Desperately Seeking Just justice justness justitude Justinia

I am frustrated.

I want sweet thirds; the music I play (from Frescobaldi to J.S. Bach) could easily not ever stumble across a horrible built in wolftone.

My Korg machine with Werkmeister and several other tunings does not satisfy anymore. I don't want to deal with anything that has 19 in it (did I read some esoteric reference to 19 as a sacred or mystically powerful number recently?).

Please Help Please Help with something like F# to A, D to F#, and so on...

Dana

--- In tuning@yahoogroups.com, "apollo992" <apollo992@...> wrote:
>
> I am frustrated.
>
> I want sweet thirds; the music I play (from Frescobaldi to J.S. Bach) could easily not ever stumble across a horrible built in wolftone.
>
> My Korg machine with Werkmeister and several other tunings does not satisfy anymore. I don't want to deal with anything that has 19 in it (did I read some esoteric reference to 19 as a sacred or mystically powerful number recently?).
>
> Please Help Please Help with something like F# to A, D to F#, and so on...
>
> Dana

Hi Dana,
All numbers are sacred, especially when you play good music with them.
What's wrong with 19 ? too much Pythagorean for your music may be ?
But the Bulgarian singers and others make wonders with 19.

I am not sure to understand precisely your problem, but if you encounter a wolf fifth, you are using a meantone tuning obviously.
May be you know already what follows, but in case you don't :

1) try to see all the fifths you use in all the pieces of your recital, and if by chances there are some fifths you do not use, try to transpose the tuning to another key so that the wolf doesn't show up. This will solve the problem.
2) if all the 12 fifths are used, the only possibility left is to use a well-tempered tuning, that will show less sweet thirds in some places.
3) then you can also sometimes optimize a WT tuning by a proper transposition to another key, depending on the pieces you play.
4) the question of which Meantone, or WT to use depends of your repertoire but also of your own taste.

Does that help ?
- - - -
Jacques

Apollo>"I want sweet thirds; the music I play (from Frescobaldi to J.S. Bach)
could easily not ever stumble across a horrible built in wolftone."

   Well, here's a 7-tone irregularly tempered scale that attempts to maximize the consonance of all major and minor thirds to an error of only under 3 cents off perfect:

1
70.2066668
191.48007828336293
382.9601565667259
574.466627673006
695.7400391416815
887.2201174250444
1078.7265885313248
2/1

--- On Tue, 2/15/11, apollo992 <apollo992@...> wrote:

From: apollo992 <apollo992@...>
Subject: [tuning] Desperately Seeking Just       justice justness justitude Justinia
To: tuning@yahoogroups.com
Date: Tuesday, February 15, 2011, 8:50 AM

   
     
     
      I am frustrated.

I want sweet thirds; the music I play (from Frescobaldi to J.S. Bach) could easily not ever stumble across a horrible built in wolftone.

My Korg machine with Werkmeister and several other tunings does not satisfy anymore.  I don't want to deal with anything that has 19 in it (did I read some esoteric reference to 19 as a sacred or mystically powerful number recently?).

Please Help      Please Help       with something like F# to A, D to F#, and so on...

Dana

--- In tuning@yahoogroups.com, "apollo992" <apollo992@...> wrote:
>
> I am frustrated.
>
> I want sweet thirds; the music I play (from Frescobaldi to J.S. Bach) could easily not ever stumble across a horrible built in wolftone.

I'm not sure what your question is, but have you tried 1/4 comma meantone?

http://xenharmonic.wikispaces.com/Regular+Temperaments

http://xenharmonic.wikispaces.com/JustIntonation

lots of information and tunings to try with many example compositions.

On Tue, Feb 15, 2011 at 11:50 AM, apollo992 <apollo992@...> wrote:

>
>
> I am frustrated.
>
> I want sweet thirds; the music I play (from Frescobaldi to J.S. Bach) could
> easily not ever stumble across a horrible built in wolftone.
>
> My Korg machine with Werkmeister and several other tunings does not satisfy
> anymore. I don't want to deal with anything that has 19 in it (did I read
> some esoteric reference to 19 as a sacred or mystically powerful number
> recently?).
>
> Please Help Please Help with something like F# to A, D to F#, and so on...
>
> Dana
>
>
>

    One hulking question I have...especially to George Secor and those into irregular temperaments, is as there any specific well temperaments which have better accuracy than 1/4 comma meantone for the same dyads quarter comma meantone seeks to optimize and what are they (Scala files and generator maps included, if possible)?

   I used to think 1/4 comma meantone was the "gold standard" for 5-limit dyadic accuracy...until John posted his temperaments and Igs said well-temperament, and not quarter comma meantone, was the most competitive alternative to John's temperaments. 

--- On Tue, 2/15/11, genewardsmith <genewardsmith@...t> wrote:

From: genewardsmith <genewardsmith@...>
Subject: [tuning] Re: Desperately Seeking Just justice justness justitude Justinia
To: tuning@yahoogroups.com
Date: Tuesday, February 15, 2011, 12:40 PM

--- In tuning@yahoogroups.com, "apollo992" <apollo992@...> wrote:

>

> I am frustrated.

>

> I want sweet thirds; the music I play (from Frescobaldi to J.S. Bach) could easily not ever stumble across a horrible built in wolftone.

I'm not sure what your question is, but have you tried 1/4 comma meantone?

You could try "Adaptive JI", or Hermode tuning. If you want a fixed tuning system, something's got to give--I suggest you see Paul Erlich's paper "A Middle Path" if you want to understand where the difficulties of musical scales come from and how they may be dealt with:

http://eceserv0.ece.wisc.edu/~sethares/paperspdf/Erlich-MiddlePath.pdf

To put it bluntly, if you want a fixed 12-note scale with sweet thirds, the best you can do is 1/4-comma or 1/3-comma meantone, but that will mean some keys are going to be messed up. The only way to do better than this for common-practice music is to add more notes to the octave or to use the Adaptive JI described above.

Or you could contact Marcel DeVelde, he's been trying to figure out how to tune common-practice music in JI for years (though I don't think you will like his results if you want all your chords to have sweet 5-limit thirds).

-Igs

--- In tuning@yahoogroups.com, "apollo992" <apollo992@...> wrote:
>
> I am frustrated.
>
> I want sweet thirds; the music I play (from Frescobaldi to J.S. Bach) could easily not ever stumble across a horrible built in wolftone.
>
> My Korg machine with Werkmeister and several other tunings does not satisfy anymore. I don't want to deal with anything that has 19 in it (did I read some esoteric reference to 19 as a sacred or mystically powerful number recently?).
>
> Please Help Please Help with something like F# to A, D to F#, and so on...
>
> Dana
>

On Tue, Feb 15, 2011 at 6:04 PM, cityoftheasleep
<igliashon@...> wrote:
>
> You could try "Adaptive JI", or Hermode tuning. If you want a fixed tuning system, something's got to give--I suggest you see Paul Erlich's paper "A Middle Path" if you want to understand where the difficulties of musical scales come from and how they may be dealt with:

Paul also had a pretty good idea for an adaptive-JI algorithm - if you
take 62 notes of quarter-comma meantone, the result looks a lot like
two 31-note chains that lie a quarter comma away from one another.
This is kind of like doing adaptive-JI around 31-tet, I guess you
could say.

> http://eceserv0.ece.wisc.edu/~sethares/paperspdf/Erlich-MiddlePath.pdf
>
> To put it bluntly, if you want a fixed 12-note scale with sweet thirds, the best you can do is 1/4-comma or 1/3-comma meantone, but that will mean some keys are going to be messed up.

There's also the woolhouse optimal one, which works out to 7/26-comma.

-Mike

MikeB>"There's also the woolhouse optimal one, which works out to 7/26-comma."
Dare I ask
A) How does woolhouse optimization work?
B) What's the generator for 1/3rd comma meantone (in cents)?

On Tue, Feb 15, 2011 at 9:16 PM, Michael <djtrancendance@...> wrote:
>
> MikeB>"There's also the woolhouse optimal one, which works out to 7/26-comma."
> Dare I ask
> A) How does woolhouse optimization work?

I believe that, in the 5-limit, it's the squared-error optimal tuning
for the 3 5-limit consonances: 3/2, 5/4, and 6/5. There's a long, long
page on it here:

http://www.tonalsoft.com/monzo/woolhouse/essay.aspx

I studied this in some detail a while ago but have forgotten all of
the nuances. For the 7-limit, what I've been doing is just working out
the squared-error optimal tuning for 3/2, 5/4, 6/5, 7/5, and 7/6. You
could hypothetically pick any dyads you want and just optimize it
around that.

> B) What's the generator for 1/3rd comma meantone (in cents)?

It's pretty much identical to 19-tet's fourth (or fifth), but let's
work it out. So for 1/3-comma meantone, three fourths should put you
at an exact 6/5 (octave-equivalent), just like in how 1/4 comma
meantone, four fifths puts you at an exact 5/4 (octave-equivalent).
Think about this for a second if it doesn't make immediate sense. But
mathematically, the derivation of the generator is simple:

1) Start with 6/5 * 2/1, or 12/5.
2) You know that three generators will get you to 12/5, meaning that g^3 = 12/5.
3) Therefore, g = (12/5)^(1/3).
4) So what is g in cents? cents(g) = 1200*log(g)/log(2) =
1200*log((12/5)^(1/3))/log(2)
5) Since log(x^y) = y * log(x), the 1/3 comes outside of the log and
you get 1/3 * 1200 * log(12/5)/log(2)
6) This equals 505.214 cents. In comparison, 19-equal's fourth is
505.263 cents. That is a difference of 0.049 cents.

There is just no point pretending that 1/3-comma meantone is somehow
distinct from 19-equal, except for on a purely mathematical level.

-Mike

I am sorry Apollo (I feel rather like I have been taken up on high into Olympus), but as long as you use only 12 pitches in the octave and want sweet thirds/sixths, there is no way to escape the wolf fifth. Some theorists over the centuries, most estimable persons in their way, have tried to favor both the thirds/sixths and the fifths in some keys, at the expense of making others even worse than in 12-tone equal. Personally these efforts do not much interest me. They certainly are not adequate for playing Baroque music with an extended key range, which is why they were finally rejected.

Music of the middle Baroque such as you play usually employed between 13 and 16 keys. It was very common to split the G#/Ab accidental, but with only one split accidental, the outer keys still suffered from the wolf, either in 1/4 meantone (the most common tuning) or in 1/3 comma meantone (the second most common; the most common in Naples). Handel had a famous organ built with 16 pitches in the octave, dividing all the accidentals except F#. It is often asserted that this was tuned in 1/4 comma meantone.

You will have to resort to 19-tone equal temperament (aurally indisguishable from 1/3 comma meantone), with a 19-tone instrument, to both escape a wolf fifth and achieve good thirds and sixths throughout all the keys. Even then it is necessary to stretch the octave by about 1/512 octave to make the twelfths sweet enough for my taste. Technically, you might be able to do without the E#/Fb and B#/Cb accidentals for Baroque music, but I would not recommend omitting them, as they are highly necessary for later music, though sometimes not without problems if the music was written explicitly for 12-tone equal.

I believe that 19 is considered a sacred number in some corners of the world, but less so than 12, which is associated with the twelve Greek gods, the twelve tribes of the Hebrews, the twelve disciples of Khrishna/Christ, etc. However, the Greeks added 7 non-Olympian gods, Poseidon, Hades, Persephone, Prometheos, Demeter, Dionysos and the Dioscouroi (Godlike-youths, counted as one) to arrive at 19 Great Gods. 20 they regarded as accurst, as one of the Dioscouroi had to die during six months of the year. 7 is also considered a sacred number, partly because of the diatonic scale, which is the only heptatonic scale with consonant triads on six of the seven modal degrees, and is thus of unique importance in harmony. I have proven that it is the only such scale, using tree-diagrams, although the most usual form of the minor is NOT a diatonic but a chromatic mode of a chromatic scale.

22 is a sacred number to the Indians because of the 22 srutis. Some westerners have been interested in 22-tone equal temperament because of this. But the temperament actually exaggerates the comma instead of distributing it, so it has very poor thirds/sixths in the cycle of fifths.

My recommendation is to have a 19-tone harpsichord built, preferably with two manuals. Each of the five accidentals must be split and two new accidentals inserted between E and F, and between B and C. This, and this alone, will really give you what you want. 31-tone equal is also adequate for Baroque music, but you would have to have 26 of its 31 degrees available, and that is not really practical; such an instrument is unplayably complex. Also, 19-tone equal is what singers are most comfortable with.

I am not a great Bach scholar. It does seem to me however that in his keyboard works he was less interested in smooth harmony than any western composer until the 20th century; his keyboard works do not sound much better in 19-tone equal than in 12-tone equal, just more clean-cut and biting. With Frescobaldi, it is a different matter entirely; like most of the Italians, smooth harmony was vital to his music, and you will be astonished how much better his music sounds in the 19-tone equal temperament. Gesualdo scarcely exists in the 12-tone equal; he wrote his music in 19-tone equal; his master Nenna was an advocate of this tuning. It would appear that Lassus wrote his famous Prophetiae Sibyllarum for 19-tone equal as well.

Gregg Gibson

--- In tuning@yahoogroups.com, "gregggibson" <gregggibson@...> wrote:
> Gesualdo scarcely exists in the 12-tone equal; he wrote his music in 19-tone equal; his master Nenna was an advocate of this tuning. It would appear that Lassus wrote his famous Prophetiae Sibyllarum for 19-tone equal as well.

Do you have citations for this? Gesualdo seems to have been influenced by Vicentino, but how do you go from there to 19et? Lassus and the musica reservata style in general, once again, could be related to 19et, or it could be related to 31et (which would probably make the most sense), but why should it be related to any equal division?

Advocacy by Pomponio Nenna of 19edo would be interesting to establish if you have actual evidence for it; it could have come by way of Costeley I presume.

WOW! Did I get a response! It is a pleasure to come home at last to people who understand my frustrations, even if there seems to be no simple answers. When I buy professional harpsichord CD's, or listen on YouTube, INTONATION DOES NOT SEEM TO BE AN ISSUE. Yet when I sit down to play, either at home on my harpsichord or at the nearby college (Wright State University) where I hang out and give the daily 0645 concert on the grand piano, intonation seems to overwhelm all other aspects of music making. Maybe the problem is that I am a reformed violist.

You're not the Gregg that I knew long ago at Taos Chamber Music, are you?

Dana

--- In tuning@...m, "gregggibson" <gregggibson@...> wrote:
>
>
> I am sorry Apollo (I feel rather like I have been taken up on high into Olympus), but as long as you use only 12 pitches in the octave and want sweet thirds/sixths, there is no way to escape the wolf fifth. Some theorists over the centuries, most estimable persons in their way, have tried to favor both the thirds/sixths and the fifths in some keys, at the expense of making others even worse than in 12-tone equal. Personally these efforts do not much interest me. They certainly are not adequate for playing Baroque music with an extended key range, which is why they were finally rejected.
>
> Music of the middle Baroque such as you play usually employed between 13 and 16 keys. It was very common to split the G#/Ab accidental, but with only one split accidental, the outer keys still suffered from the wolf, either in 1/4 meantone (the most common tuning) or in 1/3 comma meantone (the second most common; the most common in Naples). Handel had a famous organ built with 16 pitches in the octave, dividing all the accidentals except F#. It is often asserted that this was tuned in 1/4 comma meantone.
>
> You will have to resort to 19-tone equal temperament (aurally indisguishable from 1/3 comma meantone), with a 19-tone instrument, to both escape a wolf fifth and achieve good thirds and sixths throughout all the keys. Even then it is necessary to stretch the octave by about 1/512 octave to make the twelfths sweet enough for my taste. Technically, you might be able to do without the E#/Fb and B#/Cb accidentals for Baroque music, but I would not recommend omitting them, as they are highly necessary for later music, though sometimes not without problems if the music was written explicitly for 12-tone equal.
>
> I believe that 19 is considered a sacred number in some corners of the world, but less so than 12, which is associated with the twelve Greek gods, the twelve tribes of the Hebrews, the twelve disciples of Khrishna/Christ, etc. However, the Greeks added 7 non-Olympian gods, Poseidon, Hades, Persephone, Prometheos, Demeter, Dionysos and the Dioscouroi (Godlike-youths, counted as one) to arrive at 19 Great Gods. 20 they regarded as accurst, as one of the Dioscouroi had to die during six months of the year. 7 is also considered a sacred number, partly because of the diatonic scale, which is the only heptatonic scale with consonant triads on six of the seven modal degrees, and is thus of unique importance in harmony. I have proven that it is the only such scale, using tree-diagrams, although the most usual form of the minor is NOT a diatonic but a chromatic mode of a chromatic scale.
>
> 22 is a sacred number to the Indians because of the 22 srutis. Some westerners have been interested in 22-tone equal temperament because of this. But the temperament actually exaggerates the comma instead of distributing it, so it has very poor thirds/sixths in the cycle of fifths.
>
> My recommendation is to have a 19-tone harpsichord built, preferably with two manuals. Each of the five accidentals must be split and two new accidentals inserted between E and F, and between B and C. This, and this alone, will really give you what you want. 31-tone equal is also adequate for Baroque music, but you would have to have 26 of its 31 degrees available, and that is not really practical; such an instrument is unplayably complex. Also, 19-tone equal is what singers are most comfortable with.
>
> I am not a great Bach scholar. It does seem to me however that in his keyboard works he was less interested in smooth harmony than any western composer until the 20th century; his keyboard works do not sound much better in 19-tone equal than in 12-tone equal, just more clean-cut and biting. With Frescobaldi, it is a different matter entirely; like most of the Italians, smooth harmony was vital to his music, and you will be astonished how much better his music sounds in the 19-tone equal temperament. Gesualdo scarcely exists in the 12-tone equal; he wrote his music in 19-tone equal; his master Nenna was an advocate of this tuning. It would appear that Lassus wrote his famous Prophetiae Sibyllarum for 19-tone equal as well.
>
> Gregg Gibson
>

Mike wrote:

> Paul also had a pretty good idea for an adaptive-JI algorithm - if you
> take 62 notes of quarter-comma meantone, the result looks a lot like
> two 31-note chains that lie a quarter comma away from one another.

Supposing we want to "regularly" chain the two chains -- i.e. keep it 2D but treat it like 3D.

A different approach is that of Vicentino. In 1555, he suggested a 36-tone tuning, which contains a 19-tone quarter-comma meantone chain from Gb to B# together with a "quarter-comma higher" 17-tone chain from Gb to A#. So if we mark the tones of the latter chain with, let's say, a "+" sign, then the ordinary 4-chord major comma pump would be played like this:
C-E-G+, C+-E+-A, D-F+-A+, B-D+-G, C-E-G+

Obviously, 5-limit JI is a 3D system and so is Vicentino's tuning. So now the question remains how much we really care about the proper sound of the original 5-limit chords, considering the fact that standard staff notation is 2D and doesn't offer secondary accidentals which would be independent of sharps/flats. If something like a 3D notation existed, then the written comma pump would indeed finish with a different chord than it started -- I mean something like this:
C-E<-G, C-E<-A<, D<-F-A<, B<<-D<-G<, C<-E<<-G<

Petr

--- In tuning@yahoogroups.com, Petr Parízek <petrparizek2000@...> wrote:

> Obviously, 5-limit JI is a 3D system and so is Vicentino's tuning.

Vicentino's system is just an arrangement of 1/4-comma meantone, so it's 2D.

Gene wrote:

> Vicentino's system is just an arrangement of 1/4-comma meantone, so it's > 2D.

Meantone tempering is 2D because it only tempers out one interval within the 3D system called "5-limit JI". However, Vicentino's improvement aims to have pure 3/2 wherever possible, which effectively results in a 3D system -- i.e. the period is 2/1, one generator is the 4th root of 5 (chained as many times as desired), another is 3/2 (used only once in his version; but recall he didn't need things like pure 9/8 which would require using it more than once). This means that the mapping for 3/1 is <1, 0, 1> while 5/1 is mapped to <0, 4, 0>.

Petr

I wrote:

> Mike wrote:
>
>> Paul also had a pretty good idea for an adaptive-JI algorithm - if you
>> take 62 notes of quarter-comma meantone, the result looks a lot like
>> two 31-note chains that lie a quarter comma away from one another.
>
> Supposing we want to "regularly" chain the two chains -- i.e. keep it 2D > but treat it like 3D.

Another instance where a 2D system is treated like 3D is the Indian Shruti tuning which is essentially a 22-tone chain of fifths but the "default" version of the C major scale (if C is the tonic) is actually "C-D-Fb-F-G-A-Cb-C".

Petr

I wrote:

> Meantone tempering is 2D because it only tempers out one interval within > the 3D system called "5-limit JI". However, Vicentino's improvement aims > to have pure 3/2 wherever possible, which effectively results in a 3D > system -- i.e. the period is 2/1, one generator is the 4th root of 5 > (chained as many times as desired), another is 3/2 (used only once in his > version; but recall he didn't need things like pure 9/8 which would > require using it more than once). This means that the mapping for 3/1 is > <1, 0, 1> while 5/1 is mapped to <0, 4, 0>.

Actually, in the full 19+17-tone version, Vicentino uses two different pitches of Gb, which would mean the other generator to be the fourth root of 81/80. Then the mapping for 3/1 would come out as <1, 1, 1>. Had he found the number 9 as important as the lower ones, he might consider using the other generator more than once -- for example, 9/8 and 10/9 come out as <-1, 2, 2> and <-1, 2, -2>, respectively.

Petr

Jacques,

Bless you for answering. To put it more simply, I am too dumb and or lazy to even read much less understand all the information on this site. I appreciate you all (in this community) and share the problems, but what I want is a magic pill so I don't hear the problems anymore (I can't even listen to Nineteenth Century Romanitic music nowdays) or a magic gadget like a next generation Korg tuner.

A drug, yeah, that's it, I want a nephethe to forget. The more I tune (and I spend more time tuning these days than playing) the worse the problem.

Thanks, Dana

--- In tuning@yahoogroups.com, "Jacques Dudon" <fotosonix@...> wrote:
>
>
> --- In tuning@yahoogroups.com, "apollo992" <apollo992@> wrote:
> >
> > I am frustrated.
> >
> > I want sweet thirds; the music I play (from Frescobaldi to J.S. Bach) could easily not ever stumble across a horrible built in wolftone.
> >
> > My Korg machine with Werkmeister and several other tunings does not satisfy anymore. I don't want to deal with anything that has 19 in it (did I read some esoteric reference to 19 as a sacred or mystically powerful number recently?).
> >
> > Please Help Please Help with something like F# to A, D to F#, and so on...
> >
> > Dana
>
>
> Hi Dana,
> All numbers are sacred, especially when you play good music with them.
> What's wrong with 19 ? too much Pythagorean for your music may be ?
> But the Bulgarian singers and others make wonders with 19.
>
> I am not sure to understand precisely your problem, but if you encounter a wolf fifth, you are using a meantone tuning obviously.
> May be you know already what follows, but in case you don't :
>
> 1) try to see all the fifths you use in all the pieces of your recital, and if by chances there are some fifths you do not use, try to transpose the tuning to another key so that the wolf doesn't show up. This will solve the problem.
> 2) if all the 12 fifths are used, the only possibility left is to use a well-tempered tuning, that will show less sweet thirds in some places.
> 3) then you can also sometimes optimize a WT tuning by a proper transposition to another key, depending on the pieces you play.
> 4) the question of which Meantone, or WT to use depends of your repertoire but also of your own taste.
>
> Does that help ?
> - - - -
> Jacques
>

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
>
> One hulking question I have...especially to George Secor and those into irregular temperaments, is as there any specific well temperaments which have better accuracy than 1/4 comma meantone for the same dyads quarter comma meantone seeks to optimize and what are they (Scala files and generator maps included, if possible)?
>
> I used to think 1/4 comma meantone was the "gold standard" for 5-limit dyadic accuracy...until John posted his temperaments and Igs said well-temperament, and not quarter comma meantone, was the most competitive alternative to John's temperaments.

Since you're asking about well-temperament vs. 1/4-comma meantone, I'm assuming that you're restricting your options (for the purposes of this discussion) to tunings in which Didymus' comma (80:81) vanishes, in which case I think you're correct in thinking of 1/4-comma meantone as the 5-limit "gold standard" -- provided you extend your chain of 5ths sufficiently to provide all the sharps and flats (and doubles) required by your music (which, extended sufficiently, also gives you an excellent 7-limit consonances). If you extend meantone to 31 tones and tweak your fifths slightly narrower, you'll end up with 31-equal, giving you free modulation. (If that's too many tones, then try 19-equal.)

For a competitive alternative to John's temperaments, however, you'll have to limit yourself to 12 tones per octave. The original message (from "apollo992") mentioned "sweet thirds" for "Frescobaldi to J.S. Bach" without wolves. This calls for a high-contrast circulating temperament, usable in all keys, some keys having better thirds than others, with as many good keys as possible. This will require tempering some of the 5ths wider than Pythagorean and some of the 3rds by an amount greater than Pythagorean (21.5 cents), so we're looking for something other than a well-temperament (as Jorgensen and others have used the term). The French term _temperament ordinaire_ is commonly used for this sort of tuning, but IMO the worst thirds can get a bit extreme.

After 40 years of working on various kinds of circulating temperaments, I've come up with a temperament extraordinaire (a term Margo Schulter and I independently coined for something we like better than _temperaments ordinaires_) in which the fifths are all tempered less than 5 cents, major thirds do not exceed 11:14, and the major and minor triads are all proportional-beating. See the last half of this message:
/tuning/topicId_88708.html#88894

A key sentence in that message is that "there are now 6 major triads with the major 3rd tempered less than 8 cents" (Bb, F, C, G, D, and A); 2 others (Eb and E) sound similar to 12-equal, 2 others (Ab and B) like Pythagorean, and the remaining two (F# and C#) pay the dues. You may not want to play a piece in the key of F# or C# major, but OTOH those triads won't howl at you like they do in Eb-to-G# meantone. I dare you to put this tuning on a baroque organ (for testing, a synthetic one will do) and see if it doesn't knock your socks off.

--George

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:

> Vicentino's system is just an arrangement of 1/4-comma meantone,

As it contains 1/4-comma intervals, I don't think that's true.

-Carl

Fascinating.  All the fifths seem to be very near 698 cents, but alternate to slightly different values.  How did you calculate this?

>"This calls for a high-contrast circulating temperament, usable in all
keys, some keys having better thirds than others, with as many good keys
as possible. This will require tempering some of the 5ths wider than
Pythagorean and some of the 3rds (??FLAT??) by an amount greater than Pythagorean
(21.5 cents)"

   Makes sense...but how did you come up with that conclusion IE why is it that way?

--- On Thu, 2/17/11, gdsecor <gdsecor@yahoo.com> wrote:

From: gdsecor <gdsecor@...>
Subject: [tuning] Re: Desperately Seeking Just justice justness justitude Justinia
To: tuning@yahoogroups.com
Date: Thursday, February 17, 2011, 11:33 AM

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:

>

> One hulking question I have...especially to George Secor and those into irregular temperaments, is as there any specific well temperaments which have better accuracy than 1/4 comma meantone for the same dyads quarter comma meantone seeks to optimize and what are they (Scala files and generator maps included, if possible)?

>

> I used to think 1/4 comma meantone was the "gold standard" for 5-limit dyadic accuracy...until John posted his temperaments and Igs said well-temperament, and not quarter comma meantone, was the most competitive alternative to John's temperaments.

Since you're asking about well-temperament vs. 1/4-comma meantone, I'm assuming that you're restricting your options (for the purposes of this discussion) to tunings in which Didymus' comma (80:81) vanishes, in which case I think you're correct in thinking of 1/4-comma meantone as the 5-limit "gold standard" -- provided you extend your chain of 5ths sufficiently to provide all the sharps and flats (and doubles) required by your music (which, extended sufficiently, also gives you an excellent 7-limit consonances). If you extend meantone to 31 tones and tweak your fifths slightly narrower, you'll end up with 31-equal, giving you free modulation. (If that's too many tones, then try 19-equal.)

For a competitive alternative to John's temperaments, however, you'll have to limit yourself to 12 tones per octave. The original message (from "apollo992") mentioned "sweet thirds" for "Frescobaldi to J.S. Bach" without wolves. This calls for a high-contrast circulating temperament, usable in all keys, some keys having better thirds than others, with as many good keys as possible. This will require tempering some of the 5ths wider than Pythagorean and some of the 3rds by an amount greater than Pythagorean (21.5 cents), so we're looking for something other than a well-temperament (as Jorgensen and others have used the term). The French term _temperament ordinaire_ is commonly used for this sort of tuning, but IMO the worst thirds can get a bit extreme.

After 40 years of working on various kinds of circulating temperaments, I've come up with a temperament extraordinaire (a term Margo Schulter and I independently coined for something we like better than _temperaments ordinaires_) in which the fifths are all tempered less than 5 cents, major thirds do not exceed 11:14, and the major and minor triads are all proportional-beating. See the last half of this message:

/tuning/topicId_88708.html#88894

A key sentence in that message is that "there are now 6 major triads with the major 3rd tempered less than 8 cents" (Bb, F, C, G, D, and A); 2 others (Eb and E) sound similar to 12-equal, 2 others (Ab and B) like Pythagorean, and the remaining two (F# and C#) pay the dues. You may not want to play a piece in the key of F# or C# major, but OTOH those triads won't howl at you like they do in Eb-to-G# meantone. I dare you to put this tuning on a baroque organ (for testing, a synthetic one will do) and see if it doesn't knock your socks off.

--George

No my friend, I have never been in Taos. My stomping grounds are South Carolina, Georgia and Europe. Some people confuse me with the Gregg Gibson who is a rock singer. We are of about the same age, but I have never met him.

I am very pleased that the responses here have been of assistance to you. The best references to temperament are usually in Latin, Italian, French or German, and not everyone has time to acquire so many languages. Even I do not yet read German just for pleasure. English was only a minor language in this subject until the 20th century. Zarlino is however slowly being translated into English, and I believe someone has done Salinas as well, although I have not seen it; my Latin is good enough to use the original. Mersenne was translated long ago, but I have never seen it; I do have the original however, which makes fascinating reading.

I should not have said that Nenna and the Neapolitans used the 19-tone equal; they used the 1/3 comma meantone, which is aurally identical to the 19-tone equal; the divergence in the intervals is on the order of a cent. Any good musical theorist will have calculated and studied this somewhere in his notes. It is as easy as multiplying fractions and converting their product to logarithms and then to cents or to savarts.

As you are probably aware, in the cycle of fifths, the major sixth is three fifths above the prime or beginning tone, and the major third is four fifths above the prime. The minor third is three fifths below the prime, and the minor sixth is four fifths below the prime. Therefore the most obvious way to redistribute the comma is to diminish the fifth by 1/3 comma or 1/4 comma, and these are the systems that were first invented and used. No one knows who actually invented these tunings; we only know who first described them. Salinas states that he was the inventor of the 1/3 comma meantone, but Zarlino assures us that it was in use before Salinas, and Salinas gracefully conceded that this was probably the case, in a famous passage of his de Musica.

A fifth augmented even slightly will exacerbate the comma, resulting in thirds and sixths still worse than in Pythagorean Intonation.

It is now generally admitted that the Neapolitan School employed 1/3 comma meantone in preference to the 1/4 comma meantone. The kingdom was full of 19-tone keyboards at a time when 12- or 13-tone keyboards were the rule elsewhere. This is a commonplace of historical musicology. A few of them are still on display in Naples museums.

It is sometimes asserted, after Ellis as I recall, that 19-tone keyboards could just as easily have been tuned in 1/4 comma meantone. This would however have reintroduced the wolf. Moreover, there is no particular reason to select 19 of the 31 slightly unequal degrees of the 1/4 meantone. 17 or 26 are much more logical selections for Baroque music.

I am always puzzled when theorists treat 1/3 comma meantone as an oddity , and question whether it was ever used. I have read and translated Costeley's preface, and Mersenne's references to Titelouze. There is no doubt that 1/3 comma meantone was in current use, at least in France and Italy. Zarlino assures us that both were in common use at Rome and in Italy in general.

I believe Costeley was mentioned by another correspondent.

As will shortly be seen, Guillaume Costeley (pronounce: coat-lay) expressly states that he used an octave divided into equal 1/3 tones, and also expressly refers to 19 pitches in the octave, in a preface of 1570, a year before Salinas mathematically defined the 1/3 comma mesotonic, which is aurally absolutely identical to the 19-tone equal. Probably this tuning was 'in the air', and both men independently discovered slightly different ways of defining the same system. Costeley finds the 19-tone equal far superior to any other tuning.

The relevant passage is printed in Les Maîtres Musiciens de la Renaissance Française, and precedes Costeley's celebrated chromatic motet "Seigneur Dieu Ta pitié". I give my translation, posted on this list circa 1998:

[After a greeting to his friends, and a general praise of music, there follows:]

"Now I do not doubt that you gentlemen find it strange that I should have exceeded in certain of my compositions the usual limits of the tones [i.e. the diatonics plus C#, Eb, F#, G# or Ab, & Bb - he implies that he has used Db, D#, E#, Fb, Gb,A#, B# & Cb in some of his compositions, which exceed the usual limits of the pythagorean tuning hitherto prevailing] of which usual limits I am not ignorant.

"I reply to these possible criticisms, first, that I have wished to provide the most excellent choristers of our Most Christian, Magnanimous and most Royally Born King of FRANCE (whom may God long preserve among us) with all that which might most please our Master. But I have done this [i.e. added these new tones to music] without ever going out of the key [this phrase admits of differing interpretations, but probably means that by the use of 1/3 tones he can more exactly preserve the mode when it has been modulated to a new key], and withal so that I might render our music more airy [he probably means, more harmonious, or perhaps less polyphonic and more dramatic, or else more freely modulated].

"As for the song that follows, I composed it 12 long years ago as an experiment, to render more practical an idea that I had, which I hoped should give a sweeter and more agreeable music than the diatonic, provided it were well and skillfully handled. This new music has its voices separated by intervals of one-third tone [instead of by unequal fractions of a tone, or else by more or less equal semitones]. And this [possibility] points up how far from perfection are the designs of our organs and spinets, inasmuch as they have but 7 diatonics and 5 accidentals in the octave, whereas perfection requires not 5, but 12 accidentals, which a good workman [with a skilful] design can introduce into the keyboard without making it unplayably complex. And when by these equal 1/3 tones we dispose the diatonics and accidentals in their natural order, we possess a marvelously new and pleasing instrument, without which the song which I have composed for it cannot be played.

"By using this 19-tone instrument tuned by 1/3 tones, we can always modulate [détonner] without discord [for the requisite accidental is always present, just as it is on the 12-tone keyboards, but with far less sweetness in the harmony, and the requisite accidental is by no means available in the 1/4 comma mesotonic, unless it be carried to an unmanageable 31 tones in the octave]. For we can always lower or raise a note by 1/3 or 2/3 of a tone as needed [to fit the new key].

"There is no further need to speak of semitones, for in this tuning there are none [he means, the chromatic and diatonic semitones are no longer equal, the former being but half the width of the latter]. Our lutes as usually tuned suffer from the same imperfection as our keyboard instruments, although by its natural sweetness even the most delicate ears rarely find anything amiss with it. [He means perhaps that good players adjust the unequal semitones of the mesotonic as needed, it being remembered that the lute's tones are not prolonged, which tends to mask bad harmony - or perhaps he even means that lute-players used an approximation of 12-tone equal temperament, though this is very doubtful.] Therefore the perfect music such as I have suggested has not been more practiced on the lute than on organs or spinets [despite the lute's movable frets], for it imperatively requires the use of all the 1/3 tones. Well-played violins have the advantage over the above-mentioned instruments in this regard, inasmuch as they can be played justly without the division of the octave into any particular intervals.

"Now the true difference between flats and sharps, between flats and naturals, or between sharps and naturals, is 1/3 tone. For example, between Bb and B is 1/3 tone, and between Eb and E is 1/3 tone again. But on the other hand between F# and G is 2/3 tone. I have marked this distinction whenever necessary for the sake of clarity. For most musicians and singers have hitherto confounded sharps and flats. But only when a G for example is twice flatted, is it the same as F#.

"As for all other information regarding this matter, I leave it sirs, to your most reliable and equitable judgement, which will permit you to benefit from my labors, both now and in the future. And in this spirit I pray God that He may keep you ever in His peace.
Paris, 1 January 1570

[End of Costeley's Preface]

Gregg Gibson

I am replying to my own post, to clarify and elaborate upon something I said.

>
> I am always puzzled when theorists treat 1/3 comma meantone as an oddity , and question whether it was ever used. I have read and translated Costeley's preface, and Mersenne's references to Titelouze. There is no doubt that 1/3 comma meantone was in current use, at least in France and Italy. Zarlino assures us that both were in common use at Rome and in Italy in general.

for "both were in common use" read "both 1/3 and 1/4 comma meantone were in common use"

Anyone who clearly understands how 1/4 comma meantone was invented, must also admit that 1/3 meantone was invented at virtually the same moment.

For consider: the musicians of the 16th century sucked in a knowledge of Pythagorean Intonation with their mother's milk, with its cycle of fifths. Anyone dissatisfied with the thirds and/or sixths of this system, would immediately look at the cycle of fifths. He would first find the major sixth three fifths above the prime, e.g.

D-A-E-B

He would note that the Pythagorean Major Sixth is sharper or wider than the 5/3 major sixth.

And he would then multiply 3/2 three times in succession to get 27/8, or within the reference octave 27/16. But he wants 5/3, the consonant major third. He would then take the difference of 27/16 and 5/3, i.e. he would divide 27/16 by 5/3, or multiply 27/16 by 3/5, yielding 81/80, the comma. Hence the major sixth in the cycle of fifths is one comma too wide. But the major sixth is separated from the prime by three fifths, so each fifth must be narrowed or diminished by 1/3 comma so that the whole comma may be distributed among the three fifths, and the major sixth narrowed by one comma, so that it becomes just.

Therefore it is the 1/3 comma meantone that was first discovered. Only later would the 1/4 comma meantone have been discovered, after someone noticed (or remembered from common tuning practice) that the fourth fifth above the prime is the major third.

Of course, one might argue that the theorists were more concerned about the major third than the major sixth, but this is questionable; they were concerned about all the thirds and sixths.

It is true that, all other things being equal, it might seem more attractive to temper a fifth by 1/4 comma than by 1/3 comma. But all other things are NOT equal. The 1/3 comma meantone very nearly closes the cycle of fifths after only 19 degrees, whereas the 1/4 comma meantone requires 31 degrees to do so, and these degrees are by no means so nearly equal. So the dreaded wolf is eliminated much more conveniently by 1/3 comma meantone than by any other tuning, except of course the 12-tone equal, but the harmony of that system is very poor because of its bad thirds and sixths.

Also, it may be noted that in most timbres it is easier to tune major sixths pure using the beats, than to tune the major thirds pure.

There is also the matter of the habits of thought of 16th century musicians as regards the accidentals. Musicians had been acquainted with 7 sharps and 7 flats since the 14th century, and 1/3 comma meantone remains reassuringly within this system. But double sharps and double flats were outside musicians' usual frame of reference, and the 1/4 comma meantone of course introduces this unfamiliar concept.

Therefore it is scarcely surprising that some musicians would have welcomed 1/3 comma meantone, while initially giving 1/4 meantone the cold shoulder. Vicentino was considered a madman, but Nenna founded a whole school of 1/3 meantone proponents.

However, the course of musical history finally favored 1/4 meantone, or rather a 12 to 17 tone selection of this tuning. It is interesting to speculate why this occurred.

For one thing, the Neapolitan Kingdom and Rome fell under Spanish influence and the Inquisition, and began to decline economically, while as the result of the 30-Years War Germany was ruined and France became the dominant European power for centuries. But France too was relatively backward in tuning lore compared to the Italians, despite Costeley and Titelouze. Mersenne preferred the just intervals, and could never fully understand why temperament was necessary at all, or desirable. He refers readers interested in temperament to Zarlino and Salinas. Finally, Sauveur, after considering 1/3, 1/4, 1/5 and 1/6 comma meantone, opted for 1/5 meantone, and his choice, together with 1/6 comma, became the usual tuning of the 18th century. 1/3 comma meantone seemed too great a temperament of the fifth, and the Renaissance interest in 19-key chromaticism was replaced by the modulated diatonicism of Haydn, with its self-confinement into 12 keys only. This effectively obscured the virtues of the 1/3 meantone, until even 1/4 comma meantone died out everywhere except in England, where however it survived almost till the end of the 19th century, perhaps because of the very ancient English preference for just thirds and sixths.

Underlying all this was the confirmed prejudice of keyboard players in favor of 12-tone instruments. As Helmholtz once commented, the convenience of musicians was permitted to take precedence over everything else... to the great discomfiture of those same musicians, who ended up with the horrible harmonies of the 12-tone equal.

Compared to the main voices on this list, I am relatively clueless about
these issues as well, apollo. This system of Vicentino is interesting to me,
though. I was wondering on a lark if there might happen to be a tuning based
on a 19+20 chain? The reason I ask is basically this: the lunar month is
29.5 days, so two months are 29+30 days (I number them 1 to 29 and 0 to 29).
20 days are defined as "special" (10 in each month) for this construction,
and thus 19+20 pitches might be associated. (Also, with 19, of course, a
tone might be associated with each year of the Metonic cycle.)

--- In tuning@yahoogroups.com, "gdsecor" <gdsecor@...> wrote:

> I dare you to put this tuning on a baroque organ (for testing, a synthetic one will do) and see if it doesn't knock your socks off.

I'm not going to dare anyone anything, but if you do a least squares optimization starting from either Werckmeister III or Vallotti-Young, the two best known circulating temperaments, you end up with the same "lesfip" scale:

! val-werck.scl
Vallotti-Young and Werckmeister III, 10 cents lesfip
12
!
87.16412
192.01793
296.87174
384.03586
502.60070
582.31031
696.70413
792.01793
887.33173
1001.72555
1081.43517
1200.00000

Daniel Nielsen <nielsed@...> wrote:
> Compared to the main voices on this list, I am relatively
> clueless about these issues as well, apollo. This system
> of Vicentino is interesting to me, though. I was
> wondering on a lark if there might happen to be a tuning
> based on a 19+20 chain? The reason I ask is basically
> this: the lunar month is 29.5 days, so two months are
> 29+30 days (I number them 1 to 29 and 0 to 29). 20 days
> are defined as "special" (10 in each month) for this
> construction, and thus 19+20 pitches might be associated.
> (Also, with 19, of course, a tone might be associated
> with each year of the Metonic cycle.)

Vicentino had two systems, manifested as two tunings of his
archicembalo (souped-up harpsichord). This instrument had
two manuals (keyboards) one of 19 notes and the other 17
notes to the octave. Each was a conventional (Halberstadt)
layout with split keys.

The first tuning, and the one most associated him, was
essentially 31 note equal temperament spread over the two
manuals. The 19 note manual was tuned to a meantone
chain. (The exact tuning isn't specified, so you could
call it 1/4 comma meantone or a subset of 31-equal.) 12
notes of the other manual were tuned to a different
meantone chain that you can think of as an extension of the
31-equal tuning. His notation distinguished between the
two manuals, but he also used equivalent spellings that
show he was thinking of a 31 note scale. The tuning of the
other 5 keys of the second manual is unclear.

The second tuning, and the one mentioned in this thread,
had both manuals tuned to meantone but the offset between
the two manuals chosen such that a root played on the first
and a fifth played on the second gave a true 3:2 perfect
fifth. Because quarter comma meantone has a true 5:4 major
third, it's possible to get just intonation on this
arrangement. But some meantone intervals are still going
to be outside just intonation. It's what we call a rank 3,
contorted system. The point of it is probably that your
root motions can follow the first manual, with the fifths
on the second manual to keep pure triads. That makes it a
simple form of adaptive temperament. Vicentino didn't
explain this, though. I think the first person to suggest
it, based on what Vicentino did say, was Paul Erlich.

Graham

--- In tuning@yahoogroups.com, "gregggibson" <gregggibson@...> wrote:

> It is now generally admitted that the Neapolitan School employed 1/3 comma meantone in preference to the 1/4 comma meantone. The kingdom was full of 19-tone keyboards at a time when 12- or 13-tone keyboards were the rule elsewhere. This is a commonplace of historical musicology. A few of them are still on display in Naples museums.

That they used 19 note keyboards does not prove they used them so as to circulate. Do you have a cite for that?

> It is sometimes asserted, after Ellis as I recall, that 19-tone keyboards could just as easily have been tuned in 1/4 comma meantone.

And he's right, hence the need for a cite. If someone said the minor thirds were tuned justly, that would suffice.

This would however have reintroduced the wolf. Moreover, there is no particular reason to select 19 of the 31 slightly unequal degrees of the 1/4 meantone. 17 or 26 are much more logical selections for Baroque music.

17 is a good deal more irregular, and 26 is a lot of notes; if you use that many, why not 31?

> I am always puzzled when theorists treat 1/3 comma meantone as an oddity , and question whether it was ever used. I have read and translated Costeley's preface, and Mersenne's references to Titelouze. There is no doubt that 1/3 comma meantone was in current use, at least in France and Italy. Zarlino assures us that both were in common use at Rome and in Italy in general.

What, specifically, did they say? Thanks, by the way, for the extended quotation from Costeley.

Thanks, Graham, for taking the time to provide that intelligible, concise,
and full explanation about Vicentino tuning history. BTW, your site led me
to this list last year, so thanks for that as well.

Dan N

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, "gdsecor" <gdsecor@> wrote:
>
> > I dare you to put this tuning on a baroque organ (for testing, a synthetic one will do) and see if it doesn't knock your socks off.
>
> I'm not going to dare anyone anything, but if you do a least squares optimization starting from either Werckmeister III or Vallotti-Young, the two best known circulating temperaments, you end up with the same "lesfip" scale:
>
> ! val-werck.scl
> Vallotti-Young and Werckmeister III, 10 cents lesfip
> 12
> !
> 87.16412
> 192.01793
> 296.87174
> 384.03586
> 502.60070
> 582.31031
> 696.70413
> 792.01793
> 887.33173
> 1001.72555
> 1081.43517
> 1200.00000

Hi Gene,

I'll take this as a challenge. I've plugged the numbers into a spreadsheet, and I've also tried it out in Scala. My reaction: Hmm, not bad, but it could be better.

Here are the things that could be improved upon:

1) The major thirds in the F, C, and G major triads are narrower than just. That lowers the error of the minor thirds in the F & C major triads, but why bother when 1/4-comma meantone triads (the "gold standard") would have done just fine, without paying the price in my next point.

2) The fifths on G and D are over 6.6 cents narrow and beat noticeably faster than 1/4-comma fifths.

3) There are only 5 major triads that are noticeably better than 12-equal triads; I have 6 in my 5/23-comma temperament extraordinaire.

4) There are 3 "worst" major triads, in which the major thirds are tempered an amount significantly greater than Pythagorean (i.e., by more than about 25 to 28 cents); I have only 2 in my 5/23-TX.

Both val-werck & 5/23-TX have 2 major triads apiece sounding similar to 12-equal and 2 sounding similar to Pythagorean, so those 4 triads apiece are a tossup. The bottom line is: if you use 5/23-TX instead of val-werck, you get a better-than-12-equal major triad (on A) instead of a very bad one (on B). (And I didn't even need to bring up proportional beating, of v-w has only one triad that even comes close: D major with brat of -3.996.)

I rest my case.

--George

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
>
> Fascinating. All the fifths seem to be very near 698 cents, but alternate to slightly different values. How did you calculate this?

I've pasted the Scala file here for reference:

! Secor5_23TX.scl
!
George Secor's synchronous 5/23-comma temperament extraordinaire
12
!
62/59
66/59
70/59
591/472
631/472
331/236
353/236
745/472
395/236
631/354
221/118
2/1

Notice that all of the numbers in the denominators are multiples of 59. Arriving at the above ratios was a process of trial & error using a spreadsheet in which I input the approximate cents values for each tone in the temperament and a trial denominator for the ratios. The spreadsheet calculated the numerator for the fraction for each tone that would yield a result closest to the cents value for that tone and the beat ratios (or brats) for all the major & minor triads produced by that set of ratios. Lower-number denominators tended to favor simple-number brats but didn't give very close approximations of the target pitches and usually resulted in one or more fifths excessively tempered (by over 6 or 7 cents). Increasing the denominator got the tuning closer to the target pitches more frequently, but didn't result in simple-number brats as often. 472 turned out to be the magic number that gave both simple brats and fifths within the desired range of size.

> --- On Thu, 2/17/11, gdsecor <gdsecor@...> wrote:
> >"This calls for a high-contrast circulating temperament, usable in all
> keys, some keys having better thirds than others, with as many good keys
> as possible. This will require tempering some of the 5ths wider than
> Pythagorean and some of the 3rds (??FLAT??)

The major 3rds are all wider than just and the minor 3rds all narrower than just.

> by an amount greater than Pythagorean
> (21.5 cents)"
>
> Makes sense...but how did you come up with that conclusion IE why is it that way?

If you have a chain of just fifths, then the resulting major and minor thirds are Pythagorean, 21.5 cents wide and narrow, respectively. If you make even one of the fifths in the chain wider than just, then the amount of error in the thirds increases (i.e., you have gone beyond Pythagorean).

--George

I would like to comment further on the tuning of early 19 tone harpsichords.

Whenever anyone hears of a 19-tone keyboard being built in the 19th or 20th centuries, he always assumes (correctly) that the maker wanted to experiment with the 19-tone equal temperament, or what amounts to the same thing melodically at least, 1/3 comma meantone.

But whenever these same musicologists meet with a 19-tone harpsichord from the 16th or 17th centuries, they always seem to immediately regurgitate the phrase "must have been tuned in the 1/4 comma meantone". Now most musicologists know little of temperament, and must take their opinions at second hand. No shame in that, unless the source used happens to be biassed, ignorant or untrustworthy. The first occurence of this opinion that I have found is in an article by Alexander Ellis, the English translator of Helmholtz. I do not have the article to hand, and will leave it to interested readers to locate, if they care to do so. I consulted it many years ago at the University of Georgia Music Library. In an appendix to Helmholtz added by the translator, he dismisses 19-tone equal with the terse comment that its fifths are out of the question. He did not bother to inquire if this minor defect could be remedied, but that is not so surprising; few theorists dared suggest tempering the octave until the 20th century, however slightly. Ellis, though quite influential, does not appear to have had much knowledge of early tunings; perhaps he could not read Latin and Italian fluently enough to consult the original sources.

Of course it is absurd to assert that 19-tone keyboards were ever built with 1/4 meantone in mind. I will now explain why.

The first 19-tone harpsichord occurring in musical literature was built to the order of Gioseffo Zarlino. He was a close friend of Francisco Salinas, from the 1550s to the 1580s. Now anyone who has read Salinas' _de musica_ is aware that the main subject of this work is Spanish folk song. This folk song is highly chromatic, and Salinas wanted to discover a tuning system which could exactly render it, and permit him to notate it. He therefore turned to the study of temperament. He discovered that 1/3 comma meantone, which he initially supposed to be his own invention, divided the octave into nineteen very nearly equal 1/3 tones, and that this system was alone adequate for his purposes of precisely rendering the chromatic complexities of Spanish folk song. He also suggested, echoing Aristoxenos, that ancient Greek singers of the enharmonic genus could not have accurately sung 1/4 tones, and must have substituted 1/3 tones instead. Salinas also studied the 1/4 comma meantone, but showed little interest in it, presumably because its 1/5 tones are too narrow to be sung. He was not hostile to it however, noting its just major thirds with approval. But it was not to his purpose.

Now Salinas, we may be sure, discussed these matters with Zarlino; if he did not, Zarlino would surely have read _de musica_ for it was almost as famous in its day as Zarlino's own works. Its Latin is elegant and clear, and a pleasure to read. And the Italian Zarlino was aware, unlike the provincial Spaniard Salinas, that 1/3 and 1/4 comma meantone had long been in use in Italy on 12- and 13-tone keyboards. Salinas says that Zarlino so informed him, and Zarlino also refers to the matter.There is nothing like reading the original sources to understand the past!

But apparently no one had as yet thought of having a 19-tone keyboard made, perhaps because no one was fully aware that 1/3 comma meantone does indeed very nearly close the cycle of fifths after 19 degrees, dividing the octave into 1/3 tones.

Zarlino proceeded to command such a keyboard to be made. I use the word `command' advisedly; Zarlino, the Choirmaster of St Marks in Venice, was the second most celebrated musician of all Europe, after only Palestrina himself. Zarlino tells us that he did so in order to investigate Salinas' discovery, and to have a perfect instrument. But this strongly implies that he intended it to be tuned in 1/3 comma meantone, with no wolves, and hence `perfect'. That is what this adjective usually means in Renaissance Latin and Italian, when applied to instruments. He would scarcely have thought of ordering such an instrument PRIMARILY to investigate 1/4 comma meantone, for a 19-tone selection of 1/4 comma meantone does not eliminate the wolf, and is not otherwise by any means a natural one, as I will soon show.

Now when the famed Zarlino ordered such a 19-tone keyboard, and had an engraving of it made to be inserted in his latest work of musical theory, all Europe drew its collective breath.
Soon every musician lay awake at night, dreaming of obtaining the perfect harpsichord or clavi-organ. And this is true even though Zarlino himself was troubled slightly by the flat fifths of the 1/3 comma meantone, though not so troubled that he did not hasten to propose his own compromise between 1/3 comma meantone and 1/4 comma meantone, 2/7 comma meantone. Of course, he never thought of stretching the octave to improve the fifths of 1/3 comma meantone, at the cost of a negligible deterioration of the fourth; tempering the octave, however slightly, was outside his world-view, and one gets the impression that he was not as rigorous a thinker as Salinas. After all, he was basically an artist, not a mathematician.

Now I turn to why it is logically most unlikely that anyone would order a 19-tone keyboard in order to tune it in 1/4 comma meantone.

First, I observe that 19-tone instruments fit 1/3 comma meantone like a glove, because in that tuning the augmented third and the diminished fourth, e.g. C-E# & C-Fb, are only a few cents apart, so that they can be confounded with each other, as also are the augmented seventh and the diminished octave, e.g. C-B# & C-Cb. This effectively reduces the number of accidentals from 14 (7 flats and 7 sharps) to 12.

But this is NOT the case for 1/4 comma meantone, Where these two pairs differ by about 35 cents in width. Hence anyone desiring access to all the first group of accidentals (single sharps and single flats) of the 1/4 comma meantone would certainly not order a 19-tone keyboard, but a 21-tone keyboard. I will note in passing that the most heavily used keys of Handel's time require 16 or perhaps 17 notes of the 1/4 comma meantone; the rarer keys raise this to 26 or perhaps 27; obviously one cannot be exact in such a fluid matter as key ranges.

Second, I again observe that a 19-tone selection of the 1/4 comma meantone makes no sense as far as eliminating the wolf, whereas of course 1/3 comma meantone has no wolf, when carried out to 19 tones.

Third, there is a great deal of evidence that when musicians really did use 1/4 comma meantone, they were content merely to "push the wolf away from the door" by slowly and reluctantly splitting first the G#/Ab accidental, and then others, but almost never beyond 16 tones in the octave. Only the advocates and users of 1/3 comma meantone had any compelling reason to go as high as 19 tones at one stroke, and then to be content to stay at 19.

Fourth, in most timbres it is much easier to tune the just major sixth by beats than the just major third, or even the just major tenth. This may be why musicians who used 1/4 comma meantone, with its just major thirds, were not enthusiastic about tuning more than 12 or 13 notes in the octave, whereas musicians who used 1/3 comma meantone were not discouraged, at least not much, by tuning 19 notes in the octave.

Finally, I would like to comment on a tendency I have noted among musical theorists to regard 1/4 comma meantone as much more usual than it really was. They seem to assume that since 12-tone equal has become almost universal during the last century or so, earlier centuries must have had a dominant tuning at all times. This is not at all true. Within a century or so after the invention of 1/3 and 1/4 comma meantone, there were a great many tunings in Europe other than the Pythagorean; by 1700 Sauveur assures us that French musicians preferred 1/5 comma meantone, and by the eighteenth century there were dozens of more-or-less usual tunings, equal and unequal. Also, because only 1/4 comma meantone results in a tone exactly intermediate in width between the minor tone 10:9 and the major tone 9:8, there is sometimes a disposition to regard it as somehow more `orthodox' than the other meantone temperaments. This is however unjustified; the older theorists were little concerned with the exact size of the tone; the consonances were what worried them.

From the very beginnings of temperament, the world has been largely divided into those who want to somehow tweak 12 tones in the octave into sounding a little better, and those who are willing and eager to make the jump to 19 actual tones in the octave. In my opinion there is no way to make a silk purse out of a sow's ear, and we should make the jump to 19. Terrestrial brains appear to have made this jump hundreds of thousands, or even hundreds of millions of years ago, because, as I have but very recently discovered, it is about 30 times more efficient to express the 19-tone equal temperament in binary code than any other tuning. Music should follow the native tuning of the brain.

Gregg Gibson

--- In tuning@yahoogroups.com, "gregggibson" <gregggibson@...> wrote:

> Terrestrial brains appear to have made this jump hundreds of thousands, or even hundreds of millions of years ago, because, as I have but very recently discovered, it is about 30 times more efficient to express the 19-tone equal temperament in binary code than any other tuning. Music should follow the native tuning of the brain.

You made a nice case, but at the very end you seemingly run off the rails. I suspect leaving neurophysiology and evolutionary biology to the experts would be a good plan, but if this "binary code" can be expressed mathematically without dragging the Cretaceous period into it, there are people here who speak math.

--- In tuning@yahoogroups.com, "gregggibson" <gregggibson@...> wrote:
> ...
> From the very beginnings of temperament, the world has been largely divided into those who want to somehow tweak 12 tones in the octave into sounding a little better, and those who are willing and eager to make the jump to 19 actual tones in the octave. In my opinion there is no way to make a silk purse out of a sow's ear, and we should make the jump to 19. ...
>
> Gregg Gibson

Greg, lest you get the mistaken impression from my recent messages on 12-tone circulating temperaments that I'm in the former group, let me make it clear that I'll take 19-equal over a circle of 12 fifths (of whatever kind) any day.

In my view, the microtonal world has been largely divided into those who would make a microtonal keyboard by tacking extra keys onto a conventional keyboard and those who would follow Bosanquet's lead in using a generalized keyboard with the property of transpositional invariance. For an example, see Erv Wilson's design for a 19-tone clavichord that was built in the 1970's for Scott Hackleman (on the first page of this link):
http://anaphoria.com/Clavichord19-17-22.pdf

The drawing suggests that additional key surfaces (for E# and B# as dashed hexagons) can be added to the key levers for Fb and Cb, but I would recommend that 10 additional surfaces be added in the same manner for 5 double sharps and 5 double flats in order to provide transpositional invariance in virtually all keys.

--George

Thought for the day: The world is divided into two kinds of people: those who divide the world into two kinds of people, and those who don't.

Although I do naturally have settled opinions on temperament, I do not have a deep emotional involvement in these matters. I will not issue a fatwa against you if you do not conform to my own preferences. The only people that I find annoying are the academic worshippers of the 12-tone equal like Barbour, who wrote a book on the equal temperaments, and then confessed that he had never bothered to listen to any of them... everyone knew 12-tone equal was perfect. Apparently he only studied temperament as an empty academic exercise. Alas, his book is still quoted as an authority.

There is however a certain limited aspect of the brain's binary code that favors 12-tone equal almost as much as 19-tone equal, but I confess I am not eager to point this out, for well I know with what joy our conservatories will welcome such a finding, while ignoring everything else.

The division in the microtonal community is perhaps best described as the old quarrel between the Pythagoreans, who love ratios, and Aristoxenians, who are skeptical that these ratios really exist immanently in the brain, and would rather speak of divisions of the tone or of the octave. The melodic limen strongly favors the latter school, while the beating phenomenon somewhat favors the former, but in most timbres only for some members of the senario and perhaps a septimal or two. If pressed I would describe myself as a moderate Aristoxenian. At one time I pored over the ratios with as much devotion as anyone could wish, however.

I have been working on a theory that attempts to transcend this division by directly examining the binary code used by the deep brain. It would appear from the evidence that the brain uses both the ratios and the melodic limen, but not the exact ratios; it prefers a fifth of about 696 cents, for example. Also, from quite a different perspective, the 19-tone equal with a stretched octave happens to fit this binary code very exactly, much more so than any other equal temperament through 55-tone equal.

As far as keyboards are concerned, I have the impression that neither of the two approaches you mention is entirely adequate for the 19-tone equal. One really wants the Holy Graal: a 19-tone equal keyboard that can be played faster and easier than the traditional 12-tone design, so that a concert virtuoso is satisfied with it. I have recently invented a design which does seem to be more easily manipulated than the Renaissance solution of just splitting accidentals. Of course, it would be unwise to publish such a design until it is patented. I can say however that it is about halfway between General Bosanquet's keyboard and one of Wilson's designs. I have tried at least a score of 19-tone designs over the years, and I believe this is perhaps the best solution.

By the way, I will take this opportunity to add to my post on Alexander Ellis' dislike of the 1/3 comma meantone. Everything and everyone having anything to do with it he treated as a personal enemy. He went so far as to publish an article arguing that Salinas did not know what he was taking about regarding 1/3 comma meantone, and suggesting that he was really describing 1/6 comma meantone instead. But as others have pointed out before me, for example in the Journal of Music Theory, it was Ellis himself who was in error - his knowledge of Latin was rudimentary. Salinas description is absolutely clear, indeed a model of patient clarity, and nobody but Ellis has ever had any trouble understanding it. I have the distinct impression that Ellis was one of those researchers who confuse ideas with people, and resort to all sorts of tricks to discredit an idea they don't like. His translation of Helmholtz is unreliable; for those with shaky German it is safer to use the French translation as a crib instead. But perhaps I am falling into my own case of personal dislike. Elis was an ugly fellow too.

Ellis also invented the cent, which is heavily biassed in favor of the 12-tone equal, and succeeded in persuading the anglophone world to use it in preference to the savart, which is of course is in much more direct contact with the ear's logarithmic hearing. Ever since we have had to confront musicians convinced that the cents somehow `prove' that there are 12 equal semitones in the octave. Helmholtz by the way is said to have become very disturbed at the liberties Ellis took with his German.

In short, Ellis... bad boy, bad boy. But his appendix to Helmholtz is full of interesting material, if one treats it with caution.

Gregg Gibson

What exactly is this "binary code" that you keep speaking about?

-Mike

On Sun, Feb 20, 2011 at 2:17 AM, gregggibson <gregggibson@...> wrote:
>
> I have been working on a theory that attempts to transcend this division by directly examining the binary code used by the deep brain. It would appear from the evidence that the brain uses both the ratios and the melodic limen, but not the exact ratios; it prefers a fifth of about 696 cents, for example. Also, from quite a different perspective, the 19-tone equal with a stretched octave happens to fit this binary code very exactly, much more so than any other equal temperament through 55-tone equal.

--- In tuning@yahoogroups.com, "gregggibson" <gregggibson@...> wrote:

> By the way, I will take this opportunity to add to my post on Alexander Ellis' dislike of the 1/3 comma meantone. Everything and everyone having anything to do with it he treated as a personal enemy. He went so far as to publish an article arguing that Salinas did not know what he was taking about regarding 1/3 comma meantone, and suggesting that he was really describing 1/6 comma meantone instead.

Which involved him in the preposterous claim that Salinas could not have meant 1/3 comma was "languid", because that better fits 1/6 comma.

> Ellis also invented the cent, which is heavily biassed in favor of the 12-tone equal, and succeeded in persuading the anglophone world to use it in preference to the savart, which is of course is in much more direct contact with the ear's logarithmic hearing.

Cents are logarithms and so are savarts. Logarithms are all equally logarithmic, though logs base 10^(1/1000) (savarts) might be preferred to logs base 2^(1/1200) (cents) if you are a base ten fanatic. For musical purposes, 2 is more important than 10 and dividing an octave into an integer number of parts should be preferred. You could even argue it's in more direct contact with how hearing works to do it that way, I suppose, since octaves are more significant than ratios of 10.

--- In tuning@yahoogroups.com, "gregggibson" <gregggibson@...> wrote:
>
> Although I do naturally have settled opinions on temperament, I do not have a deep emotional involvement in these matters. I will not issue a fatwa against you if you do not conform to my own preferences. The only people that I find annoying are the academic worshippers of the 12-tone equal like Barbour, who wrote a book on the equal temperaments, and then confessed that he had never bothered to listen to any of them... everyone knew 12-tone equal was perfect. Apparently he only studied temperament as an empty academic exercise. Alas, his book is still quoted as an authority.

Yes, and then other folks read these things and probably figure that, except for more "historically accurate" performances of old music, alternative tunings, are, for the most part, a waste of time. This is what virtually all of here are up against.

I remember reading one comment about 19-equal, probably Barbour's, where the writer offered the opinion that, judging by the numbers, the major 3rds will sound "insipid in the extreme".

> There is however a certain limited aspect of the brain's binary code that favors 12-tone equal almost as much as 19-tone equal, but I confess I am not eager to point this out, for well I know with what joy our conservatories will welcome such a finding, while ignoring everything else.

I don't quite know what to make of this. I think 12-equal tends to be favored for its diatonic melodic properties, but 19-equal for its diatonic harmonic properties. However, 19-equal also provides new melodic intervals that can trump 12, including an alternate (higher) leading tone that makes a wide dissonant third with the dominant that very effectively resolves (both melodically & harmonically) to the tonic tone by 63 cents. Another thing that can be done in 19, but not in 12, is a rendering of the ancient Greek enharmonic genus.

> The division in the microtonal community is perhaps best described as the old quarrel between the Pythagoreans, who love ratios, and Aristoxenians, who are skeptical that these ratios really exist immanently in the brain, and would rather speak of divisions of the tone or of the octave. The melodic limen strongly favors the latter school, while the beating phenomenon somewhat favors the former, but in most timbres only for some members of the senario and perhaps a septimal or two. If pressed I would describe myself as a moderate Aristoxenian. At one time I pored over the ratios with as much devotion as anyone could wish, however.

Isn't the observation that the most acoustically consonant intervals are expressed as simple-number ratios (due to the elimination of beats) sufficient grounds for affirming that ratios are the single, most significant in determining which intervals are most consonant? Is not the justification for dividing the octave into 12, 19, or 31 parts based on the fact that these are the ratios that we're attempting to approximate? Although composers such as Igs and Herman Miller have chosen to concentrate on divisions of the octave that do not closely approximate these ratios for their compositions, the rest of us here have no quarrel with them, because we believe that virtually any kind of tuning can be used to produce worth-while music, and that the skill of the composer in working with the tonal materials is a greater factor for success than the tuning itself.

> I have been working on a theory that attempts to transcend this division by directly examining the binary code used by the deep brain.

I don't understand this. Are you claiming to have deciphered the circuitry in the part of the neural network of the human brain that processes musical sounds?

> It would appear from the evidence that the brain uses both the ratios and the melodic limen, but not the exact ratios; it prefers a fifth of about 696 cents, for example.

Personally, I would have to disagree with this last statement. I find slightly wide fifths (of around 703 to 705 cents) preferable to narrow ones. One of my favorite tunings has fifths tempered wide by ~1.6237 cents in 3 separate chains: one for 3-limit intervals, the second one 5:4 above the first (for ratios of 5), and the third 7:4 above the first (for ratios of 7, 11, and 13). No 15-limit consonance (i.e., an interval ratio expressed using any two integers 15 or lower in value, or inversion thereof) is tempered by more than 3.25 cents. The tuning sounds very much like just intonation (complete with what sounds for all the world like periodicity buzz in isoharmonic chords), and the melodic properties are excellent (IMO).

> Also, from quite a different perspective, the 19-tone equal with a stretched octave happens to fit this binary code very exactly, much more so than any other equal temperament through 55-tone equal.

???

> As far as keyboards are concerned, I have the impression that neither of the two approaches you mention is entirely adequate for the 19-tone equal.

For the benefit of others, it would be good to mention that it appears that you're replying to this message:
/tuning/topicId_96310.html#96408

> One really wants the Holy Graal: a 19-tone equal keyboard that can be played faster and easier than the traditional 12-tone design, so that a concert virtuoso is satisfied with it. I have recently invented a design which does seem to be more easily manipulated than the Renaissance solution of just splitting accidentals. Of course, it would be unwise to publish such a design until it is patented.

I'd say that it's not worth the time, effort, and $$$ to patent a microtonal keyboard. You'll never recoup your investment, because no one will use it if they have to pay you to do so, as long as there are alternatives free for the asking.

> I can say however that it is about halfway between General Bosanquet's keyboard and one of Wilson's designs. I have tried at least a score of 19-tone designs over the years, and I believe this is perhaps the best solution.

It sounds as if whatever your're advocating has transpositional invariance (which is good), but differs from the Bosanquet and Wilson designs in the shape & dimensions of the keys (which I hope are essentially all the same size & shape) and perhaps in the properties of the key surfaces (touch-coding via differences in texture being desirable, as are a key surface that's not perfectly flat (i.e., either slightly convex or concave), so that the player is able to discern by feel the portion of the key on which the finger is making contact. I already have a key shape intermediate between Bosanquet's and Wilson's in my diagram of a keyboard designed for tunings and octave divisions in the pajara temperament class (which have tones in two chains of wide fifths, with the chains 600 cents apart):
/tuning-math/files/secor/kbds/KbPaj34-46.gif
(You must be a member of the tuning-math group to access this file. Most of my files are there, because that's where space was available at the time I began uploading them.)

Since both touch-coding and convex key surfaces have already been used on the Motorola Scalatron with generalized keyboard (1975, using elliptical keys, which I expect you won't find in a patent search, because it wasn't considered worthwhile to attempt a patent), I can't imagine what feature(s) you could have come up with that might be patentable unless it's some particular *combination* of features that those of us who have designed keyboard might regard as too obvious to be suitable for a patent. If it's only a variation on Bosanquet that consists of a particular key size & shape, then there's probably a gray area regarding how close to your key dimensions someone else may come without infringing on your potential patent. Several decades ago Erv Wilson released quite a few drawings showing considerable variation of size and proportion for hexagonal keys, so the idea of modifications on Bosanquet's original idea involving variations in size, proportion, and key shape are by this time rather obvious and IMO undeserving of a patent.

> By the way, I will take this opportunity to add to my post on Alexander Ellis' dislike of the 1/3 comma meantone. Everything and everyone having anything to do with it he treated as a personal enemy. He went so far as to publish an article arguing that Salinas did not know what he was taking about regarding 1/3 comma meantone, and suggesting that he was really describing 1/6 comma meantone instead. But as others have pointed out before me, for example in the Journal of Music Theory, it was Ellis himself who was in error - his knowledge of Latin was rudimentary. Salinas description is absolutely clear, indeed a model of patient clarity, and nobody but Ellis has ever had any trouble understanding it. I have the distinct impression that Ellis was one of those researchers who confuse ideas with people, and resort to all sorts of tricks to discredit an idea they don't like. His translation of Helmholtz is unreliable; for those with shaky German it is safer to use the French translation as a crib instead. But perhaps I am falling into my own case of personal dislike. Elis was an ugly fellow too.

It's been a long time since I've read Ellis's translation of Helmholz (I have the book), so for now I'll take your word for it, because I've seen enough errors and half-truths on the part of so-called "authorities" that I would have no trouble believing that.

> Ellis also invented the cent, which is heavily biassed in favor of the 12-tone equal, and succeeded in persuading the anglophone world to use it in preference to the savart, which is of course is in much more direct contact with the ear's logarithmic hearing. Ever since we have had to confront musicians convinced that the cents somehow `prove' that there are 12 equal semitones in the octave.

Cents and savarts (or heptamerides, or degrees of 301-EDO) are both logarithmic units of measure, so there is no merit in arguing that one has any advantage over the other on the basis of the way we hear pitches. However, I quite agree with your observation of the 12-equal bias inherent in cents. As I will demonstrate below, there are distinct advantages in thinking outside the 12-tone squirrelcage.

The historical advantage in using savarts was that you could use a four-place base-10 log table to look up the number of savarts in a ratio (expressed as a decimal), since an octave (with ratio 2/1) is exactly 301 savarts and the base-10 logarithm of 2 is .3010 (to 4 decimal places). However, with the wide availability of computers and scientific calculators, this is no longer necessary.

A more significant advantage in using savarts is that, because 301-EDO is a fairly good division for approximating harmonics and just ratios (it's 17-limit consistent), you can round the values for many just ratios off to whole numbers and use them in calculations without the rounding errors that occur with whole-number cents.

The chief problem with savarts is that, rounded to whole numbers, they lack the precision of cents. For example, if I want to compare the interval 8505:8192 (the apotome-complement of 35:36) with 26:27, I would not be able to use whole-number savarts, since these two ratios differ by only ~0.4 cents -- much less than 1 savart. (This is not a contrived example; these are actual ratios that came up in the process of my devising symbols for microtonal accidentals.) Since I would have to use savarts with decimal places, I would then lose the advantage of having a unit of interval measure that can be expressed as whole numbers.

Fortunately, there is a better alternative. My own nomination for a unit of measure is the "mina" (short for "schismina", a term Dave Keenan coined to represent a class of interval smaller than a schisma). One mina is 1/2460th of an octave (alternatively defined as 1/233rd of an apotome), which is slightly less than 1/2 cent. Since 2460-EDO is 27-limit consistent and is far more accurate (in both absolute and relative terms) than lesser octave divisions, it allows many complicated interval calculations free of rounding errors. In the foregoing example, 8505:8192 is 133.075 minas and 26:27 is 133.942 minas, which round to 133 and 134 minas, respectively. It's no accident that the decimal figures are so close to integers: the 2460 division, first suggested by Gene Ward Smith, is highly consistent at the 27 limit.

Since 2460 is divisible by 12, a semitone of 12-equal is exactly 205 minas. Didymus' comma (80:81) is 44.088 minas, a number which, when rounded to 44, is divisible by 4, so the tones of 1/4-comma meantone can also be expressed in whole-number minas.

The usefulness of the mina as a unit of measure doesn't stop there. 2460 is also divisible by 41, so the tones of 41-EDO can be expressed exactly in whole minas, and they happen to be multiples of 60. Since 41 also happens to be an excellent division (15-limit consistent), it's possible to express intervals as degrees of 41 plus leftover minas in exactly the same manner that one writes hours and minutes of time. For example:

Ratio Minas [degrees of 41]:[minas]
2:3 1439.008 23:59 - rounded from 23:59.008
4:5 791.943 13:12 - rounded from 13:11.943
5:6 647.065 10:47 - rounded from 10:47.065
4:7 1986.093 33:06 - rounded from 33:06.093
80:81 44.088 0:44 - rounded from 0:44.088

Thus you can round off the numbers in column 2 to whole minas (1439.008 to 1439), or you can round the numbers in column 3 (23:59.008 to 23:59 or, more roughly, to 24), since both the 2460 and 41 divisions allow interval calculations to be performed (within certain limits) without rounding errors. Of course, these limits are more generous with minas than with degrees of 41. (Note that 23:59 is *not* the decimal number 23.59. You must read it as you would read hours and minutes of time, observing that it's only one mina less than 24:00.)

In case you didn't already know, Dave Keenan and I have also devised a system of microtonal accidentals (called Sagittal) capable of distinguishing pitches differing by a single mina. It consists of a single set of symbols that can be used to notate just intonation (to at least the 31 limit), a huge number of EDO's, many different temperament classes, and other kinds of tunings. Although designed for a conventional staff, the Sagittal accidentals may also be used in notations that use staves having other than 7 positions per octave.

> Helmholtz by the way is said to have become very disturbed at the liberties Ellis took with his German.
>
> In short, Ellis... bad boy, bad boy. But his appendix to Helmholtz is full of interesting material, if one treats it with caution.

Thanks, Gregg. I'll take that under advisement.

--George

--- In tuning@yahoogroups.com, "gdsecor" <gdsecor@...> wrote:

> I remember reading one comment about 19-equal, probably Barbour's, where the writer offered the opinion that, judging by the numbers, the major 3rds will sound "insipid in the extreme".

Barbour didn't even like the major third of 22 much.

> Another thing that can be done in 19, but not in 12, is a rendering of the ancient Greek enharmonic genus.

Not to mention keemun, negri, godzilla, triton and magic/muggles temperaments.

> > I have been working on a theory that attempts to transcend this division by directly examining the binary code used by the deep brain.
>
> I don't understand this. Are you claiming to have deciphered the circuitry in the part of the neural network of the human brain that processes musical sounds?

> Personally, I would have to disagree with this last statement. I find slightly wide fifths (of around 703 to 705 cents) preferable to narrow ones.

I concur, on most days. But narrower fifths are more mellow in sound, which you might want.

> One of my favorite tunings has fifths tempered wide by ~1.6237 cents in 3 separate chains: one for 3-limit intervals, the second one 5:4 above the first (for ratios of 5), and the third 7:4 above the first (for ratios of 7, 11, and 13).

Have you ever worked with mystery temperament?

> > I can say however that it is about halfway between General Bosanquet's keyboard and one of Wilson's designs. I have tried at least a score of 19-tone designs over the years, and I believe this is perhaps the best solution.

Bosanquet's brother was an admiral, but you probably mean Thomas Perronet Thompson.

> Fortunately, there is a better alternative.

Several better ones, in fact. You might also mention 612, which is divisible not only by 12 but also 17, which you ought to like. Then there's Woolhouse's 730, and Woolhouse has priority in this matter. 730 is not 29-limit consistent, but its patent val does score better in the 29 limit than anything smaller, including consistent divisions. 882et is the first to beat it.

Still, for general xenharmonic purposes it's hard to beat the mina, except for the little fact that cents are already established. I'd use it if not for that.

On Mon, Feb 21, 2011 at 1:53 PM, gdsecor <gdsecor@...> wrote:
>
> > There is however a certain limited aspect of the brain's binary code that favors 12-tone equal almost as much as 19-tone equal, but I confess I am not eager to point this out, for well I know with what joy our conservatories will welcome such a finding, while ignoring everything else.
>
> I don't quite know what to make of this. I think 12-equal tends to be favored for its diatonic melodic properties, but 19-equal for its diatonic harmonic properties. However, 19-equal also provides new melodic intervals that can trump 12, including an alternate (higher) leading tone that makes a wide dissonant third with the dominant that very effectively resolves (both melodically & harmonically) to the tonic tone by 63 cents. Another thing that can be done in 19, but not in 12, is a rendering of the ancient Greek enharmonic genus.

12-equal is also favored for music that makes use of the beautiful
colors that emerge when 128/125 vanishes. As we were discussing in the
other thread about major thirds vs diminished fourths, this often
turns up in subtle ways that you don't even realize at first. A cheesy
and obvious example of this is if you're playing major chords that you
keep transposing up in major thirds, expecting to hit the octave again
after three iterations. A less cheesy and less obvious example of this
is a scale like C C# D# E F# G# A C, which is a fusion of C D# E F# G#
A B C and C Db Eb Fb Gb Ab Bbb C.

> Personally, I would have to disagree with this last statement. I find slightly wide fifths (of around 703 to 705 cents) preferable to narrow ones. One of my favorite tunings has fifths tempered wide by ~1.6237 cents in 3 separate chains: one for 3-limit intervals, the second one 5:4 above the first (for ratios of 5), and the third 7:4 above the first (for ratios of 7, 11, and 13). No 15-limit consonance (i.e., an interval ratio expressed using any two integers 15 or lower in value, or inversion thereof) is tempered by more than 3.25 cents. The tuning sounds very much like just intonation (complete with what sounds for all the world like periodicity buzz in isoharmonic chords), and the melodic properties are excellent (IMO).

I find the fifths of 17-equal to be far more pleasant than the fifths
of 19-equal. 22-equal is where it gets bit too far out in my opinion.

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> 12-equal is also favored for music that makes use of the beautiful
> colors that emerge when 128/125 vanishes.

Not everyone likes those "beautiful colors".

On Mon, Feb 21, 2011 at 3:33 PM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > 12-equal is also favored for music that makes use of the beautiful
> > colors that emerge when 128/125 vanishes.
>
> Not everyone likes those "beautiful colors".

Some people don't even like the "beautiful colors" that emerge when
81/80 vanishes.

-Mike

>
> I remember reading one comment about 19-equal, probably Barbour's, where the writer offered the opinion that, judging by the numbers, the major 3rds will sound "insipid in the extreme".

Yes, long before I read that he had finally confessed never to have listened to the temperaments he presumed to judge in print, his strange statements led me to suspect his honesty. GG

>
> > There is however a certain limited aspect of the brain's binary code that favors 12-tone equal almost as much as 19-tone equal, but I confess I am not eager to point this out, for well I know with what joy our conservatories will welcome such a finding, while ignoring everything else. GG
>
> I don't quite know what to make of this. I think 12-equal tends to be favored for its diatonic melodic properties, but 19-equal for its diatonic harmonic properties. However, 19-equal also provides new melodic intervals that can trump 12, including an alternate (higher) leading tone that makes a wide dissonant third with the dominant that very effectively resolves (both melodically & harmonically) to the tonic tone by 63 cents. Another thing that can be done in 19, but not in 12, is a rendering of the ancient Greek enharmonic genus.

I am sorry, but it is premature for me to explain further on this forum. After I have published my theory, I will be glad to answer your questions about it, time permitting. I will however observe that to my ears 12-tone equal sounds very mistuned even in diatonic melody. GG

>
> > The division in the microtonal community is perhaps best described as the old quarrel between the Pythagoreans, who love ratios, and Aristoxenians, who are skeptical that these ratios really exist immanently in the brain, and would rather speak of divisions of the tone or of the octave. The melodic limen strongly favors the latter school, while the beating phenomenon somewhat favors the former, but in most timbres only for some members of the senario and perhaps a septimal or two. If pressed I would describe myself as a moderate Aristoxenian. At one time I pored over the ratios with as much devotion as anyone could wish, however. GG
>
> Isn't the observation that the most acoustically consonant intervals are expressed as simple-number ratios (due to the elimination of beats) sufficient grounds for affirming that ratios are the single, most significant in determining which intervals are most consonant? Is not the justification for dividing the octave into 12, 19, or 31 parts based on the fact that these are the ratios that we're attempting to approximate? Although composers such as Igs and Herman Miller have chosen to concentrate on divisions of the octave that do not closely approximate these ratios for their compositions, the rest of us here have no quarrel with them, because we believe that virtually any kind of tuning can be used to produce worth-while music, and that the skill of the composer in working with the tonal materials is a greater factor for success than the tuning itself.

But of course you understand that the presence of beats is dependent upon the presence of the relevant partial pairs; they are by no means invariably present. Few timbres will have all the partials present necessary even to define all members of the senario by beats. For example, if the eighth or fifth partial is weak, or their multiples, then 8:5 will NOT be surrounded by audible beats. This is one of the basic flaws with Helmholtz' theory; he himself recognized this, and offered his theory of consonance in a somewhat diffident manner. The modern just intonationists, with their worship of the ratios, seized upon Helmholtz' theory for their own purposes. GG

It is true that the senario favors the 19- and 31-tone equal among others, but this is by no means the only consideration. The melodic limen of 55-60 cents favors the 19-tone equal, and no other tuning. And the brain's binary code also strongly favors it, quite apart from the simple ratios. GG

By the way, another correspondent asserts that cents are logarithms. They are not. Neither are savarts. Only logarithms are logarithms. One can of course divide the octave into any artificial number of measuring degrees that one desires by the use of a conversion factor, but the resulting artificial division is not a logarithm. GG

>
> > I have been working on a theory that attempts to transcend this division by directly examining the binary code used by the deep brain. GG
>
> I don't understand this. Are you claiming to have deciphered the circuitry in the part of the neural network of the human brain that processes musical sounds?

Yes. GG

>
> > Ellis also invented the cent, which is heavily biassed in favor of the 12-tone equal, and succeeded in persuading the anglophone world to use it in preference to the savart, which is of course is in much more direct contact with the ear's logarithmic hearing. Ever since we have had to confront musicians convinced that the cents somehow `prove' that there are 12 equal semitones in the octave. GG
>
> Cents and savarts (or heptamerides, or degrees of 301-EDO) are both logarithmic units of measure, so there is no merit in arguing that one has any advantage over the other on the basis of the way we hear pitches. However, I quite agree with your observation of the 12-equal bias inherent in cents. As I will demonstrate below, there are distinct advantages in thinking outside the 12-tone squirrelcage.

I did not say that the savart is a logarithm. But because its conversion factor is 1000, it is less likely to confuse beginners, who often do not fully realize that we measure intervals in logarithms, so that we may replace multiplication and division of ratios by addition and subtraction. It is incidentally somewhat more logical to use base 2 logarithms, since the octave equals 2, not 10. But this is really trivial of course. GG

>
> The historical advantage in using savarts was that you could use a four-place base-10 log table to look up the number of savarts in a ratio (expressed as a decimal), since an octave (with ratio 2/1) is exactly 301 savarts and the base-10 logarithm of 2 is .3010 (to 4 decimal places).

Yes, but there are not EXACTLY 301 savarts in the octave; there are more digits. Usually at least 8 digits are used. The errors quickly become serious otherwise. GG

>
> A more significant advantage in using savarts is that, because 301-EDO is a fairly good division for approximating harmonics and just ratios (it's 17-limit consistent), you can round the values for many just ratios off to whole numbers and use them in calculations without the rounding errors that occur with whole-number cents.

The so-called 17-limit is an artefact, not a reality, because the brain has no way to identify such intervals, save in the unlikely event that such high partial pairs are present. But such timbres are very harsh and unmusical. Personally I always carry cents and savarts out to at least 8 digits. GG

>
> The chief problem with savarts is that, rounded to whole numbers, they lack the precision of cents. For example, if I want to compare the interval 8505:8192 (the apotome-complement of 35:36) with 26:27, I would not be able to use whole-number savarts, since these two ratios differ by only ~0.4 cents -- much less than 1 savart. (This is not a contrived example; these are actual ratios that came up in the process of my devising symbols for microtonal accidentals.) Since I would have to use savarts with decimal places, I would then lose the advantage of having a unit of interval measure that can be expressed as whole numbers.

Other than for avoiding calculation errors, there is no particular point in greater accuracy than the differential limen however. No one can hear a cent difference. Even the savart is near the edge of differential perception. GG

< Although composers such as Igs and Herman Miller have chosen to <concentrate on divisions of the octave that do not closely approximate these ratios for their compositions, the rest of us here have no quarrel with them, because we believe that virtually any kind of tuning can be used to produce worth-while music, and that the skill of the composer in working with the tonal materials is a greater factor for success than the tuning itself.

These are followers of Stockhausen. You have no quarrel with them? I doubt that everyone here views such experiments with enthusiasm. Although I do not wish to persecute such experiments, they are entirely in contradiction with common musical experience. No composer of the past, however open-minded, would have accepted such an idea. They would have assumed the theorist who proposed it to be insane. I personally do not feel that strongly, but it is a very strange idea, and entirely outside my musical world-view. GG

Such experiments should never be funded by the public, in my opinion. They are really just a form of public exhibitionism. What you are saying is that there is no such thing as being "in tune", but that all tunings are of more or less equal worth. I would disagree with such an assertion. GG

I would strongly disagree that individual genius can overcome even a tuning that, for example, has no consonances. This is rather like saying that Virgil could have written the AEneid without vowels, or better, without an alphabet, or better still, without a language at all. GG

Your opinion is not uncommon among microtonalists, although very uncommon among musicians. It arises perhaps as a reaction against the overwhelming dominance of the 12-tone equal. In order for music to affect the emotions, it must be memorable, and conform at least roughly to the brains's own innate tuning system. Music written in say, 13-tone equal is just random noise. It can have no more musical effect than street noises. Again, I do not say that such extreme ideas are entirely harmful. Perhaps a period of wild liberty is needed to break up the 12-tone equal mould. GG

--- In tuning@yahoogroups.com, "gregggibson" <gregggibson@...> wrote:

> I am sorry, but it is premature for me to explain further on this forum.

Then it was premature to mention it on this forum.

> But of course you understand that the presence of beats is dependent upon the presence of the relevant partial pairs; they are by no means invariably present. Few timbres will have all the partials present necessary even to define all members of the senario by beats. For example, if the eighth or fifth partial is weak, or their multiples, then 8:5 will NOT be surrounded by audible beats.

Most timbres, such as the human voice or most orchestral instruments, are well-supplied with partials.

>And the brain's binary code also strongly favors it, quite apart from the simple ratios. GG

Sorry, if it is "premature" to explain this so-called "binary code" then dragging it out in an argument is an argumentative fallacy; a combination of argument from authority and appeal to the future. What it boils down to is that it is bullshit.

> By the way, another correspondent asserts that cents are logarithms. They are not. Neither are savarts.

If you want to see what a real appeal to authority looks like, let me try one out: I am a mathematician. You are not. I have a PhD in mathematics. You do not. I know what a logarithm is, and you should trust me on that topic.

Did it work?

>It is incidentally somewhat more logical to use base 2 logarithms, since the octave equals 2, not 10. But this is really trivial of course. GG

It's more logical to use base 2 but not more logical to use base 2^(1/1200)?

> Other than for avoiding calculation errors, there is no particular point in greater accuracy than the differential limen however. No one can hear a cent difference. Even the savart is near the edge of differential perception. GG

People can hear it in the context of harmony.

>> And the brain's binary code also strongly favors it, quite
>> apart from the simple ratios. GG
>
> Sorry, if it is "premature" to explain this so-called
> "binary code" then dragging it out in an argument is an
> argumentative fallacy; a combination of argument from authority
> and appeal to the future. What it boils down to is that it is
> bullshit.

I see 13 years have made no blemish on Gregg's theories.

-Carl

--- In tuning@yahoogroups.com, "gregggibson" <gregggibson@...> wrote:
> These are followers of Stockhausen. You have no quarrel with them? I doubt that everyone here views such experiments with enthusiasm. Although I do not wish to persecute such experiments, they are entirely in contradiction with common musical experience. No composer of the past, however open-minded, would have accepted such an idea. They would have assumed the theorist who proposed it to be insane. I personally do not feel that strongly, but it is a very strange idea, and entirely outside my musical world-view. GG
>

Excuse ME? I could not possibly be less a follower of Stockhausen. I am a rock musician, I play guitar, I write pop and electronic music, and I am certainly not insane. Moreover, I think my music speaks for itself on behalf of my sanity. I would be more than happy to provide you some of my music written in tunings like 11-EDO, 16-EDO, 20-EDO, even 23-EDO. It may not be, stylistically, to your taste, but you will see that these tunings can be applied to a musical idiom other than atonality, serialism, or other varieties of abstract experimental avant-garde "noise". IJ

> Such experiments should never be funded by the public, in my opinion. They are really just a form of public exhibitionism. What you are saying is that there is no such thing as being "in tune", but that all tunings are of more or less equal worth. I would disagree with such an assertion. GG
>

No one is saying any such thing. What I am saying (and have said, and will continue to say even on my death-bed) is that *no tuning is devoid of some musical or expressive purpose*. "In tune" and "out of tune" refer only to the audible quality of beatlessness which is itself merely a dimension of timbre. If you think that only beatless or near-beatless intervals (such as those of the senario) can be used to create non-artificial human folk music, you have also exposed your ignorance of the musical traditions of many non-Western cultures. IJ

> I would strongly disagree that individual genius can overcome even a tuning that, for example, has no consonances. This is rather like saying that Virgil could have written the AEneid without vowels, or better, without an alphabet, or better still, without a language at all. GG
>

Nonsense. There is no such thing as tuning with *no* consonances. I know this because I have been seeking just such a tuning for years and have been unable to find it. You may find tunings with poorly-tuned consonances, but they cannot be eliminated all together short of using a scale of 3 or 4 notes dividing a significantly-stretched octave. I challenge you to prove me wrong, if you can. Even 13-EDO contains 7/6, 9/5, 12/7, and the octave. Regardless, consonance in music is *not* analogous to language in literature! If it were, that would imply that that dissonance in music is analogous to gibberish (or perhaps silence?) in literature, which is patently nonsense. Would you compare Stravinsky to the babblings of a feral child? IJ

> Music written in say, 13-tone equal is just random noise. It can have no more musical effect than street noises. Again, I do not say that such extreme ideas are entirely harmful. Perhaps a period of wild liberty is needed to break up the 12-tone equal mould. GG
>

And here you are demonstrating the same variety of ignorance as Barbour, whom you have just thoroughly lampooned for criticizing that which he has not heard. "No more musical effect than street noises"? Clearly you have heard not the music of John Lyle Smith, or Easley Blackwood, or David Finnamore, or Herman Miller, or myself. All have composed music in 13-EDO that *CERTAINLY* has more musical effect than street noises. And NONE follow in the tradition of Stockhausen. IJ

-Igs

On Tue, Feb 22, 2011 at 1:10 AM, gregggibson <gregggibson@...> wrote:
>
> By the way, another correspondent asserts that cents are logarithms. They are not. Neither are savarts. Only logarithms are logarithms. One can of course divide the octave into any artificial number of measuring degrees that one desires by the use of a conversion factor, but the resulting artificial division is not a logarithm. GG

Gregg, man, this is some classic stuff you're posting here.

-Mike

As a new guy and non-musician who hasn't been much contaminated by
music that doesn't conform to the innate 19-EDO structure of the brain
(unless, I suppose, that's the effect of a lifetime of 12-EDO), I have
to say that these assertions don't fit my own recent experience.

The odd-EDO music I've listened to is somewhat alien to me, but not
non-musical; certainly not "street noise". It may give the kind of
effect that Schoenberg was thinking of when he quoted Stefan George,
saying, "I feel the air of another planet." There's much of it I don't
like, and I don't need to apologize for that -- it's alien, after all,
as is a fair amount of Schoenberg's music -- but there are some things
that are striking or interesting or otherwise musical and worthwhile.

My 2 cents.

Regards,
Jake

On 2/22/11, cityoftheasleep <igliashon@...> wrote:
> --- In tuning@yahoogroups.com, "gregggibson" <gregggibson@...> wrote:
>> These are followers of Stockhausen. You have no quarrel with them? I doubt
>> that everyone here views such experiments with enthusiasm. Although I do
>> not wish to persecute such experiments, they are entirely in contradiction
>> with common musical experience. No composer of the past, however
>> open-minded, would have accepted such an idea. They would have assumed the
>> theorist who proposed it to be insane. I personally do not feel that
>> strongly, but it is a very strange idea, and entirely outside my musical
>> world-view. GG
>>
>
> Excuse ME? I could not possibly be less a follower of Stockhausen. I am a
> rock musician, I play guitar, I write pop and electronic music, and I am
> certainly not insane. Moreover, I think my music speaks for itself on
> behalf of my sanity. I would be more than happy to provide you some of my
> music written in tunings like 11-EDO, 16-EDO, 20-EDO, even 23-EDO. It may
> not be, stylistically, to your taste, but you will see that these tunings
> can be applied to a musical idiom other than atonality, serialism, or other
> varieties of abstract experimental avant-garde "noise". IJ
>
>> Such experiments should never be funded by the public, in my opinion. They
>> are really just a form of public exhibitionism. What you are saying is
>> that there is no such thing as being "in tune", but that all tunings are
>> of more or less equal worth. I would disagree with such an assertion. GG
>>
>
> No one is saying any such thing. What I am saying (and have said, and will
> continue to say even on my death-bed) is that *no tuning is devoid of some
> musical or expressive purpose*. "In tune" and "out of tune" refer only to
> the audible quality of beatlessness which is itself merely a dimension of
> timbre. If you think that only beatless or near-beatless intervals (such as
> those of the senario) can be used to create non-artificial human folk music,
> you have also exposed your ignorance of the musical traditions of many
> non-Western cultures. IJ
>
>> I would strongly disagree that individual genius can overcome even a
>> tuning that, for example, has no consonances. This is rather like saying
>> that Virgil could have written the AEneid without vowels, or better,
>> without an alphabet, or better still, without a language at all. GG
>>
>
> Nonsense. There is no such thing as tuning with *no* consonances. I know
> this because I have been seeking just such a tuning for years and have been
> unable to find it. You may find tunings with poorly-tuned consonances, but
> they cannot be eliminated all together short of using a scale of 3 or 4
> notes dividing a significantly-stretched octave. I challenge you to prove
> me wrong, if you can. Even 13-EDO contains 7/6, 9/5, 12/7, and the octave.
> Regardless, consonance in music is *not* analogous to language in
> literature! If it were, that would imply that that dissonance in music is
> analogous to gibberish (or perhaps silence?) in literature, which is
> patently nonsense. Would you compare Stravinsky to the babblings of a feral
> child? IJ
>
>> Music written in say, 13-tone equal is just random noise. It can have no
>> more musical effect than street noises. Again, I do not say that such
>> extreme ideas are entirely harmful. Perhaps a period of wild liberty is
>> needed to break up the 12-tone equal mould. GG
>>
>
> And here you are demonstrating the same variety of ignorance as Barbour,
> whom you have just thoroughly lampooned for criticizing that which he has
> not heard. "No more musical effect than street noises"? Clearly you have
> heard not the music of John Lyle Smith, or Easley Blackwood, or David
> Finnamore, or Herman Miller, or myself. All have composed music in 13-EDO
> that *CERTAINLY* has more musical effect than street noises. And NONE
> follow in the tradition of Stockhausen. IJ
>
> -Igs
>
>
>
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Just wanted to drop in to say - there is certainly nothing wrong with
Stockhausen. He has some interesting ideas just like *most* free
thinkers.

This seems to be another conformist 1984 thread that I'm deleting.
Good luck Igs.

Chris

On Tue, Feb 22, 2011 at 9:43 AM, cityoftheasleep
<igliashon@sbcglobal.net> wrote:

> >
>
> And here you are demonstrating the same variety of ignorance as Barbour, whom you have just thoroughly lampooned for criticizing that which he has not heard. "No more musical effect than street noises"? Clearly you have heard not the music of John Lyle Smith, or Easley Blackwood, or David Finnamore, or Herman Miller, or myself. All have composed music in 13-EDO that *CERTAINLY* has more musical effect than street noises. And NONE follow in the tradition of Stockhausen. IJ
>
> -Igs

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> >> And the brain's binary code also strongly favors it, quite
> >> apart from the simple ratios. GG
> >
> > Sorry, if it is "premature" to explain this so-called
> > "binary code" then dragging it out in an argument is an
> > argumentative fallacy; a combination of argument from authority
> > and appeal to the future. What it boils down to is that it is
> > bullshit.
>
> I see 13 years have made no blemish on Gregg's theories.
>
> -Carl
>

I seem to have inadvertently provoked all sorts of heartrending outrage in certain quarters. I was going to post a few more things on my vacation, but perhaps I had better refrain. I do not want to cause anyone to have a nervous breakdown on my account. I am actually a very easygoing, mild-mannered sort of fellow, and not at all the apoplectic Adolf Hitler that the poor misguided advocates of such things as the 13-tone equal now imagine me to be.

However, it is quite true that I am convinced that, while they may have native musical talent and be thoroughly lovable fellows in real life, they are very incompetent musical theorists. Can one imagine either a classical or rock or Indian or Arab musician composing without consonances? I mean any one of them with a following? It is all very silly, even pathetic. But there is no need to combat it and upset these poor people further. There will never be a public for such 'music', outside a few academic refuges.

I cannot resist commenting briefly on Arnold Schoenberg, however. In a famous interview, he was asked why he wanted to use 12-tone equal at all. His system implies that dissonances and consonances are to be treated in exactly the same fashion, so why use 12-tone equal, which has at least two very good consonances, the fifth and octave?

Poor Schoenberg was thoroughly nonplussed. After sputtering and going red in the face, he finally confessed he did not know the answer to the question, but that he had to use what was available for the orchestra - 12-tone instruments.

Even more ironically, the serialists now venerate the 12-tone equal more than most, again without logical reason. They do not have the courage of their convictions.

Gregg Gibson

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Tue, Feb 22, 2011 at 1:10 AM, gregggibson <gregggibson@...> wrote:
> >
> > By the way, another correspondent asserts that cents are logarithms. They are not. Neither are savarts. Only logarithms are logarithms. One can of course divide the octave into any artificial number of measuring degrees that one desires by the use of a conversion factor, but the resulting artificial division is not a logarithm. GG
>
> Gregg, man, this is some classic stuff you're posting here.
>
> -Mike
>

And in return, I would like to again compliment you on your idea for a nineteen-tone keyboard. That Joseph Yasser happened to have had the idea before you does not detract from your ability to 'think outside the box', as they say.

You should not be afraid of mathematics however. You are intelligent enough to easily master the subject. The mathematics of tuning and temperament is really rather simple, once you understand the CONCEPTS behind the math.

Some mathematicians do not really want the hoi polloi to understand these concepts clearly; they prefer to pose as gods, whose ideas are beyond the understanding of common mortals.

Gregg Gibson

Gregg,

> I cannot resist commenting briefly on Arnold Schoenberg, however. In a famous interview, he was asked why he wanted to use 12-tone equal at all. His system implies that dissonances and consonances are to be treated in exactly the same fashion, so why use 12-tone equal, which has at least two very good consonances, the fifth and octave?
>
> Poor Schoenberg was thoroughly nonplussed. After sputtering and going red in the face, he finally confessed he did not know the answer to the question, but that he had to use what was available for the orchestra - 12-tone instruments.

I really like this anecdote. I didn't understand how you could really be "atonal" with these twelve notes, and Bernstein, in his "Unanswered Question" lectures, says essentially the same thing.

That said, if being atonal means you treat dissonances and consonances the same, then that implies you need at least some consonances and some dissonances, right? Otherwise you may as well use power drills and ping-pong balls.

Regards,
Jake

--- In tuning@yahoogroups.com, "gregggibson" <gregggibson@...> wrote:
>I am actually a very easygoing, mild-mannered sort of fellow, and not at all the apoplectic Adolf Hitler that the poor misguided advocates of such things as the 13-tone equal now imagine me to be.
>

Gregg, buddy! A little early in the discourse for Godwin's law, isn't it, dear boy?

> However, it is quite true that I am convinced that, while they may have native musical talent and be thoroughly lovable fellows in real life, they are very incompetent musical theorists. Can one imagine either a classical or rock or Indian or Arab musician composing without consonances? I mean any one of them with a following? It is all very silly, even pathetic. But there is no need to combat it and upset these poor people further. There will never be a public for such 'music', outside a few academic refuges.
>

LOL. Oh man, this is hilarious! Is this that "trolling" thing I've been hearing so much about lately?

I gotta love it when someone dismisses all of my music and theoretical work with an arrogant wave of the hand, without having listened to or read any of it. I love being lumped in with, of all people, the atonalists, when in reality my music and my theorizing is about as far from that school of thought as possible. It's so much more fun than when someone actually takes the time to level a legitimate critique at my work.

We should invite Marcel back just so he and Greggy-boy here can have at it, since both have clearly done all their research right and have unlocked the key to music. That would be a fun flame-war to watch!

Poor Gregg...I truly feel pity for someone who has spent 13 years on research and learned so little of what is possible in music, so woefully ignorant of how broadly humans can perceive tonality. Well, perhaps in another 13 years, when he has finally published his discoveries, a few more academics will be swayed in favor of 19-TET. Though I suspect in another 13 years, it may also be abundantly clear for what sort of microtonal music there is and is not a "public"...LOL.

-Igs

Hi Gregg,

I have no idea what you're talking about. That notwithstanding, a cent
is a logarithmic unit of measure.

-Mike

On Tue, Feb 22, 2011 at 9:32 PM, gregggibson <gregggibson@...> wrote:
>
> And in return, I would like to again compliment you on your idea for a nineteen-tone keyboard. That Joseph Yasser happened to have had the idea before you does not detract from your ability to 'think outside the box', as they say.
>
> You should not be afraid of mathematics however. You are intelligent enough to easily master the subject. The mathematics of tuning and temperament is really rather simple, once you understand the CONCEPTS behind the math.
>
> Some mathematicians do not really want the hoi polloi to understand these concepts clearly; they prefer to pose as gods, whose ideas are beyond the understanding of common mortals.

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Tue, Feb 22, 2011 at 1:10 AM, gregggibson <gregggibson@...> wrote:
> >
> > By the way, another correspondent asserts that cents are logarithms. They are not. Neither are savarts. Only logarithms are logarithms. One can of course divide the octave into any artificial number of measuring degrees that one desires by the use of a conversion factor, but the resulting artificial division is not a logarithm. GG
>
> Gregg, man, this is some classic stuff you're posting here.
>
> -Mike
>

Hi again Mike. If this is too elementary for you, perhaps it will be useful to others.

Note to moderator: I am posting this here and not on the math site, because it is very simple, but not always clearly understood even by musical theorists. Cents are NOT logarithms. They are an artificial quantity derived from logarithms.

Nature divides the octave 2/1 or 2:1 into about 0.30103 units, because that is the logarithm of 2. That is to say, one must raise 10 to about the 0.30103 power to make it equal 2.

This is a simplification, but the ear hears logarithmically, in powers of 2, so nature really divides the octave into one unit, because 2 raised to the first power equals 2. But it is customary to use powers of 10 because they are more readily available in tables and now on calculators.

Therefore if one wants to divide the octave into say, 1200 units, one has the equation:

0.30103 x = 1200

or:

log 2 x = 1200

x is the conversion factor. What this really means is merely: By what number x must we multiply the logarithm of 2 in order to make it equal 1200?

Using very simple algebra, we find this number x to be:

x = 1200 / log 2

or about 3986.314

And for savarts: x = 301.01 / log 2

or exactly 1000. In fact the savart is DEFINED as the logarithm multiplied by 1000.

Now if you want to know the cents in a given ratio, first take the logarithm of that ratio, and then multiply it by 1200/log 2. For example, to find the cents in 3/2:

log (3/2) multiplied by 1200/log 2 = about 701.955 cents

Be sure to enter the parentheses on your calculator:

LOG (3 divided by 2) times 1200 divided by LOG 2

To find string lengths or vibration numbers for a given interval in cents, we reverse the process, proceeding from cents or savarts back to logarithms and then finally back to simple numbers or fractions.

For example, to find the string length of 701.955 cents:

701.955 / (1200/log 2) = 0.176091

This last number is a logarithm, that is to say an exponent or power of 10.

We therefore raise 10 to the 0.176091 power to get back to the ratio of 1.5 or 3/2. This can be used to find the vibration numbers.

The string length will be the inverse of 3/2 or 2/3. That is to say, if one desires to know the part of an open string that will sound the just fifth 3/2 or 3:2, one takes its inverse, or 2/3.

Gregg Gibson

--- In tuning@yahoogroups.com, "gregggibson" <gregggibson@...> wrote:

> Note to moderator: I am posting this here and not on the math site, because it is very simple, but not always clearly understood even by musical theorists. Cents are NOT logarithms. They are an artificial quantity derived from logarithms.

If you like, I could explain it in a way deliberately designed to prevent hoi polloi from understanding the explanation. This explanation could involve the composition of morphisms in the category of topological groups, if you think that sounds obscure and imposing enough.

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> Hi Gregg,
>
> I have no idea what you're talking about. That notwithstanding, a cent
> is a logarithmic unit of measure.
>
> -Mike
>
> On Tue, Feb 22, 2011 at 9:32 PM, gregggibson <gregggibson@...> wrote:
> >
> > And in return, I would like to again compliment you on your idea for a nineteen-tone keyboard. That Joseph Yasser happened to have had the idea before you does not detract from your ability to 'think outside the box', as they say.
> >
> > You should not be afraid of mathematics however. You are intelligent enough to easily master the subject. The mathematics of tuning and temperament is really rather simple, once you understand the CONCEPTS behind the math.
> >
> > Some mathematicians do not really want the hoi polloi to understand these concepts clearly; they prefer to pose as gods, whose ideas are beyond the understanding of common mortals.

Cents are not logarithms. But cents are derived from logarithms via a conversion factor, and so could indeed be called a logarithmic unit of measure.

Gregg Gibson

On Tue, Feb 22, 2011 at 10:55 PM, gregggibson <gregggibson@...> wrote:
>
> Hi again Mike. If this is too elementary for you, perhaps it will be useful to others.
>
> Note to moderator: I am posting this here and not on the math site, because it is very simple, but not always clearly understood even by musical theorists. Cents are NOT logarithms. They are an artificial quantity derived from logarithms.
>
> Nature divides the octave 2/1 or 2:1 into about 0.30103 units, because that is the logarithm of 2. That is to say, one must raise 10 to about the 0.30103 power to make it equal 2.

Uh, since when is log base 10 "natural?"

> This is a simplification, but the ear hears logarithmically, in powers of 2, so nature really divides the octave into one unit, because 2 raised to the first power equals 2. But it is customary to use powers of 10 because they are more readily available in tables and now on calculators.

It's "customary" to use powers of 2 when talking about things in which
it makes sense to use base 2 logs.

> Therefore if one wants to divide the octave into say, 1200 units, one has the equation:
>
> 0.30103 x = 1200
>
> or:
>
> log 2 x = 1200
>
> x is the conversion factor. What this really means is merely: By what number x must we multiply the logarithm of 2 in order to make it equal 1200?
>
> Using very simple algebra, we find this number x to be:
>
> x = 1200 / log 2
>
> or about 3986.314

This is ridiculous. I can't tell if you're not understanding the basic
flaw in the math here, or if you're just trolling to get a rise out of
us. You imposed some ridiculous requirement about only base 10 logs
being Natural, presumably because Nature gave us Ten Fingers. Then,
from that and the realization that you can convert from one log to
another by multiplying by a constant, you have now determined that
only base 10 logs are Actual Logarithms. Why?

-Mike

On Tue, Feb 22, 2011 at 11:15 PM, gregggibson <gregggibson@...> wrote:
>
> Cents are not logarithms. But cents are derived from logarithms via a conversion factor, and so could indeed be called a logarithmic unit of measure.

A cent is a base 1.00057779 logarithm, where 1.00057779 ~ 2^(1/1200).

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Tue, Feb 22, 2011 at 11:15 PM, gregggibson <gregggibson@...> wrote:
> >
> > Cents are not logarithms. But cents are derived from logarithms via a conversion factor, and so could indeed be called a logarithmic unit of measure.
>
> A cent is a base 1.00057779 logarithm, where 1.00057779 ~ 2^(1/1200).
>
> -Mike
>

Not so. By such a device any quantity can be made into any other quantity. Whoever is feeding you your information should be ashamed of himself.

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Tue, Feb 22, 2011 at 10:55 PM, gregggibson <gregggibson@...> wrote:
> >
> > Hi again Mike. If this is too elementary for you, perhaps it will be useful to others.
> >
> > Note to moderator: I am posting this here and not on the math site, because it is very simple, but not always clearly understood even by musical theorists. Cents are NOT logarithms. They are an artificial quantity derived from logarithms.
> >
> > Nature divides the octave 2/1 or 2:1 into about 0.30103 units, because that is the logarithm of 2. That is to say, one must raise 10 to about the 0.30103 power to make it equal 2.
>
> Uh, since when is log base 10 "natural?"
>
> > This is a simplification, but the ear hears logarithmically, in powers of 2, so nature really divides the octave into one unit, because 2 raised to the first power equals 2. But it is customary to use powers of 10 because they are more readily available in tables and now on calculators.
>
> It's "customary" to use powers of 2 when talking about things in which
> it makes sense to use base 2 logs.
>
> > Therefore if one wants to divide the octave into say, 1200 units, one has the equation:
> >
> > 0.30103 x = 1200
> >
> > or:
> >
> > log 2 x = 1200
> >
> > x is the conversion factor. What this really means is merely: By what number x must we multiply the logarithm of 2 in order to make it equal 1200?
> >
> > Using very simple algebra, we find this number x to be:
> >
> > x = 1200 / log 2
> >
> > or about 3986.314
>
> This is ridiculous. I can't tell if you're not understanding the basic
> flaw in the math here, or if you're just trolling to get a rise out of
> us. You imposed some ridiculous requirement about only base 10 logs
> being Natural, presumably because Nature gave us Ten Fingers. Then,
> from that and the realization that you can convert from one log to
> another by multiplying by a constant, you have now determined that
> only base 10 logs are Actual Logarithms. Why?
>
> -Mike
>

Oh dear, this is mere gibberish that you are speaking. Perhaps you really do have a problem with even simple mathematics. Also, you are extremely rude and immature. I will have no more to do with you.

On Tue, Feb 22, 2011 at 11:43 PM, gregggibson <gregggibson@...> wrote:
>
> Not so. By such a device any quantity can be made into any other quantity. Whoever is feeding you your information should be ashamed of himself.

Hello Gregg,

This is actually how existence happens to work. Try plugging numbers
into a calculator and see for yourself.

Cheers,
Mike

Gregg wrote:

> Oh dear, this is mere gibberish that you are speaking. Perhaps you really > do have a problem with even simple
> mathematics. Also, you are extremely rude and immature. I will have no > more to do with you.

Dear Gregg,

before you start insulting others, please be so kind and read (at least briefly) these words:
http://en.wikipedia.org/wiki/Common_logarithm
http://en.wikipedia.org/wiki/Natural_logarithm
http://en.wikipedia.org/wiki/Binary_logarithm

Can you see my point? The first example is the logarithm with base 10, the second is the logarithm with base "e", the third is the logarithm with base 2. All of these are logarithmical measurements, only their bases are different. First to note, the logarithm with base 10 is not called the "natural logarithm" since this term is used for the base "e" logarithm. Secodn, there's also the binary logarithm, which corresponds to octaves in hearing frequency ratios. This means there's no need to use logarithms with base 10 for musical purposes when there's actually a more useful thing, the base 2 (or binary) logarithm. Now, if you multiply this by 1200 or if you use a logarithm of base 2^(1/1200), you get cents. If you can, get your calculator, try it and you'll see we were right. Because binary logarithm is a logarithmical unit, then also a 12-EDO semitone is a logarithmical unit and so are cents. This is because of the definition of logarithms which has been agreed upon centuries ago so there's not too much of a point in trying to change the definition.

Petr

--- In tuning@yahoogroups.com, "gregggibson" <gregggibson@...> wrote:
>It is all very silly, even pathetic. But there is no need to combat it and upset these poor people further. There will never be a public for such 'music', outside a few academic refuges.
>

I should also like to add, as you seem to be on your way out the door, that one seems to *me* to be pathetic is to spend such an enormous amount of one's life-energy trying to validate a musical tuning based on any merit other than the music that can be made with it. Any superiority of 19-TET should be obvious in the sound of it, and music alone should be sufficient to make the case for it. If there is some fundamental relationship to 19-TET and the "binary code of the brain", why is this relationship not overwhelmingly apparent when we hear music in 19? How could 19 have failed to become the universal tuning of mankind, if your theories are correct? That any individual might prefer to explore or work in any other tuning, this tiny fact is enough to refute all of your work. So it is a pity to see how much time you have wasted in defense of a patently-untenable position.

-Igs

Gene,

Thank you. Your response, in combination with the paragraph you were responding to, wins the Internet today.

Cheers,
Jon

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, "gregggibson" <gregggibson@> wrote:
>
> > Note to moderator: I am posting this here and not on the math site, because it is very simple, but not always clearly understood even by musical theorists. Cents are NOT logarithms. They are an artificial quantity derived from logarithms.
>
> If you like, I could explain it in a way deliberately designed to prevent hoi polloi from understanding the explanation. This explanation could involve the composition of morphisms in the category of topological groups, if you think that sounds obscure and imposing enough.
>

--- In tuning@yahoogroups.com, "jonszanto" <jszanto@...> wrote:

> Thank you. Your response, in combination with the paragraph you were responding to, wins the Internet today.

Thanks, Jon. I hope I don't need to add I really could have given the threatened "explanation".

--- In tuning@yahoogroups.com, "gregggibson" <gregggibson@...> wrote:
> > > Note to moderator: I am posting this here and not on the math site, because it is very simple, but not always clearly understood even by musical theorists.

Sorry, Gregg, but what you wrote in that post is not entirely correct.

> > > Nature divides the octave 2/1 or 2:1 into about 0.30103 units, because that is the logarithm of 2. That is to say, one must raise 10 to about the 0.30103 power to make it equal 2.

The second sentence is, of course, correct but the first is not.

> [in reply to Mike] Oh dear, this is mere gibberish that you are speaking. Perhaps you really do have a problem with even simple mathematics. Also, you are extremely rude and immature. I will have no more to do with you.

Gregg, you may have a problem with the way Mike said it, but what he said was quite correct. Logarithms to base 10 are not the only kind of logarithm - did you look at the wikipedia pages that Petr suggested? To be sure they are called "common logarithms" but they are not called "natural logarithms" - the latter refers to something else - logarithms to base "e" which have much more general mathematical application than those to base 10.

Steve.

Hi Jake,

My position / observation is that true atonal music is hard to make.
You do have of course the full spectrum from totally tonal (one note
only that never varies) to X number of notes - and here is the crucial
part - played in such a way that none has precedence over the others.
And this matter of precedence is the really hard part to do even if
you are not using functional harmony because the melodic and harmonic
relationships will imply tonality to some varying degree depending on
the composition.

And as for the power drills and ping-pong balls - not a bad idea.

Chris

On Tue, Feb 22, 2011 at 10:25 PM, Jake Freivald <jdfreivald@...> wrote:

> That said, if being atonal means you treat dissonances and consonances
> the same, then that implies you need at least some consonances and some
> dissonances, right? Otherwise you may as well use power drills and
> ping-pong balls.
>
> Regards,
> Jake

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
> Thanks, Jon. I hope I don't need to add I really could have given the threatened "explanation".

You don't. :)

This conversation cracks me up. I think Gregg is yanking all of our chains.

I was going to let it lie, but Mario asked about it again, and this is more
fun than doing what I'm supposed to be doing.

Gregg said:
> Nature divides the octave 2/1 or 2:1 into about 0.30103 units,
> because that is the logarithm of 2. That is to say, one must
> raise 10 to about the 0.30103 power to make it equal 2.

Note that Gregg says that Nature *divides the octave into 0.30103 units*.
Not that it breaks the octave into multiple parts, each of which is .30103
in size, but that it "divides the octave into 0.30103 units".

Think about what that means. By analogy, consider taking a banana and
dividing it into three units, i.e., cutting it into thirds. 1 banana / 3
units = 1/3 banana / unit.

Now, if you take a banana and *divide it into 0.30103 units, *what have you
done, really? Using the same math, 1 banana / 0.30103 units = 3.32 bananas /
unit.

Instead of dividing something into a larger number of units, you are
"dividing" something into a *smaller* number of units. Which is to say, you
haven't divided anything. You have assembled it. Each unit is a
Frankenstein's superbanana that is the conglomeration of over three bananas.

Okay, that's pretty funny. But ignore the obvious language problem / math
misconception for a moment and let's talk about logarithms.

Logarithms can have any base. Gregg is working in base 10. But someone could
just as easily use 13.

That person would say, "Nature divides the octave 2/1 or 2:1 into about
0.270238 units, because one must raise 13 to about the 0.270238 power to
make it equal 2. (I.e., 0.270238 is the log (base 13) of 2." Again, ignore
the gibberish about "dividing" the octave into fractional units. The math in
this paragraph is essentially equivalent to what Gregg posited.

And it's equally useless. The choice of 13 was arbitrary, just as Gregg's
choice of 10 was arbitrary. What does the 10 or 13 *actually tell us* about
anything? To what do they correlate?

Nothing in particular.

The so-called natural logarithm seems less arbitrary because it has the word
"natural" in it. But that doesn't correlate to anything interesting, either.

More than either 10 or 13, it would make more sense to say that Nature
divides the octave 2/1 into about 1 unit, because that is the log (base 2)
of 2. That is to say, one must raise 2 to the 1 power to make it equal 2.
One must raise 2 to the 2 power to make it equal 4. One must raise it to the
3 power to make it equal 8.

It seems boring, but it's precisely this correlation -- the fact that 1, 2,
and 3 are integers instead of something like 0.30103 -- that makes it a
non-arbitrary measurement. Our ears take these intervals and make them sound
"the same". Each unit is a successive octave.

Got that? If you use 2 as the logarithm base, each unit is an octave.

*Now* we're talking. But that unit isn't a very precise measure, so let's
make it finer-grained. Let's, say, divide that 1 unit into, say, 1200
mini-units -- which neatly corresponds with each 12-EDO semitone having 100
mini-units.

The fact that it's 100 mini-units (as opposed to 500 or 235 or whatever) is
somewhat arbitrary, but it's enough mini-units to provide reasonable
precision, and few enough that a difference of a few mini-units is
detectable.

Since it's 100 mini-units per semitone, let's call these mini-units "cents".

Now the choice is clear:

You can use an arbitrary system endorsed by only one person on the planet
that makes not-particularly-usable units based on powers of 10.

Or you can use the common system that makes cents.

Regards,
Jake

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, "gdsecor" <gdsecor@> wrote:
> ...
[Gregg wrote:]
> > > I have been working on a theory that attempts to transcend this division by directly examining the binary code used by the deep brain.
> >
> > I don't understand this. Are you claiming to have deciphered the circuitry in the part of the neural network of the human brain that processes musical sounds?
>
> > Personally, I would have to disagree with this last statement. I find slightly wide fifths (of around 703 to 705 cents) preferable to narrow ones.
>
> I concur, on most days. But narrower fifths are more mellow in sound, which you might want.

I believe that it's open for debate and that no one yet has the final word, at least until Gregg decides to share his deep "inside" information. :-)

> > One of my favorite tunings has fifths tempered wide by ~1.6237 cents in 3 separate chains: one for 3-limit intervals, the second one 5:4 above the first (for ratios of 5), and the third 7:4 above the first (for ratios of 7, 11, and 13).
>
> Have you ever worked with mystery temperament?

No, but I'd be surprised if it's as accurate as what I'm using. My fifths are exactly (504/13)^(1/9), which is ~703.5787 cents, and all of the 15-limit consonances are within 3.25 cents of just.

> > > I can say however that it is about halfway between General Bosanquet's keyboard and one of Wilson's designs. I have tried at least a score of 19-tone designs over the years, and I believe this is perhaps the best solution.
>
> Bosanquet's brother was an admiral, but you probably mean Thomas Perronet Thompson.
>
> > Fortunately, there is a better alternative.
>
> Several better ones, in fact. You might also mention 612, which is divisible not only by 12 but also 17, which you ought to like. Then there's Woolhouse's 730, and Woolhouse has priority in this matter. 730 is not 29-limit consistent, but its patent val does score better in the 29 limit than anything smaller, including consistent divisions. 882et is the first to beat it.

Of course, 612 is great for the 5-limit, but it falls apart at 17. I disqualify 730 for its excessive amount of error (37% of degree), which causes ratios containing 7^2 as a factor to fail the rounding test. I also want an extremely low error (<1%) for prime 3, because ratios of interest frequently have relatively high powers of 3 as factors; the 5-schisma contains 3^8, where 882 fails. Minas can express tones in a Pythagorean chain well beyond 3^50 without rounding errors.

> Still, for general xenharmonic purposes it's hard to beat the mina, except for the little fact that cents are already established. I'd use it if not for that.

Yeah, even I'm not using them yet. I intend to introduce them in The Book and to use them throughout, with the corresponding number of cents in parentheses. I expect that it will be an uphill battle to get them accepted, but perhaps no worse than trying to get folks to use something other than 12.

--George

--- In tuning@yahoogroups.com, "gregggibson" <gregggibson@...> wrote:
>
> > I remember reading one comment about 19-equal, probably Barbour's, where the writer offered the opinion that, judging by the numbers, the major 3rds will sound "insipid in the extreme".
>
> Yes, long before I read that he had finally confessed never to have listened to the temperaments he presumed to judge in print, his strange statements led me to suspect his honesty. GG
>
> > > There is however a certain limited aspect of the brain's binary code that favors 12-tone equal almost as much as 19-tone equal, but I confess I am not eager to point this out, for well I know with what joy our conservatories will welcome such a finding, while ignoring everything else. GG
> >
> > I don't quite know what to make of this. I think 12-equal tends to be favored for its diatonic melodic properties, but 19-equal for its diatonic harmonic properties. However, 19-equal also provides new melodic intervals that can trump 12, including an alternate (higher) leading tone that makes a wide dissonant third with the dominant that very effectively resolves (both melodically & harmonically) to the tonic tone by 63 cents. Another thing that can be done in 19, but not in 12, is a rendering of the ancient Greek enharmonic genus.
>
> I am sorry, but it is premature for me to explain further on this forum. After I have published my theory, I will be glad to answer your questions about it, time permitting. I will however observe that to my ears 12-tone equal sounds very mistuned even in diatonic melody. GG

I should mention that not everyone who has tried 19-equal has liked it. In Joel Mandelbaum's doctoral thesis, "Multiple Division of the Octave and the Tonal Resources of 19o?=tone Temperament" (1961), he mentions (in chapter 11 on Joseph Yasser), he states that: "Yasser appears to have been the principal influence on McClure in England, who built a 19-tone harmonium. According to Schafer, who maintained a correspondence with McClure (now deceased), McClure was not altogether satisfied with the results and finally decided that the fifths and thirds were too small." Evidently McClure was already familiar with 1/4-comma meantone temperament and wouldn't settle for something not as "good" as that.

Having worked with both 19-equal and 31-equal (the virtual equivalent for 1/4-comma meantone) for several decades, I fully understand his reservations regarding 19. When I first tried it, I had already tried 1/4-comma meantone and was not able to accept 19 "right out of the box." It required a "break-in" period of about a week to get accustomed to its diatonic melodic properties, which I now regard as a sort of "accent" that the "language" of Meantone is "spoken" with in Nineteen-tone Land. Once I got used to it, I found it quite charming for most music written between 1500 and 1750 and very suitable for a blues idiom (by virtue of its augmented 2nds and 6ths). It also tends to lose its diatonic "accent" when used in non-diatonic scales (of which it is abundantly capable, as Gene noted).

Yet, in spite of all this, 19-equal is definitely not in my Top Five List of Favorite Tunings. However, one tuning that contains a closed circle of 19 (unequal) fifths *is* in my Top Two, because it significantly improves five of the six 7-limit consonances and also allows 15-limit harmony in 3 different keys. I have little interest in a tuning that would limit consonant intervals to the senario.

> > > The division in the microtonal community is perhaps best described as the old quarrel between the Pythagoreans, who love ratios, and Aristoxenians, who are skeptical that these ratios really exist immanently in the brain, and would rather speak of divisions of the tone or of the octave. The melodic limen strongly favors the latter school, while the beating phenomenon somewhat favors the former, but in most timbres only for some members of the senario and perhaps a septimal or two. If pressed I would describe myself as a moderate Aristoxenian. At one time I pored over the ratios with as much devotion as anyone could wish, however. GG
> >
> > Isn't the observation that the most acoustically consonant intervals are expressed as simple-number ratios (due to the elimination of beats) sufficient grounds for affirming that ratios are the single, most significant in determining which intervals are most consonant? Is not the justification for dividing the octave into 12, 19, or 31 parts based on the fact that these are the ratios that we're attempting to approximate? Although composers such as Igs and Herman Miller have chosen to concentrate on divisions of the octave that do not closely approximate these ratios for their compositions, the rest of us here have no quarrel with them, because we believe that virtually any kind of tuning can be used to produce worth-while music, and that the skill of the composer in working with the tonal materials is a greater factor for success than the tuning itself.
>
> But of course you understand that the presence of beats is dependent upon the presence of the relevant partial pairs; they are by no means invariably present. Few timbres will have all the partials present necessary even to define all members of the senario by beats. For example, if the eighth or fifth partial is weak, or their multiples, then 8:5 will NOT be surrounded by audible beats. This is one of the basic flaws with Helmholtz' theory; he himself recognized this, and offered his theory of consonance in a somewhat diffident manner. The modern just intonationists, with their worship of the ratios, seized upon Helmholtz' theory for their own purposes. GG
>
> It is true that the senario favors the 19- and 31-tone equal among others, but this is by no means the only consideration. The melodic limen of 55-60 cents favors the 19-tone equal, and no other tuning.

What do you mean by putting the melodic limen at 55-60 cents? You seem to be saying that an interval of less than 55 cents or so cannot be perceived melodically, but that can't be right. We can easily hear an interval of a comma, which is less than half that size. If you're saying that anything smaller than about 55 cents is unsuitable for use as a melodic interval, I must strongly disagree, because my own experience informs me that steps of 31-equal are melodically distinct, i.e., I clearly perceive them as being in different pitch-classes. Furthermore, I clearly perceive any two intervals of 31-equal differing in size by a single degree (e.g., a major 3rd and neutral 3rd) as being both melodically and harmonically distinct.

But wait a minute! What do you mean by "19-tone equal, and no other tuning" being favored by this melodic limen. Doesn't 22-equal also make it? The greatest liability I found with 19-equal is that it doesn't have a good dominant 7th chord: in C-E-G-Bb the top tone is too high and in C-E-G-A# it's too low. 22-equal has a very nice dominant 7th chord, with three thirds of different sizes, and it permits the chord progression C, F, G7, C with the tonic of F and the 7th of G7 to be the same pitch. What about that? (And in case you're wondering, 22-equal also didn't make my Top Five List, but 31-equal did.)

> And the brain's binary code also strongly favors it, quite apart from the simple ratios. GG
>
> By the way, another correspondent asserts that cents are logarithms. They are not. Neither are savarts. Only logarithms are logarithms. One can of course divide the octave into any artificial number of measuring degrees that one desires by the use of a conversion factor, but the resulting artificial division is not a logarithm. GG
> >
> > > I have been working on a theory that attempts to transcend this division by directly examining the binary code used by the deep brain. GG
> >
> > I don't understand this. Are you claiming to have deciphered the circuitry in the part of the neural network of the human brain that processes musical sounds?
>
> Yes. GG

<Gasp!>

> > > Ellis also invented the cent, which is heavily biassed in favor of the 12-tone equal, and succeeded in persuading the anglophone world to use it in preference to the savart, which is of course is in much more direct contact with the ear's logarithmic hearing. Ever since we have had to confront musicians convinced that the cents somehow `prove' that there are 12 equal semitones in the octave. GG
> >
> > Cents and savarts (or heptamerides, or degrees of 301-EDO) are both logarithmic units of measure, so there is no merit in arguing that one has any advantage over the other on the basis of the way we hear pitches. However, I quite agree with your observation of the 12-equal bias inherent in cents. As I will demonstrate below, there are distinct advantages in thinking outside the 12-tone squirrelcage.
>
> I did not say that the savart is a logarithm.

Nevertheless it is, whether you say so or not.

> But because its conversion factor is 1000, it is less likely to confuse beginners, who often do not fully realize that we measure intervals in logarithms, so that we may replace multiplication and division of ratios by addition and subtraction. It is incidentally somewhat more logical to use base 2 logarithms, since the octave equals 2, not 10. But this is really trivial of course. GG
>
> > The historical advantage in using savarts was that you could use a four-place base-10 log table to look up the number of savarts in a ratio (expressed as a decimal), since an octave (with ratio 2/1) is exactly 301 savarts and the base-10 logarithm of 2 is .3010 (to 4 decimal places).
>
> Yes, but there are not EXACTLY 301 savarts in the octave; there are more digits. Usually at least 8 digits are used. The errors quickly become serious otherwise. GG

Okay, I stand corrected; it's not exactly 1/301 octave. But then, why should anyone want to use an interval measure in which an octave is not exactly a whole number of units? For me, it's a deal-breaker!

> > A more significant advantage in using savarts is that, because 301-EDO is a fairly good division for approximating harmonics and just ratios (it's 17-limit consistent), you can round the values for many just ratios off to whole numbers and use them in calculations without the rounding errors that occur with whole-number cents.
>
> The so-called 17-limit is an artefact, not a reality, because the brain has no way to identify such intervals, save in the unlikely event that such high partial pairs are present. But such timbres are very harsh and unmusical.

I believe that a rational interval is entitled to be called a consonance if it's possible to distinguish by ear that ratio in just intonation from tempered approximations of that ratio by a few cents. I have observed that beating of harmonic partials is not the only way to discern whether intervals are in virtually-exact rational ratios or not. There's another factor that comes into play when those intervals are used in isoharmonic chords. In the tetrad 9:11:13:15, for example, it's easy to tell whether one of the inner tones is mistuned, even with timbres having weak 11th or 13th partials. The same should hold true for the triad 14:17:20 (try it!). How would you explain this phenomenon?

> Personally I always carry cents and savarts out to at least 8 digits. GG

You've just sealed the deal for me on minas!

> > The chief problem with savarts is that, rounded to whole numbers, they lack the precision of cents. For example, if I want to compare the interval 8505:8192 (the apotome-complement of 35:36) with 26:27, I would not be able to use whole-number savarts, since these two ratios differ by only ~0.4 cents -- much less than 1 savart. (This is not a contrived example; these are actual ratios that came up in the process of my devising symbols for microtonal accidentals.) Since I would have to use savarts with decimal places, I would then lose the advantage of having a unit of interval measure that can be expressed as whole numbers.
>
> Other than for avoiding calculation errors, there is no particular point in greater accuracy than the differential limen however. No one can hear a cent difference. Even the savart is near the edge of differential perception. GG

You can easily tell the difference between a tempered fifth in 17-EDO and one in 22-EDO (about 1 savart difference), and I think you can also tell the difference between one in 19-EDO vs. one in 31-EDO (about 2 cents difference). It would be nice if our unit of measure were smaller than what we're able to hear.

--George

--- In tuning@yahoogroups.com, "gregggibson" <gregggibson@...> wrote:
>
> < Although composers such as Igs and Herman Miller have chosen to <concentrate on divisions of the octave that do not closely approximate these ratios for their compositions, the rest of us here have no quarrel with them, because we believe that virtually any kind of tuning can be used to produce worth-while music, and that the skill of the composer in working with the tonal materials is a greater factor for success than the tuning itself.
>
> These are followers of Stockhausen. You have no quarrel with them? I doubt that everyone here views such experiments with enthusiasm. Although I do not wish to persecute such experiments, they are entirely in contradiction with common musical experience. No composer of the past, however open-minded, would have accepted such an idea. They would have assumed the theorist who proposed it to be insane. I personally do not feel that strongly, but it is a very strange idea, and entirely outside my musical world-view. GG

It's standard practice in places like Thailand and Indonesia.

> Such experiments should never be funded by the public, in my opinion. They are really just a form of public exhibitionism. What you are saying is that there is no such thing as being "in tune", but that all tunings are of more or less equal worth. I would disagree with such an assertion. GG

I didn't say that all tunings are of more or less equal worth. I said that I believe that folks around here "believe that virtually any kind of tuning can be used to produce worth-while music, and that the skill of the composer in working with the tonal materials is a greater factor for success than the tuning itself." I have heard microtonal music written in "good" tunings that I thought was poorly written and that I would not care to listen to again, and I have also heard music (e.g., pieces by Easley Blackwood, Herman Miller, and Igs) written in so-called "bad" tunings that I thought was well written, to which I have listened again numerous times, because they stretch my musical horizon with their unconventional (and unexpected) tonal relationships. For example, a tuning such as 15-EDO has three small circles of fifths, each of which includes a tempered 7/4, but not a tempered 5/4 (which you must go to a different circle to get). It's a very different world of sound, something like visiting an alien planet.

Personally, I can't imagine myself wanting to write for that sort of tuning, because I believe that there are "better" options available to me as a composer. Therefore, I don't think that all tunings are of more or less equal worth. It's more a matter that not all of us agree on which ones to value most highly.

Ah, there's one exception that I'll need to confess. I am probably the first person ever to have produced a piece in 11-EDO, in 1970, using a retuned electronic organ. (I had promised Igs that I would convert my open-reel tape recording to a sound file, but it completely slipped my mind; I'll have to get on it this weekend.) Having observed that there is not much consonance-to-dissonance contrast in the intervals of 11-EDO (except for the unison and octave), and since there is nothing approximating a perfect 5th (which makes it fairly easy to avoid implying a tonic key or tonal center), I concluded that this tuning would be much more suitable for atonal music than 12, so much so that there wouldn't even be any need to bother with tone rows: just use whatever tones you feel like using, whenever and wherever you wish. I tried it, and it worked beautifully! 12-tone serialism is a movement that chooses to ignore the fact that the various intervals of 12-equal have different acoustical properties, some being more consonant than others. With 11-EDO, by contrast, the tuning cooperates with their goal, with or without a serial technique.

> I would strongly disagree that individual genius can overcome even a tuning that, for example, has no consonances. This is rather like saying that Virgil could have written the AEneid without vowels, or better, without an alphabet, or better still, without a language at all. GG

Igs already made the point that a tuning with no consonances does not exist, but your statement can be disproved even with a tuning with minimal consonance-to-dissonance contrast.

> Your opinion is not uncommon among microtonalists, although very uncommon among musicians.

Be careful with your words! Someone may infer that microtonalists are not musicians! ;-)

> It arises perhaps as a reaction against the overwhelming dominance of the 12-tone equal. In order for music to affect the emotions, it must be memorable, and conform at least roughly to the brains's own innate tuning system. Music written in say, 13-tone equal is just random noise.

I don't think you would say that if you had actually listened to anything in 13-EDO. If you haven't, then here's your chance (Herman Miller's "Triskaidekaphobia", an exercise in 13-tone equal tuning, is a little more than halfway down the page):
http://www.io.com/~hmiller/music/index.html

You may not like the piece, "Dr. Barbour", but be careful of using hyperbole to dismiss something without a hearing. If there's any consolation, you're not the only one here who has done that; I did it myself when I first showed up here with a microtonal notation that I expected would be useful for just about any "worth-while" tuning:
/tuning/topicId_34071.html#34231
Eventually I realized that my notation would need to be expanded to handle both these weird octave divisions and also divisions with literally hundreds of tones per octave. Dave Keenan and I developed the essentials of Sagittal over the next 18 months, but it took about 5 years to address numerous additional details and concerns.

> It can have no more musical effect than street noises. Again, I do not say that such extreme ideas are entirely harmful. Perhaps a period of wild liberty is needed to break up the 12-tone equal mould. GG

Whatever it takes!

--George

"gdsecor" <gdsecor@...> wrote:
>
> --- In tuning@yahoogroups.com, "genewardsmith"
> <genewardsmith@...> wrote:
> >
> > --- In tuning@yahoogroups.com, "gdsecor" <gdsecor@>
> > wrote: ...
> > > One of my favorite tunings has fifths tempered wide
> > > by ~1.6237 cents in 3 separate chains: one for
> > > 3-limit intervals, the second one 5:4 above the first
> > > (for ratios of 5), and the third 7:4 above the first
> > > (for ratios of 7, 11, and 13).
> >
> > Have you ever worked with mystery temperament?
>
> No, but I'd be surprised if it's as accurate as what I'm
> using. My fifths are exactly (504/13)^(1/9), which is
> ~703.5787 cents, and all of the 15-limit consonances are
> within 3.25 cents of just.

Right. The worst 15-limit interval is 4.7 cents out. But
it looks simpler. It requires only two chains (of
29-equal) for the 15-limit. 29 equal steps to the octave
gives a fifth of 703.448 cents. 29 of your fifths will
differ from equal by 3.7 cents.

There are variants of Mystery that keep unequally tempered
fifths. The one I thought of way back was to cross Mystery
with Schismatic. The result is apparently an extension of
Hemifamity, and scores reasonably well as a planar
temperament.

Something resulting from 29&58&72 comes out looking better
according to my scoring system. I called it History.
Here's part of its description:

reduced mapping:
[<1, 2, 0, 0, 1, 2],
<0, 6, 0, -7, -2, 9],
<0, 0, 1, 1, 1, 1]>

tuning map:
[1200.000, 1901.904, 2786.650, 3367.763, 4152.682,
4439.506> cents

Mystery can be done as an open string tuning on a 29-EDO
guitar. Did I mention that when it was relevant?

Graham

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>

> > B) What's the generator for 1/3rd comma meantone (in cents)?
>
> It's pretty much identical to 19-tet's fourth (or fifth), but let's
> work it out. So for 1/3-comma meantone, three fourths should put you
> at an exact 6/5 (octave-equivalent), just like in how 1/4 comma
> meantone, four fifths puts you at an exact 5/4 (octave-equivalent).
> Think about this for a second if it doesn't make immediate sense. But
> mathematically, the derivation of the generator is simple:
>
> 1) Start with 6/5 * 2/1, or 12/5.
> 2) You know that three generators will get you to 12/5, meaning that g^3 = 12/5.
> 3) Therefore, g = (12/5)^(1/3).
> 4) So what is g in cents? cents(g) = 1200*log(g)/log(2) =
> 1200*log((12/5)^(1/3))/log(2)
> 5) Since log(x^y) = y * log(x), the 1/3 comes outside of the log and
> you get 1/3 * 1200 * log(12/5)/log(2)
> 6) This equals 505.214 cents. In comparison, 19-equal's fourth is
> 505.263 cents. That is a difference of 0.049 cents.

I explain it on my website using vector addition, which makes
everything much simpler:

http://tonalsoft.com/enc/number/19edo.aspx

>
> There is just no point pretending that 1/3-comma meantone is somehow
> distinct from 19-equal, except for on a purely mathematical level.

And at the opposite end of the meantone spectrum, it's exactly
the same for the extremely close relationship between 12-edo and
1/11-comma meantone.

However, it is actually sometimes useful to use the fraction-of-a-comma version instead of the EDO. For example, in Tonescape
i can set up a 31-note chain of 1/11-comma meantone and enter
a score with the notes spelled exactly as the composer wrote them.
It still comes out essentially as 12-edo, but on the Tonescape
lattice you can see the chords spelled exactly as they should be.

... Then, of course, i can simply swap out that tuning with
another 31-note tuning (say, 31-edo, or 31-out-of-55-edo),
save it as a separate file, and hear something entirely different!

-monz
http://tonalsoft.com/tonescape.aspx
Tonescape microtonal music software

--- In tuning@yahoogroups.com, "gdsecor" <gdsecor@...> wrote:
> ...
> Personally, I can't imagine myself wanting to write for that sort of tuning, because I believe that there are "better" options available to me as a composer. Therefore, I don't think that all tunings are of more or less equal worth. It's more a matter that not all of us agree on which ones to value most highly.
>
> Ah, there's one exception that I'll need to confess. I am probably the first person ever to have produced a piece in 11-EDO, in 1970, using a retuned electronic organ. (I had promised Igs that I would convert my open-reel tape recording to a sound file, but it completely slipped my mind; I'll have to get on it this weekend.) Having observed that there is not much consonance-to-dissonance contrast in the intervals of 11-EDO (except for the unison and octave), and since there is nothing approximating a perfect 5th (which makes it fairly easy to avoid implying a tonic key or tonal center), I concluded that this tuning would be much more suitable for atonal music than 12, so much so that there wouldn't even be any need to bother with tone rows: just use whatever tones you feel like using, whenever and wherever you wish. I tried it, and it worked beautifully! 12-tone serialism is a movement that chooses to ignore the fact that the various intervals of 12-equal have different acoustical properties, some being more consonant than others. With 11-EDO, by contrast, the tuning cooperates with their goal, with or without a serial technique.

Here's more information and a link to the sound file:
/makemicromusic/topicId_26090.html#26090

--George

--- In tuning@yahoogroups.com, "gdsecor" <gdsecor@...> wrote:

> I tried it, and it worked beautifully! 12-tone serialism is a movement that chooses to ignore the fact that the various intervals of 12-equal have different acoustical properties, some being more consonant than others. With 11-EDO, by contrast, the tuning cooperates with their goal, with or without a serial technique.

You know, the very same thought crossed my mind back then, but it never occurred to me to actually try it.

--- In tuning@yahoogroups.com, "gdsecor" <gdsecor@...> wrote:
>
> --- In tuning@yahoogroups.com, Michael <djtrancendance@> wrote:
> >
> > Fascinating. All the fifths seem to be very near 698 cents, but alternate to slightly different values. How did you calculate this?
>
> I've pasted the Scala file here for reference:
>
> ! Secor5_23TX.scl
> !
> George Secor's synchronous 5/23-comma temperament extraordinaire
> 12
> !
> 62/59 ! C#
> 66/59 ! D
> 70/59 ! Eb
> 591/472 ! E
> 631/472 ! F
> 331/236 ! F#
> 353/236 ! G
> 745/472 ! G#=Ab
> 395/236 ! A
> 631/354 ! Bb
> 221/118 ! B
> 2/1
>
> Notice that all of the numbers in the denominators are multiples of > 59.....

Sounds nice.
Also attend in the nominators the partially epimoric reduced chain of 5ths

C: 59 118
G: (D/3 := 11 22 44 88 176 352 <) 353 (< 177 354 := 3*C)
D: 33 66 132
A: (D*3 := 99 (E/3 := 197<) 198 <) 395 (< 396 D*3 )
E: 591
B: 221 442 884 1768 =5*353.6 (< 354.6*5 = 1773 := E*3)
F#: 331 662 ( < 663 := B*3)
C#: 31 62 124 248 496 992 (< 993 := F#*3)
G#: (C#*3 := 93 186 372 744 <) 745 = Ab
Eb: 35 70 140 280 560 1120 2240 = 5*448 (< 447*5 = 2235 := G#*3)
Bb: 631/3 = 210.333333... (> 210 := Eb*3)
F: 631
C: 59 118 236 472 944 1888 = 5*377.6 ( < 378.6*6 = 1893 := F*3)

Or when more concise summarized as division of the PC=3^12/2^19
into 11 epimoric relative subparts:

F 377.6/378.6 C 353/354 G 352/351 D 395/396 A 197/198 E 353.6/354.6 B
B 662/663 F# 992/993 C# 744/745 G# 448/447 Eb 631/630 Bb 1/1 F

Here attend especially the both wide meantonic 5ths inbetween G#-Eb-Bb and the only just residual one within Bb-F respectively.

Critical quest:
Did you really intend the 5th among A-E
should deviate from JI about
1200C * ln(197 / 198)) / ln(2) = ~-8.76576075...Cents flattend down?
That's at least more off than about PC^(1/3).

bye
A.S.

--- In tuning@yahoogroups.com, "Andy" <a_sparschuh@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, "gdsecor" <gdsecor@> wrote:
> >
> > --- In tuning@yahoogroups.com, Michael <djtrancendance@> wrote:
> > >
> > > Fascinating. All the fifths seem to be very near 698 cents, but alternate to slightly different values. How did you calculate this?
> >
> > I've pasted the Scala file here for reference:
> >
> > ! Secor5_23TX.scl
> > !
> > George Secor's synchronous 5/23-comma temperament extraordinaire
> > 12
> > !
> > 62/59 ! C#
> > 66/59 ! D
> > 70/59 ! Eb
> > 591/472 ! E
> > 631/472 ! F
> > 331/236 ! F#
> > 353/236 ! G
> > 745/472 ! G#=Ab
> > 395/236 ! A
> > 631/354 ! Bb
> > 221/118 ! B
> > 2/1
> >
> > Notice that all of the numbers in the denominators are multiples of > 59.....
>
> Sounds nice.
> Also attend in the nominators the partially epimoric reduced chain of 5ths
>
> C: 59 118
> G: (D/3 := 11 22 44 88 176 352 <) 353 (< 177 354 := 3*C)
> D: 33 66 132
> A: (D*3 := 99 (E/3 := 197<) 198 <) 395 (< 396 D*3 )
> E: 591
> B: 221 442 884 1768 =5*353.6 (< 354.6*5 = 1773 := E*3)
> F#: 331 662 ( < 663 := B*3)
> C#: 31 62 124 248 496 992 (< 993 := F#*3)
> G#: (C#*3 := 93 186 372 744 <) 745 = Ab
> Eb: 35 70 140 280 560 1120 2240 = 5*448 (< 447*5 = 2235 := G#*3)
> Bb: 631/3 = 210.333333... (> 210 := Eb*3)
> F: 631
> C: 59 118 236 472 944 1888 = 5*377.6 ( < 378.6*6 = 1893 := F*3)
>
> Or when more concise summarized as division of the PC=3^12/2^19
> into 11 epimoric relative subparts:
>
> F 377.6/378.6 C 353/354 G 352/351 D 395/396 A 197/198 E 353.6/354.6 B
> B 662/663 F# 992/993 C# 744/745 G# 448/447 Eb 631/630 Bb 1/1 F
>
> Here attend especially the both wide meantonic 5ths inbetween G#-Eb-Bb and the only just residual one within Bb-F respectively.
>
> Critical quest:
> Did you really intend the 5th among A-E
> should deviate from JI about
> 1200C * ln(197 / 198)) / ln(2) = ~-8.76576075...Cents flattend down?
> That's at least more off than about PC^(1/3).

Please check your figures.

A = 395/236
E = 591/472
produces a fifth of 697.567 cents, which is 4.388 cents smaller than 3/2.

--George

--- In tuning@yahoogroups.com, "gdsecor" <gdsecor@...> wrote:

> > F 377.6/378.6 C 353/354 G 352/351 D 395/396 A
> > ????? A 197/198 E ??????? (was wrongly overtaken)
> > E 353.6/354.6 B
> > B 662/663 F# 992/993 C# 744/745 G# 448/447 Eb 631/630 Bb 1/1 F

> Please check your figures.
>
> A = 395/236
> E = 591/472
> produces a fifth of 697.567 cents,
> which is 4.388 cents smaller than 3/2.

Sorry George, pardon me:
you are fully right, in detecting an error in that, because:

> A: (D*3 := 99 (E/3 := 197 394 <) 395 (< 396 D*3 )
> E: 591 := 197*3

hence my wrong considered 5th A-E has here to corrected and replaced by ....A 394/395 E .... as denotated above in the full-detailed derivation. Of course now that agrees yours own calculation:

1200C * ln(394 / 395) / ln(2) = ~-4.38842833... flattend down.

Consequenty i must retrive my unept doubts, and conclude affirmative:
The most common 3rds and 5ths do stay remainig within
the usual meantonic range of deviations,
while the remote keys appear to be sounding almost pythagorean.

please excuse and forgive me my carelessness
bye
Andy

At 07:16 AM 2/23/11 -0000, cityoftheasleep wrote:

>I should also like to add, as you seem to be on your way out the door,
that one seems to *me* to be pathetic is to spend such an enormous amount
of one's life-energy trying to validate a musical tuning based on any merit
other than the music that can be made with it. Any superiority of 19-TET
should be obvious in the sound of it, and music alone should be sufficient
to make the case for it.

Phonologists have measured the difference in pitch between the tones in
various languages. Speakers of standard English use four tones, of which
the lowest three correspond to do-re-mi.

Rigorous comparative research from other languages might tend to validate
the psychological basis of one tuning over another.

Beco dos Gatinhos
<http://www.rev.net/~aloe/music/pitch.html>

Hi Gregg,

I hope you are still around and reading this ... i only stop in at the tuning list once every few months these days.

Can you (or anyone else who knows) please cite the source of the interview
with Schoenberg which you paraphrase here?
Thanks.

-monz
http://tonalsoft.com/tonescape.aspx
Tonescape microtonal music software

--- In tuning@yahoogroups.com, "gregggibson" <gregggibson@...> wrote:
>
> <snip>
>
> I cannot resist commenting briefly on Arnold Schoenberg, however. In a famous interview, he was asked why he wanted to use 12-tone equal at all. His system implies that dissonances and consonances are to be treated in exactly the same fashion, so why use 12-tone equal, which has at least two very good consonances, the fifth and octave?
>
> Poor Schoenberg was thoroughly nonplussed. After sputtering and going red in the face, he finally confessed he did not know the answer to the question, but that he had to use what was available for the orchestra - 12-tone instruments.
>
> Even more ironically, the serialists now venerate the 12-tone equal more than most, again without logical reason. They do not have the courage of their convictions.
>
> Gregg Gibson
>