamesBWatson (talk) 12:48, 10 June 2009 (UTC)

Too much jargon

Why, where, and how does this article contain too much jargon or need explanation or simplification? Hyacinth (talk) 01:49, 28 June 2009 (UTC)

Why 31?

On the french version, it talks about some crazy multiple possibilities (such as 19, 31, 43, 53)... So as I moved towards the english version, hoping for more detailed information and possibly an EXAMPLE of music composed in such intervals, I was quite surprised to see that that article doesn't even have a general title: "31 equal temperament", that doesn't even suggest if the actual translation for "tempérament par divisions multiples" is indeed "equal temperament".

So I guess, two things disappointed me: first, the article has no generalization of the notion of "temperament division" (I don't even know if you could actually call it that). And second, I can't seem to find any kind of content regarding this alternative division, not even on Google. I mean, of course some websites mention it here and there, but no serious example, discussion, or even a complete explanation.

I can imagine this subject might not be very popular, but if there is a reason for it (for instance, if all the possible combinations other than 12 don't make any kind of harmony -don't ask me), well at least write it down somewhere, let me know if it is worth it to continue look for answers...

This is not a wild critique, I am sincerely looking forward to completing the article, but I have almost no understanding of that field. I am deeply interested by the idea of alternative temperament divisions (or whatever is right to call it). I simply have a little trouble finding answers... — Preceding unsigned comment added by 220.110.185.240 (talk) 01:36, 30 September 2011 (UTC)

Additional citations

Why and where does this article need additional citations for verification? What references does it need and how should they be added? Hyacinth (talk) 00:34, 7 July 2012 (UTC)

Curiosity

It is curious that Diccionario de la Real Academia Española defines musical intervals following 31 equal temperament. The article "semítono" states: [2]. That is: semitone=one of two unequal parts to which a whole tone is divided, diatonic semitone=3 commas, chromatic semitone=2 commas. So whole tone=5 commas, perfect fourth=13 commas, perfect fifth=18 commas and perfect octave=31 commas. Burzuchius (talk) 14:27, 28 January 2013 (UTC)

Interesting! I suppose this is because 31 equal temperament can basically be equated with quarter-comma meantone, which was once a common tuning. Double sharp (talk) 12:36, 19 January 2016 (UTC)