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[ca88313]

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[georgey]

Equal temperament is my favorite well temperament because music sounds GREAT in all keys.  All other temperaments were just "Band-Aid" fixes IMO until equal temperament was invented and then became practical enough in the case of pianos to become the standard.  There was a thread on this a year or 2 ago were this was discussed.

[ca88313]

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[georgey]

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.

"the loss of diverse emotional expression" - is an opinion.  I get tons of emotion from well written and performed music played in equal temperament.

"every interval except for the octave is out of tune in comparison with the pure intervals" - "out of tune" compared to the harmonic series.  When Pythagoras invented his tuning system, he assumes the correct ratio of the perfect 5th was 3/2 = 1.5 which matches the harmonic series.  He was not able to calculate what I believe is the correct ration 2^(7/12) = 1.498307.... that is used for equal temperament. You have to temper the 5ths!  

Anyway, this is my opinion.

[ca88313]

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[georgey]

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.

I believe well temperament is where all or most of the 5ths are tempered  so music sounds good in all keys.  I believe equal temperament is not only a well temperament but is THE best well temperament.  A tempered 5th is a 5th that is slightly flattened (ratio < 1.5).  

[ca88313]

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[georgey]

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).

You are welcome to your opinion.  I decided to look this up.

The wording "There are many well temperament schemes, some nearer meantone temperament, others nearer equal temperament"  appears in the below link.  This implies to me that meantone is different from equal temperament and that well temperament is somewhere in between these 2 endpoints.

I stand by my posts, although I will admit that I am not an expert. Again - You are welcome to your opinion and interpretation of what you read.


https://en.wikipedia.org/wiki/Well_temperament

Here is my final answer to your question:  I like equal temperament (which may or may not be a well temperament) because it sounds best to me and works well for all keys IMO.  

Regards.

[georgey]

Further evidence that meantone is NOT an equal temperament:

https://en.wikipedia.org/wiki/Regular_temperament

Please note from the above link with wording shown below: "The rank-one tuning systems are equal temperaments, all of which can be spanned with only a single generator. A rank-two temperament has two generators. Hence, meantone is a rank-2 temperament."


Regular temperament is any tempered system of musical tuning such that each frequency ratio is obtainable as a product of powers of a finite number of generators, or generating frequency ratios. For instance, in 12-TET, the system of music most commonly used in the Western world, the generator is a tempered fifth (700 cents), which is the basis behind the circle of fifths.

When only two generators are needed, with one of them the octave, this is called "linear temperament". The best-known example of a linear temperaments is meantone temperament, where the generating intervals are usually given in terms of a slightly flattened fifth and the octave. Other linear temperaments include the schismatic temperament of Hermann von Helmholtz and miracle temperament.

Mathematical description

If the generators are all of the prime numbers up to a given prime p, we have what is called p-limit just intonation. Sometimes some irrational number close to one of these primes is substituted (an example of tempering) to favour other primes, as in twelve tone equal temperament where 3 is tempered to 2​19⁄12 to favour 2, or in quarter-comma meantone where 3 is tempered to 24√5 to favor 2 and 5.

In mathematical terminology, the products of these generators define a free abelian group. The number of independent generators is the rank of an abelian group. The rank-one tuning systems are equal temperaments, all of which can be spanned with only a single generator. A rank-two temperament has two generators. Hence, meantone is a rank-2 temperament.

I have a math background and can explain "abelian group", but this won't help probably.

[ca88313]

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[georgey]

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_intervals

This 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.

My guess is one can morph into another.  You can say that equal temperament is based on 2 generators A and B such that A = B (see correction below), and so is a meantone system.  If you take equal temperament and modify the note B down by .001 cents, it will no longer be equal temperament (but would sound exactly like equal temperament to the human ear) and possibly would be considered well temperament.  It would not be technically equal temperament.

My guess is we are both correct.  Thanks.

* - correction:  A^k = B where k is an integer (I think).

[ted]

..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.

[georgey]

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.

Interesting.  How about 2^(6999/12000) implying 12000 notes to the octave. 

[ted]

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.

[georgey]

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?

[ronde_des_sylphes]

The experimental composer Harry Partch used 43-note scales, incidentally. I can't help noticing that both 29 and 43 are primes.

[ca88313]

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[ca88313]

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[ted]

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?

17/29 gives the closest approximation to a perfect fifth up to a denominator of 29. Closer ones exist for numbers > 29, of course. A simple program loop or just trial will prove it. If the fifth is very close it follows by extension that the other diatonic intervals will be too. If the base is prime, then obviously no symmetric chords or scales (like augmented, whole tone, and diminished) can exist, but in practice many combinations close to them sound near enough, which can also be easily verified using a simple program and a sound sample.

The trouble for me is that any musical experience with these patterns lacks a physical component, the playing of an instrument, thereby excluding improvisation. I imagine they would offer considerable interest to algorithmic composers though.

[georgey]

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)

I've been wrong before, but I'm not sure you have proven anything here.  I agree that well temperament is not the same as equal temperament. I had suggested that equal temperament is (or maybe is) one of infinitely many theoretical well tempered systems.

I'm not sure about your cent math.  It would appear to hold true for the currency $.01 = 1 cent.  100 cents times 3 = 300 cents = $3.00.  

100 cents is 2^(100/1200) = 1.05946309........
100 cents times 12 is 12.7135713.... = $12.71 - can buy a pizza with this.  
100 cents to the 12th power is 2^(1200/1200) = 2 = 1 octave.

In music, a cent is a LOGRITHMIC unit of measure.  
3rd law of logs: logA^n = n logA.  Maybe you are correct in your usage.

Anyway, we will say you are correct here.  Isn't this all off topic though?  What is you favorite well tempered tuning and why?

Thanks.

[ted]

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:Double
best=1
For 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
      EndIf
Next;Next

[georgey]

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:Double
best=1
For 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
      EndIf
Next;Next

I believe you.  17/29 is the number.  Could also do in a spreadsheet like you said.  Thanks.

[ca88313]

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[ca88313]

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[georgey]

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.  

What is you favorite well temperament tuning and why?  I know there are many that agree with you that equal temperament is not good.  Thanks.

[ca88313]

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[georgey]

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

Sorry for missing this.  Do you have a link to a performance using this tuning?  Thanks.

[ca88313]

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[georgey]

Thanks for sharing these.  I listen to a lot of harpsichord so my ear is accustomed to hearing a lot of different tunings.  The harpsichord tuning sounded great to me on the Bach prelude.  I would have liked hearing the prelude and fugue #3 from book 1 in C# major to see how this sounded.  I assume it would sound nice since it is a well-tempered system.

The Chopin also sounded nice to me.  The standard equal temperament tuning played  2nd on the Chopin did sound out of tune to me though.  The tuner maybe did not spend enough time or maybe it is tough to tune a piano to equal temperament after being tuned to a different temperament?  

Thanks.

[ca88313]

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[hfmadopter]

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.

I'm inclined to agree. I don't play much 17th century music, so I like what one might term as a tighter Well Temperament. The 17th century Well had the key note and fifths with broader beats, you can tighten those beats up a bit and stay within the constrains of well temperament, where Equal would smooth it out too much IMO. Werckmeister III comes to mind off the top of my head as a usable Chopin tuning. I actually first discovered this tuning in my piano program Pianoteq for the C Bechstein, it too would give that waver you hear in Chpin's Funeral March.

I'm going to play around some more with Well Temperaments on my Henry F Miller19th century grand this winter. I haven't been running a humidifier yet this winter so it's tuning is all out of whack right now anyway.

[ca88313]

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