Equal temperament is mathematically perfect since it seems to solve all the problems with all the temperaments that preceded it. However, the loss of diverse emotional expression is just not acceptable as well as the fact that every interval except for the octave is out of tune in comparison with the pure intervals of just intonation. Well-temperament seems to ideally combine the best features of just intonation with equal temperament without inheriting their respective problems as analysed in my original post.

Equal temperament is a meantone temperament so your answer does not answer my question. Well temperament is an irregular temperament whereas equal temperament is a regular temperament.

Meantone temperament and well temperament are completely different. Meantone temperament involves tempering every perfect fifth by the same amount, for example, every fifth has a size of 700 cents if 12-EDO/12-TET is used. Well temperament involves tempering certain perfect fifths by different amounts which means the size of certain perfect fifths will be different. Meantone/Equal temperament is not equal to well temperament. Both tuning systems allow music to sound good in all keys but they do so via different methods.Equal temperament = Eleventh-comma meantone (syntonic comma) = Tweflth-comma meantone (pythagorean comma).

This source states that equal temperament/12-EDO/12-TET is "the most common meantone temperament of the modern era":https://en.m.wikipedia.org/wiki/List_of_meantone_intervalsThis makes sense completely because 12-TET and every other equal temperament flattens the perfect fifth by the same amount, for example, third-comma-meantone, quarter-comma meantone, 19-TET, 31-TET and 53-TET. This means that 12-TET must be a type of meantone temperament.This description seems to completely match the statement that: Meantone temperament is a musical temperament, that is a tuning system, obtained by slightly compromising the fifths in order to improve the thirds. Meantone temperaments are constructed the same way as Pythagorean tuning, as a stack of equal fifths, but in meantone each fifth is narrow compared to the perfect fifth of ratio 2:1.

..2^(7/12) = 1.498307....

Yes, that is a happy numerical approximation to be sure. It is easy to write a little program to find all such. The one which fascinated me for a while is 2^(17/29), implying 29 notes to an octave will embed approximations for the common intervals, and indeed it does. Years ago, I wrote code to produce music modulating around the associated 29 diatonic keys and sent it to various musicians. All of them thought it was just the normal twelve notes to the octave, and one or two of them had absolute pitch, so I still don't know what that implies. Maybe we all think we can hear more than we are actually able to hear. Unfortunately, 29 notes to the octave excludes any practical physical instrument but it is interesting nonetheless.

Ha ha ! There are certainly many more approximations with larger numbers but it's a bit like Mordell's equation; only the lower ones hold real interest. In terms of musical application, I have sufficient difficulty creating music with the ordinary setup, thank you very much ! It was just a arithmetical novelty, a bit like finding all the the possible chords using Polya-Burnside formulas. Not much good musically.

I should do a quick spreadsheet to confirm that your 17/29 is best. Are you sure there are not better? Can you prove this?

Unfortunately, I am correct and you are wrong on this occasion because well temperament is not the same as equal temperament.Your equation also proves that I was right when I said equal temperament is a form of meantone temperament:100 cents (minor second) x 12 (constant) = 1200 cents (octave)200 cents (major second) x 6 (constant) = 1200 cents (octave)300 cents (minor third) x 4 (constant) = 1200 cents (octave)400 cents (major third) x 3 (constant) = 1200 cents (octave)600 cents (tritone) x 2 (constant) = 1200 cents (octave)1200 cents (octave) x 1 (constant) = 1200 cents (octave)

Here you are:After 17/29, the next are 24/41 and 31/53, after which there are many, as you would expect.Global i:Int,j:Int,k:Double,i2:Double,j2:Double,best:Double,diff:Doublebest=1For i=12 To 1000 i2=i For j=1 To i j2=j k=2^(j2/i2) diff=Abs(k-1.5) If diff<best Print i2 Print j2 Print k Print "" best=diff EndIfNext;Next

You are completely right about that. I should have used logarithms and exponents. I have just been lazy and decided to practice my 12 times tables. My method does seem to work though but in an unorthodox way. I think we are still talking about tuning so every post is still on-topic even though they are stretching the true intention of the original question.

I like Thomas Young's second temperament and Bradley Lehman's Bach/Lehman 1722 due to the reasons highlighted in my original post.

Bach/Lehman 1722 makes the strings of a piano or a harpsichord twang and enables smooth transitions between keys, as evident in the first video. Young's second temperament changes the mood of Chopin's funeral march due to the alternating rate of beating of consecutive chords, as evident in the second video. Both of these qualities are more or less lost when equal temperament is used. I included these videos in one of my other posts called "Why are pieces written in C# minor so popular?" in the repertoire section of piano street and also mentioned that atonal pieces should be played on just-intoned, well-tempered and equal-tempered instruments in order to truly experience the differences between these three different types of tuning.