What Would a Meantone Successor Look Like?

I came back to this list because this is a question to which I've been giving a lot of thought lately, and XA is no place for deep, drawn-out discussion. I really want to hammer this out.

I've been doing this microtonal thing for a long time now, I have tons of experience writing music in lots of different tunings, tons of experience debating the intricacies of various theories and practices, and tons of experience trying to apply different tuning approaches to what I consider contemporary styles of music. I figure, it's time to stop meandering from tuning to tuning, throwing tons of money at luthiers to outfit me with a larger and larger variety of incompatible instruments. It's time to start really developing a praxis around one specific tuning--become fluent in its notation, get a set of matching instruments, and really stake my claim as a theorist that *this* tuning above all others embodies the direction that I think music should go in. It needs to be a tuning that I can reasonably expect other people to embrace on a decently large scale. This means it needs to work in some analogous way to the current meantone/pythagorean/12-TET paradigm, but be different in enough key ways that it can solve some compositional problems that the current paradigm totally fails at, problems which a substantive number of musicians are actually troubled by.

First, I think we all need to remember that meantone is a theoretical construct that in practice actually manifests as a variety of tuning systems. It encompasses 12-TET as manifested on the straight frets of a guitar and (perhaps) the fingerings of the brass and woodwinds (which I don't understand in-depth and are apparently somewhat intonationally-flexible); adaptive Pythagorean- or 5-limit JI with a tempered skeleton, as manifested on free-pitched instruments like voice and strings; interlaced transposed segments of the overtone series, as manifested on horns, and the stretched "equal" tunings of pianos and tuned percussion instruments. Any tuning system that would hope to succeed meantone would have to work similarly--it would need to be representable in an equal temperament of reasonable size that it could work on a guitar and (probably) a dedicated keyboard, but also not so far off from some JI system that it could not be performed by an ensemble of free-pitch instruments. It would be best if it even had some direct relationship to the overtone series, such that horn players could follow a similar method of transposing and interlacing harmonic series segments, without sounding too far off from the tempered tunings being played on the other instruments.

This is actually quite a list of constraints.

My point of departure is first to look at what problems are actually bothering composers and performers today. The obvious one is "lack of novelty"--see, for instance:

http://www.youtube.com/watch?v=krDxhnaKD7Q
http://www.youtube.com/watch?v=oOlDewpCfZQ

Musicians are starting to laugh at the fact that everything is starting to sound the same. Yay!

Another problem is that 12-TET can't really do those wacky "quarter-tones" that are all over these funky "Eastern" scales--quarter-tone frets are cropping on some Ibanez guitars, for instance:

http://www.ibanez.co.jp/oriental/Features.html

And there are guys like Trey Spruance using modified guitars to approximate some Middle Eastern scales:

http://en.wikipedia.org/wiki/Secret_Chiefs_3

And there's also the problem that leads an awful lot of people into microtonality, namely the intonational inaccuracy of 12-TET. There are just too many names to name here, so I won't even bother.

The first problem, the problem of novelty, is the easiest to solve, because literally *every* non-12-TET tuning offers *something* new. Witness the plethora of tunings that have actually been used in music by at least one composer--do we even know definitively how many tunings have been composed in by microtonalists in the last century?

The second problem, of incorporating Eastern scales, suggests tunings with neutral intervals atop some sort of Pythagorean-esque back-bone. In the ET corner, we're looking at 15, 16, 17, maaaaybe 19, 22, 24, 26, 27, 29, 31, 34, and 36. 15 and 16 are the least Pythagorean in structure, having the weakest 3/2's. 19 can pull off some pseudo-Middle-Eastern sounds with, perhaps, MODMOS's of Negri or Magic or something, but lacks the neutral 2nds and 3rds of the other tunings. 22 can do the neutral 2nd thing but not the neutral 3rd. Ditto 29. 26 is pushing the boundary of neutraldom with its wide minor 2nds and narrow major 3rds--maybe not quite there. 34 takes all its neutral intervals from 17. So, 17, 24, 27, 31, and 36 are probably the best bets for solving this problem.

The third problem, of intonational inaccuracy, is also closer to solution in all of these--if we leave out the 5th harmonic, both 17 and 24 are quite accurate on the rest of the 13-limit, with 17 having a decent edge. Among temperaments of the 2.3.7.11.13 subgroup, only 36 is more accurate, and it's not by much. 27 and 31 are both very good on the full 13-limit, 5th harmonic and all. We might as well kick 36 and 24 off the list, since 17 does so well on the 2.3.7.11.13 subgroup and is so very much simpler.

So, we are left with three ETs--17, 27, and 31. 31 is the most accurate but also the largest; questions of feasibility on physical instruments aside, it also poses a large conceptual and pedagogical challenge. It is an immense leap from 12 to 31, and the new varieties of chords, scales, accidentals, key signatures, etc. are *orders of magnitude* beyond what most musicians are accustomed to. The learning curve is very steep. People have been advocating for it for centuries, but it hasn't caught on, most likely because of this huge increase in complexity over 12-TET. It is very susceptible to that old chestnut, "why even bother with fixed pitches? Just go free intonation, if you're going bother with that many notes."

27, at only four fewer notes than 31, is susceptible to similar problems. It might appeal to some intrepid musicians, most likely virtuousos or those who use isomorphic keyboards, but for those on guitar, brass, tuned percussion, woodwinds, voice, and strings, it is unlikely to be very appealing.

This leaves 17. Economical, accurate, and lacking only in the ratios of 5. The loss of the ratios of 5 is significant, no doubt--but it is compensated for by ratios of 7, 11, and 13. It even has a half-assed sort of backwards-compatibility via the diatonic scale, which is probably good enough for rock music. If the whole purpose of adopting a new tuning is to move in a new direction, especially an Eastern-influenced one incorporating higher members of the harmonic series, then loss of full 5-limit backwards-compatibility shouldn't be a big deal.

Of course, we should also remember that 17-equal is just a convenience, a point of departure. In practice, in more organic settings, there is plenty of room for push-and-pull, through adaptive JI that might sneak some ratios of 5 back in, or through a circulating temperament such as that proposed by George Secor that gets more intervals closer to JI. Or we could take one of the nice rank-2 temperaments supported in 17, like bleu, maqamic/mohaha, huxley, or machine, and do up an optimized 17-note MOS--since 17 is already pretty accurate, it should be close enough to "play nicely" with a more optimal form of itself. The "17" is just the kernel of the system, much like how 12 is the kernel of our current intonational practices.

So that's my case for 17. I think it's pretty solid...obviously it's not the be-all, end-all of tunings, but it's the one I think is best-suited for large-scale deployment.

-Igs

I like everything you wrote, although I wouldn't rule out 19 and 22 as
well. But I honestly think we're headed into a direction in which more
than one tuning will become popular. I think there will be at least

1) one "one trick pony" novelty tuning that catches on just for fun,
like 7, 8, 9, 10, maybe BP, stuff like that
2) one more "serious" tuning that catches on, like 17, 19, 22, etc
3) some other hodgepodge of tunings that see use here and there
because they have random nice properties, like 15, 16, whatever EDO

I can't encourage you enough in picking one tuning to get really
serious about though. I think that Ron Sword's decision to do that
with 16-EDO and mavila has generated a million useful realizations.

-Mike

On Wed, Jan 4, 2012 at 1:28 AM, cityoftheasleep <igliashon@...> wrote:
>
> I came back to this list because this is a question to which I've been giving a lot of thought lately, and XA is no place for deep, drawn-out discussion. I really want to hammer this out.
>
> I've been doing this microtonal thing for a long time now, I have tons of experience writing music in lots of different tunings, tons of experience debating the intricacies of various theories and practices, and tons of experience trying to apply different tuning approaches to what I consider contemporary styles of music. I figure, it's time to stop meandering from tuning to tuning, throwing tons of money at luthiers to outfit me with a larger and larger variety of incompatible instruments. It's time to start really developing a praxis around one specific tuning--become fluent in its notation, get a set of matching instruments, and really stake my claim as a theorist that *this* tuning above all others embodies the direction that I think music should go in. It needs to be a tuning that I can reasonably expect other people to embrace on a decently large scale. This means it needs to work in some analogous way to the current meantone/pythagorean/12-TET paradigm, but be different in enough key ways that it can solve some compositional problems that the current paradigm totally fails at, problems which a substantive number of musicians are actually troubled by.
>
> First, I think we all need to remember that meantone is a theoretical construct that in practice actually manifests as a variety of tuning systems. It encompasses 12-TET as manifested on the straight frets of a guitar and (perhaps) the fingerings of the brass and woodwinds (which I don't understand in-depth and are apparently somewhat intonationally-flexible); adaptive Pythagorean- or 5-limit JI with a tempered skeleton, as manifested on free-pitched instruments like voice and strings; interlaced transposed segments of the overtone series, as manifested on horns, and the stretched "equal" tunings of pianos and tuned percussion instruments. Any tuning system that would hope to succeed meantone would have to work similarly--it would need to be representable in an equal temperament of reasonable size that it could work on a guitar and (probably) a dedicated keyboard, but also not so far off from some JI system that it could not be performed by an ensemble of free-pitch instruments. It would be best if it even had some direct relationship to the overtone series, such that horn players could follow a similar method of transposing and interlacing harmonic series segments, without sounding too far off from the tempered tunings being played on the other instruments.
>
> This is actually quite a list of constraints.
>
> My point of departure is first to look at what problems are actually bothering composers and performers today. The obvious one is "lack of novelty"--see, for instance:
>
> http://www.youtube.com/watch?v=krDxhnaKD7Q
> http://www.youtube.com/watch?v=oOlDewpCfZQ
>
> Musicians are starting to laugh at the fact that everything is starting to sound the same. Yay!
>
> Another problem is that 12-TET can't really do those wacky "quarter-tones" that are all over these funky "Eastern" scales--quarter-tone frets are cropping on some Ibanez guitars, for instance:
>
> http://www.ibanez.co.jp/oriental/Features.html
>
> And there are guys like Trey Spruance using modified guitars to approximate some Middle Eastern scales:
>
> http://en.wikipedia.org/wiki/Secret_Chiefs_3
>
> And there's also the problem that leads an awful lot of people into microtonality, namely the intonational inaccuracy of 12-TET. There are just too many names to name here, so I won't even bother.
>
> The first problem, the problem of novelty, is the easiest to solve, because literally *every* non-12-TET tuning offers *something* new. Witness the plethora of tunings that have actually been used in music by at least one composer--do we even know definitively how many tunings have been composed in by microtonalists in the last century?
>
> The second problem, of incorporating Eastern scales, suggests tunings with neutral intervals atop some sort of Pythagorean-esque back-bone. In the ET corner, we're looking at 15, 16, 17, maaaaybe 19, 22, 24, 26, 27, 29, 31, 34, and 36. 15 and 16 are the least Pythagorean in structure, having the weakest 3/2's. 19 can pull off some pseudo-Middle-Eastern sounds with, perhaps, MODMOS's of Negri or Magic or something, but lacks the neutral 2nds and 3rds of the other tunings. 22 can do the neutral 2nd thing but not the neutral 3rd. Ditto 29. 26 is pushing the boundary of neutraldom with its wide minor 2nds and narrow major 3rds--maybe not quite there. 34 takes all its neutral intervals from 17. So, 17, 24, 27, 31, and 36 are probably the best bets for solving this problem.
>
> The third problem, of intonational inaccuracy, is also closer to solution in all of these--if we leave out the 5th harmonic, both 17 and 24 are quite accurate on the rest of the 13-limit, with 17 having a decent edge. Among temperaments of the 2.3.7.11.13 subgroup, only 36 is more accurate, and it's not by much. 27 and 31 are both very good on the full 13-limit, 5th harmonic and all. We might as well kick 36 and 24 off the list, since 17 does so well on the 2.3.7.11.13 subgroup and is so very much simpler.
>
> So, we are left with three ETs--17, 27, and 31. 31 is the most accurate but also the largest; questions of feasibility on physical instruments aside, it also poses a large conceptual and pedagogical challenge. It is an immense leap from 12 to 31, and the new varieties of chords, scales, accidentals, key signatures, etc. are *orders of magnitude* beyond what most musicians are accustomed to. The learning curve is very steep. People have been advocating for it for centuries, but it hasn't caught on, most likely because of this huge increase in complexity over 12-TET. It is very susceptible to that old chestnut, "why even bother with fixed pitches? Just go free intonation, if you're going bother with that many notes."
>
> 27, at only four fewer notes than 31, is susceptible to similar problems. It might appeal to some intrepid musicians, most likely virtuousos or those who use isomorphic keyboards, but for those on guitar, brass, tuned percussion, woodwinds, voice, and strings, it is unlikely to be very appealing.
>
> This leaves 17. Economical, accurate, and lacking only in the ratios of 5. The loss of the ratios of 5 is significant, no doubt--but it is compensated for by ratios of 7, 11, and 13. It even has a half-assed sort of backwards-compatibility via the diatonic scale, which is probably good enough for rock music. If the whole purpose of adopting a new tuning is to move in a new direction, especially an Eastern-influenced one incorporating higher members of the harmonic series, then loss of full 5-limit backwards-compatibility shouldn't be a big deal.
>
> Of course, we should also remember that 17-equal is just a convenience, a point of departure. In practice, in more organic settings, there is plenty of room for push-and-pull, through adaptive JI that might sneak some ratios of 5 back in, or through a circulating temperament such as that proposed by George Secor that gets more intervals closer to JI. Or we could take one of the nice rank-2 temperaments supported in 17, like bleu, maqamic/mohaha, huxley, or machine, and do up an optimized 17-note MOS--since 17 is already pretty accurate, it should be close enough to "play nicely" with a more optimal form of itself. The "17" is just the kernel of the system, much like how 12 is the kernel of our current intonational practices.
>
> So that's my case for 17. I think it's pretty solid...obviously it's not the be-all, end-all of tunings, but it's the one I think is best-suited for large-scale deployment.
>
> -Igs

"cityoftheasleep" <igliashon@...> wrote:

> I've been doing this microtonal thing for a long time now,
> I have tons of experience writing music in lots of different
> tunings, tons of experience debating the intricacies of
> various theories and practices, and tons of experience trying
> to apply different tuning approaches to what I consider
> contemporary styles of music.

Yeah, but you don't like Bach (or haven't ever listened
to it, or something) so why should we trust you? :P

> So that's my case for 17. I think it's pretty solid...
> obviously it's not the be-all, end-all of tunings, but it's
> the one I think is best-suited for large-scale deployment.

17 is neat. I really like a lot of things about it.
But it has some kind of problem that it just can't seem
to overcome. Three entire concerts of piano music in 17
(and numerous other compositions over the years) have
practically convinced me of that. Of course it's possible
everyone just missed the right MODMOS of the right no-5s
temperament it supports, so I'll keep an open mind.

-Carl

Main requirements for a new standard system would indeed be suitability for oriental music and better support of the higher overtones. Additionally, in any case, comparably good support for the hitherto existing (western) music would be a "must". In this respect, 17edo is IMHO clearly out of question. 5-limit intervals are just too important to be neglected.

If the goal is ONE tuning for all, and 31edo is considered already too complex, there is no solution, for there is no low-number equal temperament that does not have one ore another defect.

If we drop the condition of the low number of notes, the most obvious candidates would be 53edo and 72edo, then maybe 41edo or 34edo. In a part of the academic music world (Boston microtonal society, the ekmelic music institute in Austria), there is apparently a sort of agreement towards 72edo as new standard, which meets all of the conditions stated above (support for the old standard is guaranteed in a natural way since 12edo is a subset of it - a property the other candidates do not have).

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> I can't encourage you enough in picking one tuning to get really
> serious about though.
>

That's basically what my approach is, too. When I was starting in microtonality, I thought that getting as familiar with a new tuning as I had become with 12edo, I would have to spend the rest of my life with one. First I thought this one would be 22edo - as it happened then, along came 5edo, 17edo, 19edo and 96edo, too. But currently I indeed mainly concentrate on 22edo and 17edo.
--
Hans Straub

On Wed, Jan 4, 2012 at 5:09 AM, hstraub64 <straub@...> wrote:
>
> Main requirements for a new standard system would indeed be suitability for oriental music and better support of the higher overtones. Additionally, in any case, comparably good support for the hitherto existing (western) music would be a "must". In this respect, 17edo is IMHO clearly out of question. 5-limit intervals are just too important to be neglected.
>
> If the goal is ONE tuning for all, and 31edo is considered already too complex, there is no solution, for there is no low-number equal temperament that does not have one ore another defect.

24-EDO?

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Wed, Jan 4, 2012 at 5:09 AM, hstraub64 <straub@...> wrote:
> >
> > Main requirements for a new standard system would indeed be
> > suitability for oriental music and better support of the higher
> > overtones. Additionally, in any case, comparably good support for
> > the hitherto existing (western) music would be a "must". In this
> > respect, 17edo is IMHO clearly out of question. 5-limit intervals
> > are just too important to be neglected.
> >
> > If the goal is ONE tuning for all, and 31edo is considered
> > already too complex, there is no solution, for there is no
> > low-number equal temperament that does not have one ore another
> > defect.
>
> 24-EDO?
>

Has a few points (which may partly explain its relative popularity): natural support of the hitherto existing tuning system, neutral seconds and thirds, hence at least rudimentary justice to oriental music. As for overtones, not much better than 12edo except for 11.

A few points, yes - but maybe not enough...

And oriental music will present a forever sticking point concerning lowest possible number of notes...
--
Hans Straub

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

>
> So that's my case for 17. I think it's pretty solid

That's because you assume everyone has to play music on a guitar.

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> I can't encourage you enough in picking one tuning to get really
> serious about though. I think that Ron Sword's decision to do that
> with 16-EDO and mavila has generated a million useful realizations.

Name one.

I'm beginning to wonder if part of what's screwing up microtonal music is the guitar. What if, instead of tossing out his 19edo guitar, Igs tossed out all of them, and started over with an instrument which didn't require you to play in a low-sized edo? As a bonus one on which it is impossible to shred or sound like a badass.

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

> > If the goal is ONE tuning for all, and 31edo is considered already too complex, there is no solution, for there is no low-number equal temperament that does not have one ore another defect.
>
> 24-EDO?

24 has defects you could drive a dump truck through.

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> I like everything you wrote, although I wouldn't rule out 19 and 22 as
> well. But I honestly think we're headed into a direction in which more
> than one tuning will become popular. I think there will be at least
>
> 1) one "one trick pony" novelty tuning that catches on just for fun,
> like 7, 8, 9, 10, maybe BP, stuff like that
> 2) one more "serious" tuning that catches on, like 17, 19, 22, etc
> 3) some other hodgepodge of tunings that see use here and there
> because they have random nice properties, like 15, 16, whatever EDO

I agree with all of the above. I'm certainly not going to be giving up my other tunings. But I've been trying to figure out the most "marketable" tuning, and I think the arguments for 17 are probably the most compelling.

> I can't encourage you enough in picking one tuning to get really
> serious about though. I think that Ron Sword's decision to do that
> with 16-EDO and mavila has generated a million useful realizations.

Hang on to your hat, then, because I'm gonna blow the roof off this sucker in the coming months.

-Igs

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:

> Yeah, but you don't like Bach (or haven't ever listened
> to it, or something) so why should we trust you? :P

Because the majority of the music-consuming public doesn't listen to Bach, either?

> 17 is neat. I really like a lot of things about it.
> But it has some kind of problem that it just can't seem
> to overcome. Three entire concerts of piano music in 17
> (and numerous other compositions over the years) have
> practically convinced me of that. Of course it's possible
> everyone just missed the right MODMOS of the right no-5s
> temperament it supports, so I'll keep an open mind.

Most of what's been written in 17 has totally side-stepped what I consider its two main benefits (accurate 2.3.7.11.13 harmony, and approximation of Eastern melodic forms). I'm no fan of it either, barring Chris V.'s pieces and the Blackwood etude. But the fact that enough people have been interested in 17 to write material for three entire piano concerts (and numerous other compositions) maybe says something more important than the quality of the resultant music....

-Igs

--- In tuning@yahoogroups.com, "hstraub64" <straub@...> wrote:
> Main requirements for a new standard system would indeed be suitability for oriental
> music and better support of the higher overtones. Additionally, in any case, comparably > good support for the hitherto existing (western) music would be a "must". In this respect, > 17edo is IMHO clearly out of question. 5-limit intervals are just too important to be
> neglected.

With all due respect, I'm not entirely convinced that a new standard system needs to absorb and supplant the existing one. I think that is where we have been mistaken in our previous efforts, ignoring the fact that evolution tends not toward greater complexity but toward greater diversity. 17-TET is appealing precisely because it fills a very different niche than 12-TET, and fills that niche extremely well.

> If the goal is ONE tuning for all, and 31edo is considered already too complex, there is
> no solution, for there is no low-number equal temperament that does not have one ore
> another defect.

Exactly my point.

-Igs

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

> That's because you assume everyone has to play music on a guitar.
>

Did you even read my post?? It's because I assume people want to play music on instruments (including horns, strings, non-generalized keyboards, tuned percussion, etc.), and that people don't want to learn a system of scales and harmonies that is orders of magnitude more complicated than what they are used to.

And I specifically mentioned at the end that the tuning doesn't have to be only 17-equal, but is just based on a pattern of 17 reasonably-equal notes. There's room for a 17-based adaptive 2.3.7.11.13 JI scheme, a circulating 17-note temperament, a 17-note MOS of some 2.3.7.11.13 temperament, a 17-note gamut of interlaced transposed harmonic series fragments, etc. All of these 17-note systems could play reasonably well together, just as the huge variety of 12-note systems can and do play well together in modern musical practice.

-Igs

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
> I'm beginning to wonder if part of what's screwing up microtonal music is the guitar. What > if, instead of tossing out his 19edo guitar, Igs tossed out all of them, and started over with > an instrument which didn't require you to play in a low-sized edo? As a bonus one on which > it is impossible to shred or sound like a badass.

How could a tuning be expected to succeed meantone and 12-TET in any way, if it is substantially more difficult to compose and perform with on the majority of musical instruments? It's not just the guitar, Gene.

-Igs

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

> > 24-EDO?
>
> 24 has defects you could drive a dump truck through.
>

24-EDO is the clear winner if we want Eastern scales but refuse to give up 5-limit harmony. On the 2.3.5.11.13 subgroup, it is substantially more accurate than anything lower than it, and while 29 and 31 are slightly better, only 34 is significantly better.

-Igs

--- In tuning@yahoogroups.com, "hstraub64" <straub@...> wrote:

> If the goal is ONE tuning for all, and 31edo is considered
> already too complex, there is no solution, for there is no
> low-number equal temperament that does not have one ore
> another defect.

I think the situation looks something like this:

http://lumma.org/temp/PickAnyTwo.png

-Carl

I wrote:

> I think the situation looks something like this:
> http://lumma.org/temp/PickAnyTwo.png

Bonus points for anyone who can get the additive
relationships to work all three directions and still
have the diagram make sense. Rules:

* You're allowed to write 13 and pretend it's 12.

* No EDOs larger than 99.

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> I wrote:
>
> > I think the situation looks something like this:
> > http://lumma.org/temp/PickAnyTwo.png
>
> Bonus points for anyone who can get the additive
> relationships to work all three directions and still
> have the diagram make sense. Rules:
>
> * You're allowed to write 13 and pretend it's 12.
>
> * No EDOs larger than 99.

I don't get what the "only one new tuning to learn" circle means. You have 19 and 31 in there which are meantone, but also 72 which is not meantone.

Also, I have no idea why 31 is outside the "better accuracy" circle.

Keenan

Guitars don't require you to play in a low-sized EDO. Guitars have been proven to work very well in meantone. There is, however, a school of microtonal guitarists who haven't tried meantone frettings and assume they must be too complicated.

Graham

------Original message------
From: genewardsmith <genewardsmith@...>
To: <tuning@yahoogroups.com>
Date: Wednesday, January 4, 2012 4:37:21 PM GMT-0000
Subject: [tuning] Re: What Would a Meantone Successor Look Like?

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> I can't encourage you enough in picking one tuning to get really
> serious about though. I think that Ron Sword's decision to do that
> with 16-EDO and mavila has generated a million useful realizations.

Name one.

I'm beginning to wonder if part of what's screwing up microtonal music is the guitar. What if, instead of tossing out his 19edo guitar, Igs tossed out all of them, and started over with an instrument which didn't require you to play in a low-sized edo? As a bonus one on which it is impossible to shred or sound like a badass.

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--- In tuning@yahoogroups.com, "Keenan Pepper" <keenanpepper@...> w
> I don't get what the "only one new tuning to learn"
> circle means.

Ha! I knew I'd get you with this! :)

"Only one new" means:
supports meantone, "oriental" scales, and at least one
good rank-2 system without 81/80 in its kernel.

> You have 19 and 31 in there which are meantone, but also 72
> which is not meantone.

Subsets are allowed, so 72 contains 12 (a meantone).

> Also, I have no idea why 31 is outside the "better accuracy"
> circle.

Rules for the better accuracy circle:
1. Its ET must have better accuracy than the ETs of the
other two circles
2. Its overlaps must have better accuracy than the ET of
the circle being overlapped

So, 41 > 31 and
41 > 12 or 13 and
72 > 31 and
22 > 12 or 13

-Carl

Rules for the "not too many notes" circle:
1. Must be > 12 and < 100
2. Option to use the ET written or 12 when summing
3. Overlaps must reduce the ET of the overlapped circle

Furthermore I brazenly assert that the solution I gave
is the best one.

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:

> "Only one new" means:
> supports meantone, "oriental" scales, and at least one
> good rank-2 system without 81/80 in its kernel.
[snip]
> Rules for the better accuracy circle:
> 1. Its ET must have better accuracy than the ETs of the
> other two circles
> 2. Its overlaps must have better accuracy than the ET of
> the circle being overlapped

On Wed, Jan 4, 2012 at 2:59 PM, Carl Lumma <carl@...> wrote:
>
> Rules for the "not too many notes" circle:
> 1. Must be > 12 and < 100
> 2. Option to use the ET written or 12 when summing
> 3. Overlaps must reduce the ET of the overlapped circle
>
> Furthermore I brazenly assert that the solution I gave
> is the best one.

Why not 11 instead of 13?

-Mike

On Wed, Jan 4, 2012 at 11:37 AM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> >
> > I can't encourage you enough in picking one tuning to get really
> > serious about though. I think that Ron Sword's decision to do that
> > with 16-EDO and mavila has generated a million useful realizations.
>
> Name one.

I feel like this question is a booby trap.

> I'm beginning to wonder if part of what's screwing up microtonal music is the guitar. What if, instead of tossing out his 19edo guitar, Igs tossed out all of them, and started over with an instrument which didn't require you to play in a low-sized edo? As a bonus one on which it is impossible to shred or sound like a badass.

I dunno, Chromosounds is pretty badass. There's some shredding going on in that.

But the only problem I see is that people aren't willing to spend a
few weeks playing around with lower-accuracy EDOs to let their ears
adjust. I've been trying to tell you this for months now, that it
simply stops sounding "out of tune" if you let yourself adapt and be
able to predict what's going on. You said that you reserved the right
to have whatever preferences you want, so I let it go. But now it
looks you're coming up with theories about how we're all gravitating
towards music that we actually think sounds displeasurable because
we're trying to be cool and sound like badasses.

-Mike

On Wed, Jan 4, 2012 at 11:41 AM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > > If the goal is ONE tuning for all, and 31edo is considered already too complex, there is no solution, for there is no low-number equal temperament that does not have one ore another defect.
> >
> > 24-EDO?
>
> 24 has defects you could drive a dump truck through.

Aside from the lack of 7/4, what?

-Mike

I suppose that'd be OK. My first thought was to exclude
macrotemperaments, since people are already using 12/oct
and 12-ET is far more accurate than anything smaller. -C.

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Wed, Jan 4, 2012 at 2:59 PM, Carl Lumma <carl@...> wrote:
> >
> > Rules for the "not too many notes" circle:
> > 1. Must be > 12 and < 100
> > 2. Option to use the ET written or 12 when summing
> > 3. Overlaps must reduce the ET of the overlapped circle
> >
> > Furthermore I brazenly assert that the solution I gave
> > is the best one.
>
> Why not 11 instead of 13?
>
> -Mike
>

11-EDO works wonders for the 2.7.9.11 subgroup; it's as good for that
as is 22-EDO in the full 11-limit and only a bit worse than 12-EDO in
the 5-limit. 4:7:9:11 is a great chord, especially if voiced like that
(don't play 8:9:11:14). That's rather accurate tetradic harmony with
only 11 notes per octave for a chord that I don't think anyone will
argue is too complex to hear as concordant.

I did a quick demo of a basic chord progression in 11-EDO "extended
machine" harmony a while ago just to demo that 11-EDO doesn't have to
sound inharmonic and atonal

http://www.youtube.com/watch?v=AhPjsCoMy-Q

I keep holding out to figure out how to crank the same kind of sound
out of 13-EDO, but I haven't found it yet. 13-EDO has 2.5.9.11.13, but
I haven't liked it as much as 2.7.9.11 in 11-EDO, which immediately
made sense to me. I still like 13-EDO though, I just think it's more
complex.

11-EDO is a real gem though, especially if you're willing to get away
from 3/2, which you should be.

-Mike

On Wed, Jan 4, 2012 at 4:29 PM, Carl Lumma <carl@...> wrote:
>
> I suppose that'd be OK. My first thought was to exclude
> macrotemperaments, since people are already using 12/oct
> and 12-ET is far more accurate than anything smaller. -C.

Do you have something against subgroups? You're always talking like 19 and 22 as if they're the only accurate ETs between 12 and 31. If you're willing to drop just one of harmonics 3, 5, and 7 from the 13-limit, it's possible to find temperaments equally or even more accurate than 22-TET or 19-TET without having to be so much larger. 17, 20, 21, 23, and 24 all beat 22 and 19 on certain 5-dimensional 13-limit subgroups (meaning they allow for near-Just pentadic harmonies at minimum). And some of them at their best are even better than 22 or 19 at their best.

-Igs

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, "hstraub64" <straub@> wrote:
>
> > If the goal is ONE tuning for all, and 31edo is considered
> > already too complex, there is no solution, for there is no
> > low-number equal temperament that does not have one ore
> > another defect.
>
> I think the situation looks something like this:
>
> http://lumma.org/temp/PickAnyTwo.png
>
> -Carl
>

Some guitarists get by just fine with 41-ED2, or a 64-tone JI system, or an extended irrational meantone fretting where some pairs of frets are barely a fingernail's breadth apart. The fact that some people with very strong convictions about certain tunings can adapt themselves to play certain instruments is a non-starter. Equal tunings have plenty of advantages that you're ignoring. They support multiple rank-2 temperaments, for instance, and don't require musicians to memorize which keys have which intervals. They are, in other words, easier to learn (mentally) and more versatile. The fact that 12-TET supports diminished and augmented temperaments allowed composers like Stravinsky and Tcherpnin to squeeze a lot more life out of it.

--- In tuning@yahoogroups.com, "gbreed@..." <gbreed@...> wrote:
>
> Guitars don't require you to play in a low-sized EDO. Guitars have been proven to work very well in meantone. There is, however, a school of microtonal guitarists who haven't tried meantone frettings and assume they must be too complicated.
>
>
> Graham
>
> ------Original message------
> From: genewardsmith <genewardsmith@...>
> To: <tuning@yahoogroups.com>
> Date: Wednesday, January 4, 2012 4:37:21 PM GMT-0000
> Subject: [tuning] Re: What Would a Meantone Successor Look Like?
>
>
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@> wrote:
> >
> > I can't encourage you enough in picking one tuning to get really
> > serious about though. I think that Ron Sword's decision to do that
> > with 16-EDO and mavila has generated a million useful realizations.
>
> Name one.
>
> I'm beginning to wonder if part of what's screwing up microtonal music is the guitar. What if, instead of tossing out his 19edo guitar, Igs tossed out all of them, and started over with an instrument which didn't require you to play in a low-sized edo? As a bonus one on which it is impossible to shred or sound like a badass.
>
>
>
>
> ------------------------------------
>
> You can configure your subscription by sending an empty email to one
> of these addresses (from the address at which you receive the list):
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>

My diagram's more accurate:

http://www.mikebattagliamusic.com/music/BestEDOs.html

-Mike

On Wed, Jan 4, 2012 at 2:09 PM, Carl Lumma <carl@...> wrote:

> **
>
>
> I wrote:
>
> > I think the situation looks something like this:
> > http://lumma.org/temp/PickAnyTwo.png
>
> Bonus points for anyone who can get the additive
> relationships to work all three directions and still
> have the diagram make sense. Rules:
>
> * You're allowed to write 13 and pretend it's 12.
>
> * No EDOs larger than 99.
>
> -Carl
>

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:
>
> --- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@> wrote:
>
> > That's because you assume everyone has to play music on a guitar.
> >
>
> Did you even read my post??

Yes, and as usual you assume low-complexity edos are the only resonable tunings, a thoroghly guitar-based point of view.

It's because I assume people want to play music on instruments (including horns, strings, non-generalized keyboards, tuned percussion, etc.), and that people don't want to learn a system of scales and harmonies that is orders of magnitude more complicated than what they are used to.
>
> And I specifically mentioned at the end that the tuning doesn't have to be only 17-equal, but is just based on a pattern of 17 reasonably-equal notes. There's room for a 17-based adaptive 2.3.7.11.13 JI scheme, a circulating 17-note temperament, a 17-note MOS of some 2.3.7.11.13 temperament, a 17-note gamut of interlaced transposed harmonic series fragments, etc.

All of these fall into the category of unequal scales, which you don't seem to much like. Maybe you should try some unequal 17-note scales, I have plenty I could suggest.

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

> But now it
> looks you're coming up with theories about how we're all gravitating
> towards music that we actually think sounds displeasurable because
> we're trying to be cool and sound like badasses.

It's a theory. So is boiling frog syndrome. I'm open to others.

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

> > 24 has defects you could drive a dump truck through.
>
> Aside from the lack of 7/4, what?

The 5/4 of 400 cents.

On Wed, Jan 4, 2012 at 7:29 PM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > But now it
> > looks you're coming up with theories about how we're all gravitating
> > towards music that we actually think sounds displeasurable because
> > we're trying to be cool and sound like badasses.
>
> It's a theory. So is boiling frog syndrome. I'm open to others.

The last time I rebutted your theory, you played the "I like what I
like and that's my right" card, so I stopped. But if you must know,
here's the answer to
your question. You asked this:

On Wed, Jan 4, 2012 at 11:37 AM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> >
> > I can't encourage you enough in picking one tuning to get really
> > serious about though. I think that Ron Sword's decision to do that
> > with 16-EDO and mavila has generated a million useful realizations.
>
> Name one.

The biggest single useful discovery made is that if you just stop
complaining, and spend some time with it, you get used to it and it
stops sounding out of tune. That's it. The Balinese and the Chopi have
clearly figured this one out already, and it took me about a month of
solidly immersing myself in 16-EDO to also feel the same way, and you
can figure it out too.

I view this to be a point of almost spiritual significance, because
what I'm saying is that you're all not perceiving reality as it
actually exists. You're perceiving it the way it doesn't exist. If you
just stop seeing what these tunings are NOT, and start seeing what
they ARE, then there's a million things you can do with them.
Especially 16-EDO and mavila, which has all of these awesome
properties in every single regard except harmonic accuracy.

Don't you see that you're in the throes of a terrible addiction to
accurate harmony! You and Carl, the both of you. This is an
intervention!

-Mike

On Wed, Jan 4, 2012 at 7:30 PM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > > 24 has defects you could drive a dump truck through.
> >
> > Aside from the lack of 7/4, what?
>
> The 5/4 of 400 cents.

OK, here's a question: in addition to all of this stuff that we're
doing in exploring other tuning systems, do you like music in 12-EDO?
Like a Wagner symphony or something?

-Mike

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

> OK, here's a question: in addition to all of this stuff that we're
> doing in exploring other tuning systems, do you like music in 12-EDO?
> Like a Wagner symphony or something?

Wagner only wrote one symphony and its not very good. But 12edo is OK, though it's not ideal for 5-limit harmony.

On Wed, Jan 4, 2012 at 8:25 PM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > OK, here's a question: in addition to all of this stuff that we're
> > doing in exploring other tuning systems, do you like music in 12-EDO?
> > Like a Wagner symphony or something?
>
> Wagner only wrote one symphony and its not very good. But 12edo is OK, though it's not ideal for 5-limit harmony.

Alright, so if you can tolerate that, then what's wrong with 24-EDO,
which gives you the same thing and 11 and 13 as well?

-Mike

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
> Yes, and as usual you assume low-complexity edos are the only resonable tunings, a
> thoroghly guitar-based point of view.

Not low-complexity EDOs, low-complexity tunings that have some plausible relationship to low-complexity EDOs.

> All of these fall into the category of unequal scales, which you don't seem to much like.

There is a realization that hit me like a ton of bricks recently, and that's that current intonational practices do not stick strictly to one exact intonation. There is a variety of (often instrument-specific) intonational flavors within orbit around 12-TET. Why should the "successor" of 12-TET be any different?

Let me say this as succinctly as possible: I'm not proposing that 17-TET is, all by itself, the best successor for all current intonational practices. I'm proposing a system of 17-note tunings that are in the loose gravitational field of 17-TET. That there are 17 notes, **roughly** evenly-distributed in an approximate octave, should be all that is necessary.

> Maybe you should try some unequal 17-note scales, I have plenty I could suggest.

Sure! I will probably stick to 17 for guitar, but if violins playing in adaptive Pythagorean JI can accompany a piano in stretched-octave 12-equal, then strings or synths in an unequal 13-limit 17-note tuning could probably accompany a guitar playing in 17-ET. It would be great to have some various 17-note tunings that satisfy different desiderata on hand.

-Igs

I have to agree with the immersion point Mike is making here. That is why
sometimes I get aurally confused as to what other people hear. So far I've
had this happen in 10, 11, 13, 17 to a high degree. Less so in others.

It seems akin to immersion in learning a language. And I think once you hit
that point composition suddenly is easier because the voice leading makes
more aural sense.
What I haven't done is tried this with an uneven step sized tuning (besides
meantone variants) - I'm curious if the same thing will occur. I suspect it
will in the same fashion that one can think in a scale in 12 edo be it
major, minor, modes, whole tone, etc. There is, in my brain at least, a
desire to see patterns and the immersion forces the finding of these
patterns. I think it also facilitates writing music with tonal centers.

Chris

On Wed, Jan 4, 2012 at 7:32 PM, Mike Battaglia <battaglia01@gmail.com>wrote:

> **
>
>
>
>
> The biggest single useful discovery made is that if you just stop
> complaining, and spend some time with it, you get used to it and it
> stops sounding out of tune. That's it. The Balinese and the Chopi have
> clearly figured this one out already, and it took me about a month of
> solidly immersing myself in 16-EDO to also feel the same way, and you
> can figure it out too.
>
> I view this to be a point of almost spiritual significance, because
> what I'm saying is that you're all not perceiving reality as it
> actually exists. You're perceiving it the way it doesn't exist. If you
> just stop seeing what these tunings are NOT, and start seeing what
> they ARE, then there's a million things you can do with them.
> Especially 16-EDO and mavila, which has all of these awesome
> properties in every single regard except harmonic accuracy.
>
> Don't you see that you're in the throes of a terrible addiction to
> accurate harmony! You and Carl, the both of you. This is an
> intervention!
>
> -Mike
>
>
>

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

> Alright, so if you can tolerate that, then what's wrong with 24-EDO,
> which gives you the same thing and 11 and 13 as well?

Nothing's wrong with 24edo, but you were trying to sell the idea it didn't have any weaknesses.

On Wed, Jan 4, 2012 at 9:53 PM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > Alright, so if you can tolerate that, then what's wrong with 24-EDO,
> > which gives you the same thing and 11 and 13 as well?
>
> Nothing's wrong with 24edo, but you were trying to sell the idea it didn't have any weaknesses.

I didn't say it had no weaknesses, but that it met Hans's criteria for
a good "successor" tuning to 12-EDO. The real point was that just
because 24-EDO doesn't have 7 in it doesn't mean that it doesn't
support higher-limit music.

But, tbh, I don't really care much about this idea of there being one
single successor tuning that has new stuff without breaking anything
that we already have. If that's what we want, then we don't just want
something that tempers out [-4 4 -1>, but something that tempers out
[[28 -19 12>>, because so-called "enharmonic" techniques have been in
use for a full century now. I'd rather just promote the notion that
different tunings are useful for different things.

-Mike

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
> Nothing's wrong with 24edo, but you were trying to sell the idea it didn't have any
> weaknesses.

If nothing's wrong with it, doesn't that mean by definition it doesn't have any weaknesses?

It has going for it that it loses nothing from 12-TET, and raises its accuracy by adding near-Just ratios of 11 and 13. It also already has tons of established theory in the East, and probably also the West (didn't Haba, Carrillo, and Wyschnegradsky write about it?). There are lots of hacks for existing instruments to enable them to access quarter-tones. 24 is a lot of notes, but it's less of a problem because half of them are already so ingrained in everyone.

It's a close call, really, between 17 and 24. You lose the ratios of 5, but in doing so, gain accuracy--17 is better on the 2.3.7.11.13 limit than 24 is on 2.3.5.11.13. The loss of the familiar may be as much of a feature as a bug, depending on someone's motivation. The simplicity of 17 is nice, but does it soften the learning curve more than that of 24? The notation is the same (I reckon HEWM would be used for both, or at least the ^/v accidentals for alteration by 33/32).

What is your professional opinion, Gene? Not just on the ETs, but also the world of unequal 17-note 13-limit (or 2.3.7.11.13-subgroup) temperaments vs. 24-note 13-limit (or 2.3.5.11.13-subgroup) temperaments?

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

> What is your professional opinion, Gene? Not just on the ETs, but also the world of unequal 17-note 13-limit (or 2.3.7.11.13-subgroup) temperaments vs. 24-note 13-limit (or 2.3.5.11.13-subgroup) temperaments?

My opinion is go with 24. The 2.3.5.11.13 subgroup forms a solid basis on which you can try adding in 7.

"genewardsmith" <genewardsmith@...> wrote:

> > But now it
> > looks you're coming up with theories about how we're all gravitating
> > towards music that we actually think sounds displeasurable because
> > we're trying to be cool and sound like badasses.
>
> It's a theory. So is boiling frog syndrome. I'm open to others.

Say what you will about the tenants of common practice
theory... at least it's an ethos.

-C.

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

> The Balinese and the Chopi have
> clearly figured this one out already,

Those aren't polyphonic styles:

>>The regular mapping paradigm is broadly useful, but one of
>>its uses is to generalize Western polyphonic music under
>>the assumption that it required a specialized tuning system
>>and, through cultural evolution, found one. For composers
>>interested in writing polyphonic music with melody, harmony,
>>motivic structure, etc. [snip]

(And in both cases the foremost fixed-pitch instruments are
idiophones, having short decay, high inhormonicity, or both.)

-Carl

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

> OK, here's a question: in addition to all of this stuff that
> we're doing in exploring other tuning systems, do you like music
> in 12-EDO? Like a Wagner symphony or something?
>
> -Mike

You mean the Wagner symphony, which wouldn't be in 12-EDO
(a common misconception among tuning nihilists... by the way,
McLaren runs Friday night meetings if you're free).

-Carl

On Thu, Jan 5, 2012 at 1:46 AM, Carl Lumma <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > The Balinese and the Chopi have
> > clearly figured this one out already,
>
> Those aren't polyphonic styles:

Gamelan uses the interval of the "empat" (3/2) as a consonance and
plays it at the same time deliberately. AKA, the only interval in
16-EDO that everyone keep saying to avoid is the only one that they
actually play harmonically.

> (And in both cases the foremost fixed-pitch instruments are
> idiophones, having short decay, high inhormonicity, or both.)

OK, so what? Those things also apply here

http://www.youtube.com/watch?v=egwXKQDYcvc

Forget nagoya marimbas, how about mavila marimbas?

-Mike

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

> > Those aren't polyphonic styles:
>
> Gamelan uses the interval of the "empat" (3/2) as a consonance
> and plays it at the same time deliberately. AKA, the only
> interval in 16-EDO that everyone keep saying to avoid is the
> only one that they actually play harmonically.

That doesn't make it polyphonic. This was discussed
extensively on MMM in February of last year.

> > (And in both cases the foremost fixed-pitch instruments are
> > idiophones, having short decay, high inhormonicity, or both.)
>
> OK, so what? Those things also apply here

Again, discussed extensively in the past. -Carl

On Thu, Jan 5, 2012 at 2:07 AM, Carl Lumma <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > > Those aren't polyphonic styles:
> >
> > Gamelan uses the interval of the "empat" (3/2) as a consonance
> > and plays it at the same time deliberately. AKA, the only
> > interval in 16-EDO that everyone keep saying to avoid is the
> > only one that they actually play harmonically.
>
> That doesn't make it polyphonic. This was discussed
> extensively on MMM in February of last year.

The context of this discussion is, Gene has said in the past that 675
cent fifths just sound bad and that there's no hope for them. Then,
above, he asked us to name one of the realizations that we've gained
from playing in 16-EDO, and my answer was that you get used to the
more discordant intervals eventually. The fact that Gamelan music
isn't really "polyphonic" in a Western sense doesn't change the point
that their music predominantly features a ~675 cent interval being
played harmonically, and that they've clearly gotten used to it. And,
if you try getting used to 16-EDO harmony outside of that one
interval, you can get used to that too.

> > > (And in both cases the foremost fixed-pitch instruments are
> > > idiophones, having short decay, high inhormonicity, or both.)
> >
> > OK, so what? Those things also apply here
>
> Again, discussed extensively in the past. -Carl

Link?

-Mike

Igs wrote:

>Do you have something against subgroups?

I practically invented them, so no.

> You're always talking like 19 and 22 as if they're the
> only accurate ETs between 12 and 31. If you're willing
> to drop just one of harmonics 3, 5, and 7 from
> the 13-limit, it's possible to find temperaments equally
> or even more accurate than 22-TET or 19-TET without having
> to be so much larger. 17, 20, 21, 23, and 24 all beat
> 22 and 19 on certain 5-dimensional 13-limit subgroups.

The only stuff between 12 and 30 that looks like it might
beat 19 and 22 once subgroups are considered is 17 (no 5s)
and 21 (no 3s). I might have missed something with one of
them fancy rational basis elements in it. And it depends
on how you weight the concordance of subgroups (I did it by
eye... I hope we can make some progress on this question).

In any case, no one should take the Euler diagram I posted
too seriously. Except Keenan Pepper.

-Carl

I am wondering how 41edo looks in this aspect. It apparently has a number of fine properties: good support for overtones (according to the xenwiki, it is the first edo to do some justice to mode 8 of the harmonic series); Ozan has mentioned it as possible EDO for maqams; Gene has described a possible use for indian srutis; it has even close approximations for 88cET and Bohlen-Pierce. Yet I somehow never saw much music in it, much less than the more complicated 53edo in any case. Is there a reason for that?
--
Hans Straub

On Thu, Jan 5, 2012 at 7:37 AM, hstraub64 <straub@...> wrote:
>
> I am wondering how 41edo looks in this aspect. It apparently has a number of fine properties: good support for overtones (according to the xenwiki, it is the first edo to do some justice to mode 8 of the harmonic series); Ozan has mentioned it as possible EDO for maqams; Gene has described a possible use for indian srutis; it has even close approximations for 88cET and Bohlen-Pierce. Yet I somehow never saw much music in it, much less than the more complicated 53edo in any case. Is there a reason for that?
> --
> Hans Straub

Probably just the amount of notes in it. It's easier for people
starting out to mess with a low-numbered EDO rather than something
like 41 or 46 or what not. Isomorphic keyboards, which can actually
let one play in a tuning like that, literally only became affordable
last year.

Much of what I like about 41 is also in 34 though, FWIW. And much of
what I don't like about 41 is rectified by 46. To be honest, I haven't
thought about 41 in a long time. I only cared about it way back in the
day when I thought that schismatic temperament was the answer to all
the world's problems.

In closing, if the goal for 41 is to have a tuning that doesn't
support existing music, but offers a crunchy higher-limit shredfest,
I'd much rather throw in 5 extra notes and get 46, which has a stellar
representation of the 13-limit as well as things like sensi
temperament. That's what I want!

-Mike

Magic's good. I've done stuff with Magic. Schismatic tends to optimize better than 41

Graham

------Original message------
From: hstraub64 <straub@...>
To: <tuning@yahoogroups.com>
Date: Thursday, January 5, 2012 12:37:13 PM GMT-0000
Subject: [tuning] Re: What Would a Meantone Successor Look Like?

I am wondering how 41edo looks in this aspect. It apparently has a number of fine properties: good support for overtones (according to the xenwiki, it is the first edo to do some justice to mode 8 of the harmonic series); Ozan has mentioned it as possible EDO for maqams; Gene has described a possible use for indian srutis; it has even close approximations for 88cET and Bohlen-Pierce. Yet I somehow never saw much music in it, much less than the more complicated 53edo in any case. Is there a reason for that?
--
Hans Straub

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On Thu, Jan 5, 2012 at 7:49 AM, Mike Battaglia <battaglia01@...> wrote:
>
> Much of what I like about 41 is also in 34 though, FWIW. And much of
> what I don't like about 41 is rectified by 46.

I should add that, if we're going to let things like complexity fall
by the wayside, that 46-EDO is awesome in so many ways that I don't
know why people always talk about stuff like 41 and 53 and always
completely miss it.

-Mike

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:

> The only stuff between 12 and 30 that looks like it might
> beat 19 and 22 once subgroups are considered is 17 (no 5s)
> and 21 (no 3s). I might have missed something with one of
> them fancy rational basis elements in it.

You forgot 24, on the 2.3.5.11.13 subgroup. It beats 22's best of 10.05 cents of adjusted error, with 9.58 cents of adjusted error on the above subgroup. 23 also comes very close on a few rational basis subgroups. Lots of ETs between 24 and 30 beat 22 on the no-3's, no-5's, and no-7's subgroups.

-Igs

"hstraub64" <straub@...> wrote:

> I am wondering how 41edo looks in this aspect. It apparently
> has a number of fine properties: good support for overtones
> (according to the xenwiki, it is the first edo to do some
> justice to mode 8 of the harmonic series); Ozan has mentioned
> it as possible EDO for maqams; Gene has described a possible
> use for indian srutis; it has even close approximations for
> 88cET and Bohlen-Pierce. Yet I somehow never saw much music
> in it, much less than the more complicated 53edo in any case.
> Is there a reason for that?

Hi Hans,

You can see that 41 does very well among ETs < 100

http://lumma.org/music/theory/TOPDamageOfETs.txt

46 is slightly more accurate but not enough to offset
the extra 5 notes if you do a badness calculation.

It supports miracle, magic, and schismic (rank 2)
temperaments. It is like a poor man's 53 or 72, or
a rich man's 22. It does not support meantone so
it's not good for common-practice repertoire.

Erv Wilson showed that Harry Partch's 43-tone scale
fits very well to 41-ET and related Fokker blocks.
It is as if Partch resolved pairs of pitches that differ
by one of the commas of 41 but choosing one from each
pair, but only failed to do so in two cases (giving 2
extra notes).

With the sole exception of 72, there is very little
music in any ET above 31. Bosanquet wrote some studies
in 53, and I believe Groven used a 36-note subset of 53.
Denver-area musician Chris Mohr has released an album
or two of original music in 53, which is excellent.
He uses a Starr Microzone keyboard and one of his
albums contains an opera. I don't know of anything
equivalent for 41.

-Carl

-- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> I should add that, if we're going to let things like complexity
> fall by the wayside, that 46-EDO is awesome in so many ways that
> I don't know why people always talk about stuff like 41 and 53
> and always completely miss it.

It's true, and 46 is exactly where I'm headed if 22 doesn't
do it for me. Gene is the first to point out the excellent
properties of 46 AFAIK.

I neglected to mention in my last message that Neil Haverstick
has done significant work in 34, which is larger than 31
and not 72.

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:

> I neglected to mention in my last message that Neil Haverstick
> has done significant work in 34, which is larger than 31
> and not 72.

You might as well also count Carlo Serafini's work in Carlos Gamma, which is basically indistinguishable from 34-ET. Of course, neither he nor Neil has really tapped the full 13-limit glory of the tuning. I can rarely tell the difference between Neil's 34-ET tracks and his 19-ET tracks, and because he rarely mentions which is which in his liner notes, to this day I'm not even sure about many of them. Carlo's tracks are closer to 5-limit JI for the most part, with some occasional xenharmonic moments.

-Igs

Igs wrote:

> > The only stuff between 12 and 30 that looks like it might
> > beat 19 and 22 once subgroups are considered is 17 (no 5s)
> > and 21 (no 3s). I might have missed something with one of
> > them fancy rational basis elements in it.
>
> You forgot 24, on the 2.3.5.11.13 subgroup. It beats 22's best
> of 10.05 cents of adjusted error, with 9.58 cents of adjusted
> error on the above subgroup. 23 also comes very close on a few
> rational basis subgroups. Lots of ETs between 24 and 30 beat 22
> on the no-3's, no-5's, and no-7's subgroups.

I doubt it beats it on badness, and I still don't know
what adjusted error is, but I'm willing to be corrected by
the next version of your search.

-Carl

Here's the badness rankings using your n^(5/4) method:
/tuning/files/IgliashonJones/5D%20Min%20ET%20Logflat%20Badness%20.pdf

24 indeed still beats 22, quite handily. 11 and 9 now beat 20. 12 looks terrible, even compared to 18! Wacky.

-Igs

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> Igs wrote:
>
> > > The only stuff between 12 and 30 that looks like it might
> > > beat 19 and 22 once subgroups are considered is 17 (no 5s)
> > > and 21 (no 3s). I might have missed something with one of
> > > them fancy rational basis elements in it.
> >
> > You forgot 24, on the 2.3.5.11.13 subgroup. It beats 22's best
> > of 10.05 cents of adjusted error, with 9.58 cents of adjusted
> > error on the above subgroup. 23 also comes very close on a few
> > rational basis subgroups. Lots of ETs between 24 and 30 beat 22
> > on the no-3's, no-5's, and no-7's subgroups.
>
> I doubt it beats it on badness, and I still don't know
> what adjusted error is, but I'm willing to be corrected by
> the next version of your search.
>
> -Carl
>

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote (#102422):
>
> I came back to this list because this is a question to which I've been giving a lot of thought lately, and XA is no place for deep, drawn-out discussion. I really want to hammer this out.

Hi Igs,

I'm really intrigued by your thoughts here (and also a week later, in msg. #102748), because they're very similar to ones I had after working with microtonality and alternative tunings for about 12 years. When I got my Scalatron about a year earlier, I thought that 31-equal was going to be the wave of the future, but I soon realized that it might be too many tones to be conveniently implemented on some instruments, i.e., too cumbersome and/or costly.

> I've been doing this microtonal thing for a long time now, I have tons of experience writing music in lots of different tunings, tons of experience debating the intricacies of various theories and practices, and tons of experience trying to apply different tuning approaches to what I consider contemporary styles of music. I figure, it's time to stop meandering from tuning to tuning, ... It's time to start really developing a praxis around one specific tuning--become fluent in its notation, get a set of matching instruments, and really stake my claim as a theorist that *this* tuning above all others embodies the direction that I think music should go in. It needs to be a tuning that I can reasonably expect other people to embrace on a decently large scale. This means it needs to work in some analogous way to the current meantone/pythagorean/12-TET paradigm, but be different in enough key ways that it can solve some compositional problems that the current paradigm totally fails at, problems which a substantive number of musicians are actually troubled by.

I suspect that you wouldn't be happy with just a single tuning, because I'll guarantee that you'll be frustrated by its shortcomings. Instead, I would recommend that you settle on two or three different tunings that collectively possess all of the properties you're seeking. If you work with one for a while and then work with another, I think you'll find the change refreshing.

> ... Any tuning system that would hope to succeed meantone ... would need to be representable in an equal temperament of reasonable size that it could work on a guitar and (probably) a dedicated keyboard, but also not so far off from some JI system that it could not be performed by an ensemble of free-pitch instruments. ...

I would say that we should not look for a *successor* to meantone so much as an *alternative* that could compete with it.

> Musicians are starting to laugh at the fact that everything is starting to sound the same. Yay!
>
> Another problem is that 12-TET can't really do those wacky "quarter-tones" that are all over these funky "Eastern" scales ...
>
> And there's also the problem that leads an awful lot of people into microtonality, namely the intonational inaccuracy of 12-TET. ...
>
> The first problem, the problem of novelty, is the easiest to solve, because literally *every* non-12-TET tuning offers *something* new. Witness the plethora of tunings that have actually been used in music by at least one composer--do we even know definitively how many tunings have been composed in by microtonalists in the last century?
>
> The second problem, of incorporating Eastern scales, suggests tunings with neutral intervals atop some sort of Pythagorean-esque back-bone. ...
>
> The third problem, of intonational inaccuracy, is also closer to solution in all of these--if we leave out the 5th harmonic, both 17 and 24 are quite accurate on the rest of the 13-limit, with 17 having a decent edge. Among temperaments of the 2.3.7.11.13 subgroup, only 36 is more accurate, and it's not by much. 27 and 31 are both very good on the full 13-limit, 5th harmonic and all. We might as well kick 36 and 24 off the list, since 17 does so well on the 2.3.7.11.13 subgroup and is so very much simpler.
>
> So, we are left with three ETs--17, 27, and 31. 31 is the most accurate but also the largest; questions of feasibility on physical instruments aside, it also poses a large conceptual and pedagogical challenge. ... People have been advocating for it for centuries, but it hasn't caught on, most likely because of this huge increase in complexity over 12-TET. ...
>
> 27, at only four fewer notes than 31, is susceptible to similar problems. It might appeal to some intrepid musicians, most likely virtuousos or those who use isomorphic keyboards, but for those on guitar, brass, tuned percussion, woodwinds, voice, and strings, it is unlikely to be very appealing.
>
> This leaves 17. Economical, accurate, and lacking only in the ratios of 5. The loss of the ratios of 5 is significant, no doubt--but it is compensated for by ratios of 7, 11, and 13. It even has a half-assed sort of backwards-compatibility via the diatonic scale, which is probably good enough for rock music. If the whole purpose of adopting a new tuning is to move in a new direction, especially an Eastern-influenced one incorporating higher members of the harmonic series, then loss of full 5-limit backwards-compatibility shouldn't be a big deal.
>
> Of course, we should also remember that 17-equal is just a convenience, a point of departure. In practice, in more organic settings, there is plenty of room for push-and-pull, through adaptive JI that might sneak some ratios of 5 back in, or through a circulating temperament such as that proposed by George Secor that gets more intervals closer to JI.

Bingo! You've just unlocked the door to the solution you're seeking. Reasonable-number EDO's (for my purposes, that would be <24) won't deliver the intonational accuracy you're seeking, whereas >5-limit JI with a reasonable number of tones/octave doesn't offer unlimited (or even generous) ability to transpose. When I consider the number of 12-tone circulating temperaments that have been devised over the last several centuries, I'm amazed that hardly anyone else other than myself (I know only of Margo Schulter & Joe Monzo) has worked with any sort of non-12 circulating temperament.

> Or we could take one of the nice rank-2 temperaments supported in 17, like bleu, maqamic/mohaha, huxley, or machine, and do up an optimized 17-note MOS--since 17 is already pretty accurate, it should be close enough to "play nicely" with a more optimal form of itself. The "17" is just the kernel of the system, much like how 12 is the kernel of our current intonational practices.
>
> So that's my case for 17. I think it's pretty solid...obviously it's not the be-all, end-all of tunings, but it's the one I think is best-suited for large-scale deployment.
>
> -Igs

I would consider 17 to be the ideal number of tones for a meantone alternative. However, I'd be looking for alternatives to my 17-WT, because I think the omission of prime 5 is going to be a deal-breaker for some folks. (BTW, I would insist on having an alternative 17-tone tuning with a full 15-odd limit, with less error than 31-equal, as one of my alternatives, and it would also have to offer free modulation. Impossible? Not if you think outside the box. Wait & see!)

I have several different alternative tunings in mind, which I'll propose over the course of the next several days (or weeks, depending on how much discussion is generated; I recently recovered from a 3-week illness that left me severely sleep-deprived, and I'm now having to catch up on sleep, which has not left me with much free time to follow the discussion, much less participate).

I expect to post my first meantone-alternative tuning sometime tomorrow. As you'll see, my approach to this problem will be quite different from the approach that's currently being taken on the tuning list.

--George

Hi George! Great to hear from you, as always, and happy to have broached a subject that inspired you to make a come-back here!

--- In tuning@yahoogroups.com, "gdsecor" <gdsecor@...> wrote:
> I suspect that you wouldn't be happy with just a single tuning, because I'll guarantee that > you'll be frustrated by its shortcomings. Instead, I would recommend that you settle on
> two or three different tunings that collectively possess all of the properties you're seeking. > If you work with one for a while and then work with another, I think you'll find the change > refreshing.

Oh, of course--I highly doubt I'll ever be a one-tuning kinda guy, but I would like to settle enough on a flagship tuning (or rather, family of related tunings that could all share a common music theory, despite a host of intonational differences) that I can promote and teach. I have been pruning my stable of microtonal guitars--31, 22, and 19 are all gone (much to Carl and Gene's chagrin), as well as 15 and 18, while 13, 16, 17, 20, and 23 remain, with the smallest three holding the strongest gravity for me as an instrumentalist. But I've been doing a lot of work with subgroup temperaments, and might trade out one of these for a different tuning depending on my findings. In any case, though, 17 is a tough number to beat, both among the ETs and among the unequal temperaments. Good ETs make good unequal temperaments, after all....

> I would say that we should not look for a *successor* to meantone so much as an
> *alternative* that could compete with it.

Yeah, I was using Carl's words. I doubt meantone and 12-TET will ever be replaced, and I think the key to finding an alternative is not trying to find a tuning that does what meantone does *AND* does something different, but just focus on doing really well in the areas that meantone does poorly.

> Bingo! You've just unlocked the door to the solution you're seeking. Reasonable-number > EDO's (for my purposes, that would be <24) won't deliver the intonational accuracy you're > seeking, whereas >5-limit JI with a reasonable number of tones/octave doesn't offer
> unlimited (or even generous) ability to transpose. When I consider the number of 12-tone > circulating temperaments that have been devised over the last several centuries, I'm
> amazed that hardly anyone else other than myself (I know only of Margo Schulter & Joe
> Monzo) has worked with any sort of non-12 circulating temperament.

It's really a very elegant solution, and I anticipate that having a spectrum of 17-note tunings that range from equal-but-inaccurate (allowing for the invocation of a variety of rank-2 temperaments, just like how 12-TET allows the invocation of meantone, diminished, augmented, pajara, 6-EDO, etc.) to optimized rank-2, to circulating well-temperaments that offer a large variety of near-Just harmonies, will make an overall **17-tone** meta-system more broadly appealing. Rather than focusing on finding a single tuning that tries to fit everyone's desiderata, it strikes me that finding a meta-system with a spectrum of instantiations that share a common musical logic is a much more sensible approach.

> I would consider 17 to be the ideal number of tones for a meantone alternative.

Yay! I have to say I agree, if for no other reason than my 17-tone guitar is by far the easiest to play. Easier even than 19 was, and oddly easier than 15 and 16

> However, I'd be looking for alternatives to my 17-WT, because I think the omission of
> prime 5 is going to be a deal-breaker for some folks. (BTW, I would insist on having an
> alternative 17-tone tuning with a full 15-odd limit, with less error than 31-equal, as one
> of my alternatives, and it would also have to offer free modulation. Impossible? Not if you > think outside the box. Wait & see!)

The full 15-odd-limit, less error than 31, free modulation, and only 17 notes? I'm fairly certain there's not one of us here who wouldn't be interested in such a tuning. I can't wait to see what you're about to roll out! Sleep well, take care of yourself, make a full recovery, and then blow all our minds with a new 17-tone WT.

-Igs

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:
>
> Hi George! Great to hear from you, as always, and happy to have broached a subject that inspired you to make a come-back here!

Thanks for the warm welcome-back! I've actually been here here all along, but there has been so much traffic lately that I haven't had time to read each & every message, so I may have missed a significant thing or two in the meantone-alternatives discussion. Anyway, I've resurfaced to contribute some more alternatives for your consideration.

> ...
> > I would consider 17 to be the ideal number of tones for a meantone alternative.
>
> Yay! I have to say I agree, if for no other reason than my 17-tone guitar is by far the easiest to play. Easier even than 19 was, and oddly easier than 15 and 16

Some of the alternatives I'm going to propose have 19 or 22 tones/octave, and some of them have less than 17. The one thing that they all have in common is that they're quite a bit different from what others have proposed. Hardly any of them are brand new; in fact, all of them are either from the 1970's or are based on ideas I came up with back then. I've presented some of them on this list before, but there did not seem to be much interest at the time, and I figure that it was that folks aren't much interested in answers to questions they're not asking.

> > However, I'd be looking for alternatives to my 17-WT, because I think the omission of
> > prime 5 is going to be a deal-breaker for some folks. (BTW, I would insist on having an
> > alternative 17-tone tuning with a full 15-odd limit, with less error than 31-equal, as one
> > of my alternatives, and it would also have to offer free modulation. Impossible? Not if you > think outside the box. Wait & see!)
>
> The full 15-odd-limit, less error than 31, free modulation, and only 17 notes? I'm fairly certain there's not one of us here who wouldn't be interested in such a tuning. I can't wait to see what you're about to roll out! Sleep well, take care of yourself, make a full recovery, and then blow all our minds with a new 17-tone WT.

It's over 30 years old, not a WT, and I've posted it here before (probably more than once). There's a qualification regarding that "free modulation" (which I should perhaps change to "free transposition", which may not be quite the same thing) that I don't think you will fully appreciate until I present some examples (one of which is meantone alternative 1b, below). It's important for you to understand how I arrived at the end result, so I need to take it one step at a time.

Without further ado, here's the first meantone alternative. It's 10 approximately equal tones per octave, but it's not a temperament (hence the change in subject line from "circulating temperaments" to "circulating tunings"):

1/1 13/12 7/6 5/4 4/3 45/32 3/2 13/8 7/4 15/8 2/1

You wrote (msg #102748):
> And here's why I'm suddenly interested in this: much of the divide in this group (and the community at large) centers around disagreements about the relative importance of concordance vs. simplicity.

Lo and behold, this first meantone alternative is both very simple (only 10 tones/octave) and very concordant (strictly JI)! The idea of approximating an EDO with JI (instead of vice-versa) is something I thought of in the 1970's. You may remember that I previously posted this link (more than once) to a short story I wrote to describe the rationale behind this approach, with some examples at the end:
/tuning/files/secor/blarney.txt

About the only criticism I can think of is that 10 tones is too simple. Okay. I haven't tackled the problem of approximating 15-EDO with small-number rational tones, but perhaps others reading this would like to go for it.

In case someone decides to map these tones onto a conventional keyboard and is concerned about having two vacant keys per octave, then you can add 9/8 and 5/3 to the above, which maps nicely to 12:

1/1 13/12 9/8 7/6 5/4 4/3 45/32 3/2 13/8 5/3 7/4 15/8 2/1

If you do that, you'll discover that you have a chain of fifths from 4/3 to 45/32 (F to F#, if C=1/1), with a false fifth between 9/8 and 5/3. If you temper out the 80:81 comma (which could be done by tempering only 9/8 and 5/3 to make three 1/3-comma fifths, leaving the 10-tone rational tuning intact), you then have a usable diatonic scale in two keys (1/1 and 3/2), which would allow you to play a "limited" (actually rather extensive) amount of (simple) conventional music.

In summary, with 10 tones (meantone alternative 1a) you have free transposition, and with 12 tones (meantone alternative 1b) you have a multi-purpose tuning (the 10 tones of alternative 1a, the 8 tones in a chain of fifths, and 12 tones providing 13-limit harmony with no 11's). I allow the property of free transposition to be a feature of a tuning if a majority subset of the tuning has that property.

One thing I miss in this tuning is prime 11, but I'll be sure to include that in meantone alternative 2 -- coming soon!

Stay tuned!

--George

--- In tuning@yahoogroups.com, "gdsecor" <gdsecor@...> wrote:
> Without further ado, here's the first meantone alternative. It's 10 approximately equal
> tones per octave, but it's not a temperament (hence the change in subject line from
> "circulating temperaments" to "circulating tunings"):
>
> 1/1 13/12 7/6 5/4 4/3 45/32 3/2 13/8 7/4 15/8 2/1

Hmm...I'm not sure if I agree that this one is playable in all keys. Those 20/13 and 32/21 fifths are pretty rough in comparison to the 3/2's they share an interval class with. These are wolves as harsh as any found in meantone or 5-limit JI. I would think we could spread out their error a bit more and get a well-temperament that keeps many of the desirable properties of this scale but tames the wolves a bit. Perhaps Gene might find a rank-3 or rank-4 temperament that would fit the bill. As it is, I'd say we only get 7/10 playable keys (and that's allowing the 14/9 to count as a "fifth").

> In case someone decides to map these tones onto a conventional keyboard and is
> concerned about having two vacant keys per octave, then you can add 9/8 and 5/3 to the > above, which maps nicely to 12:
>
> 1/1 13/12 9/8 7/6 5/4 4/3 45/32 3/2 13/8 5/3 7/4 15/8 2/1

I count 4 wolves in this one, some sharp, some flat. Also, the narrow 27/26 between 13/8 and 5/3 makes the scale improper (which may or may not be a big deal to some people). So I'd have to say that, like the above, free transposition isn't possible in this one, with 8/12 keys being playable, but it does look like a good candidate for well tempering, or maybe regularly tempering out one or two commas to bring it down to rank-3 or rank-4. But I'm sad to say I'm not sold on the scale as-is, for it suffers the same flaws that drove Western music away from meantone toward increasingly-equal temperaments (namely, the presence of wolves).

However, 10-ET is not a very accurate tuning in its own right, so finding a JI approximation to it should be more of a challenge than, say, 12-ET, 19-ET, 22-ET, or 17-ET. I have higher hopes for what you've done about those tunings!

-Igs

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

> Hmm...I'm not sure if I agree that this one is playable in all keys. Those 20/13 and 32/21 fifths are pretty rough in comparison to the 3/2's they share an interval class with. These are wolves as harsh as any found in meantone or 5-limit JI. I would think we could spread out their error a bit more and get a well-temperament that keeps many of the desirable properties of this scale but tames the wolves a bit.

You can smear out the error of 32/21 with 64/63 tempering, but as for 20/13, I say suck it up! Actually, an idea I like which wouldn't interest you is 676/675 tempering. Or 676/675 together with 196/195.

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

> You can smear out the error of 32/21 with 64/63 tempering, but as for 20/13, I say suck it > up! Actually, an idea I like which wouldn't interest you is 676/675 tempering. Or 676/675 > together with 196/195.

What would that do to the scale?

-Igs

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

> What would that do to the scale?

! secoralternative10.scl
!
George Secor "meantone alternative", {196/195, 676/675}-tempering in POTE tuning of 2.3.5.7.13 scale
10
!
140.00441
264.20379
388.40316
496.79750
594.80816
703.20250
843.20691
967.40629
1091.60566
1200.00000
!
!! secoralternativeJI.scl
!!
!Transversal (original JI version)
! 10
!
! 13/12
! 7/6
! 5/4
! 4/3
! 45/32
! 3/2
! 13/8
! 7/4
! 15/8
! 2/1

I'm not sure that I see any benefit to that tempering over the JI version, at least as far as improvement in number of consonances goes. It does have fewer step-sizes, though.

-Igs

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@> wrote:
>
> > What would that do to the scale?
>
> ! secoralternative10.scl
> !
> George Secor "meantone alternative", {196/195, 676/675}-tempering in POTE tuning of 2.3.5.7.13 scale
> 10
> !
> 140.00441
> 264.20379
> 388.40316
> 496.79750
> 594.80816
> 703.20250
> 843.20691
> 967.40629
> 1091.60566
> 1200.00000
> !
> !! secoralternativeJI.scl
> !!
> !Transversal (original JI version)
> ! 10
> !
> ! 13/12
> ! 7/6
> ! 5/4
> ! 4/3
> ! 45/32
> ! 3/2
> ! 13/8
> ! 7/4
> ! 15/8
> ! 2/1
>

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:
>
> --- In tuning@yahoogroups.com, "gdsecor" <gdsecor@> wrote:
> > Without further ado, here's the first meantone alternative. It's 10 approximately equal
> > tones per octave, but it's not a temperament (hence the change in subject line from
> > "circulating temperaments" to "circulating tunings"):
> >
> > 1/1 13/12 7/6 5/4 4/3 45/32 3/2 13/8 7/4 15/8 2/1
>
> Hmm...I'm not sure if I agree that this one is playable in all keys. Those 20/13 and 32/21 fifths are pretty rough in comparison to the 3/2's they share an interval class with. These are wolves as harsh as any found in meantone or 5-limit JI. ...

Ha! It looks like the leprechauns have put one over on me! ;-)

Igs, I find it a bit ironic that you, of all people, promoter of discordant EDO's, are complaining about rough intervals. 8>}

Okay, I realize that you're looking for something that's not going to offend the sensibilities of those less radical than yourself and who use instruments with harmonic partials. (You do realize that this 10-tone scale would be perfectly acceptable for a gamelan.)

> I would think we could spread out their error a bit more and get a well-temperament that keeps many of the desirable properties of this scale but tames the wolves a bit. Perhaps Gene might find a rank-3 or rank-4 temperament that would fit the bill.

I wouldn't want you (plural, meaning you, Gene, or anyone else) to knock yourselves out looking for something like that. Let's consider this an illustration of some principles I've presented and move on, those principles being:

1) The objective is to construct a multi-purpose tuning in which different subsets of tones may collectively function differently from one another.

2) Each of those subsets should consist of more than half of the tones in the tuning.

3) One of those subsets should offer free modulation or transposition, without the requirement that transposed intervals be exactly the same size.

> However, 10-ET is not a very accurate tuning in its own right, so finding a JI approximation to it should be more of a challenge than, say, 12-ET, 19-ET, 22-ET, or 17-ET. I have higher hopes for what you've done about those tunings!

Okay, then. I'll stick with tunings in which the fifths (or "fifths") aren't excessively mistuned (or "mistuned".

Having looked at the 10-tone (expandable to 12) proposal and having identified its weakest point, we can move on to something else. (Note: I've changed the subject line again, hopefully for the last time.)

Stay tuned!

--George

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:
>
> I'm not sure that I see any benefit to that tempering over the JI version, at least as far as improvement in number of consonances goes. It does have fewer step-sizes, though.

676/675 is so small, and since I've included 196/195, you can't expect much improvement for intervals like 52/45 or 135/104, and you don't get that much. However 39/28 is moved a lot closer to 7/5, and 28/15 to 13/7, because of the 196/195. However, maybe what should be done is add 105/104 to the mix. This produces a linear temperament on the 2.3.5.7.13 subgroup which is no-elevens negri. While it doesn't have the cheesy 45/44 comma of 13-limit negri, it's still a lot less accurate than just 196/195 and 676/675.

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

> Ha! It looks like the leprechauns have put one over on me! ;-)

They are wont do that... ;->

> Igs, I find it a bit ironic that you, of all people, promoter of discordant EDO's, are
> complaining about rough intervals. 8>}

Discordant EDOs have their own perverse logic that suggests divergent compositional forms and quite different stylistic approaches. I like them a lot, but I'm trying to look at things from the perspective of "compositional problems"--i.e. a meantone alternative should not just do things that meantone can't, it should do things that a decent proportion of musicians *wish* an intonation *could* do. Some of my wishes are probably not, strictly speaking, representative of a sizable proportion of musicians!

> Okay, I realize that you're looking for something that's not going to offend the sensibilities > of those less radical than yourself and who use instruments with harmonic partials. (You > do realize that this 10-tone scale would be perfectly acceptable for a gamelan.)

Perhaps! Though those 3/2's might be a bit too calm. What it might be best for is someone who wants to cover the whole spectrum from JI to gamelan-esque on a single instrument with as few notes as possible. In that regard, it is a superb tuning!

> 1) The objective is to construct a multi-purpose tuning in which different subsets of tones > may collectively function differently from one another.

Ah. That wasn't quite what I was initially proposing, but it may indeed be a desirable objective.

> 2) Each of those subsets should consist of more than half of the tones in the tuning.

Care to elaborate on the reasoning behind this?

> 3) One of those subsets should offer free modulation or transposition, without the
> requirement that transposed intervals be exactly the same size.

How could we have free modulation if we're limited to a subset of the tuning?

> Okay, then. I'll stick with tunings in which the fifths (or "fifths") aren't excessively
> mistuned (or "mistuned".

That will probably please the greatest number of people. My own boredom with the interval notwithstanding, of course!

> Having looked at the 10-tone (expandable to 12) proposal and having identified its
> weakest point, we can move on to something else. (Note: I've changed the subject line
> again, hopefully for the last time.)
>
> Stay tuned!

Great, curious to see what comes next!

-Igs

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:
>
> --- In tuning@yahoogroups.com, "gdsecor" <gdsecor@> wrote:
> ...
> > Igs, I find it a bit ironic that you, of all people, promoter of discordant EDO's, are
> > complaining about rough intervals. 8>}
>
> Discordant EDOs have their own perverse logic that suggests divergent compositional forms and quite different stylistic approaches. I like them a lot, but I'm trying to look at things from the perspective of "compositional problems"--i.e. a meantone alternative should not just do things that meantone can't, it should do things that a decent proportion of musicians *wish* an intonation *could* do. Some of my wishes are probably not, strictly speaking, representative of a sizable proportion of musicians!

Before I go any further, I'd like to summarize what we think those wishes might be. In our microtonal community different persons have differing requirements for a tonal system, and where there are multiple requirements (e.g., concordance, simplicity, transposability, harmonic limit), the order of importance will differ. The primary emphasis in our Tunings groups seems to be on transposability, which generally does not adequately address the concerns of those who give concordance and simplicity higher priority.

You said that you have higher hopes for what I've done about finding approximations to 12-ET, 19-ET, 22-ET, or 17-ET, so you should be pleased to know that this is the approach I'll be taking with my remaining proposals. Since you have 12-ET and 19-ET in that list, those proposals will support meantone, which may be important to some folks, while others may not care.

As my goal, I have set the harmonic limit at 15-odd and maximum complexity at 22 tones/octave. (I also have some options with more complexity that I'll present, as alternatives to 31-, 41-, and 46-equal.)

I have also set a goal that concordance within the 15-odd limit in the best (3 or more) keys of the tuning to be comparable to (and preferably better than) 31-equal. The rationale for wanting at least 3 keys in high concordance is that in JI it requires 8 tones for a single 15-limit otonal ogdoad (i.e., 8:9:10:11:12:13:14:15), at least 5 more tones for a second ogdoad, and at least 5 more for a third ogdoad, which puts us in the ballpark of 18 tones (tempering may reduce this number slightly). If we don't have at least as many good keys as in JI, then the effort is probably not justified.

> ...
> > 1) The objective is to construct a multi-purpose tuning in which different subsets of tones
> > may collectively function differently from one another.
>
> Ah. That wasn't quite what I was initially proposing, but it may indeed be a desirable objective.
>
> > 2) Each of those subsets should consist of more than half of the tones in the tuning.
>
> Care to elaborate on the reasoning behind this?

If you start with a tuning consisting of a tempered circle of fifths (of n tones, not necessarily equal) and desire a 15-odd harmonic limit, then as long as that circle is 22 tones or less, you're going to be missing one or more primes. The missing prime(s) can be supplied by adding more (m) tones to the tuning, but complexity increases as m gets larger. If you add enough tones, you might be able to achieve free transposition (e.g., if you end up with two circles of fifths), but that comes at the cost of doubling the complexity. A better strategy is to add the missing prime(s) only for the best keys of the tuning. One subset of tones will therefore consist of the original circle of tempered fifths (which offers free transposition), while another subset will consist of the tones that are members of one or more otonal ogdoads. The higher the percentages of tones in each subset, the more successful the result.

Those extra tones outside the circle will probably not map neatly into modulo m+n, such that the resulting set of tones will probably not be a constant structure. This is more of a theoretical than a practical problem, since I've found that there are ways of mapping such tunings to a (Bosanquet generalized) keyboard or of arranging them conveniently in lateral rows on pitched percussion.

> > 3) One of those subsets should offer free modulation or transposition, without the
> > requirement that transposed intervals be exactly the same size.
>
> How could we have free modulation if we're limited to a subset of the tuning?

Free modulation within a subset is still free modulation, with the caveat that you avoid the tones outside the subset. Think of it as enjoying a meal in a fine restaurant, where you skip one particular food that's not to your liking.

> > Okay, then. I'll stick with tunings in which the fifths (or "fifths") aren't excessively
> > mistuned (or "mistuned".
>
> That will probably please the greatest number of people. My own boredom with the interval notwithstanding, of course!
>
> > Having looked at the 10-tone (expandable to 12) proposal and having identified its
> > weakest point, we can move on to something else. (Note: I've changed the subject line
> > again, hopefully for the last time.)
> >
> > Stay tuned!
>
> Great, curious to see what comes next!

Me too! I was trying something new this past weekend and was pleasantly surprised with the result. I'll post it after I've had more time to mess around with it.

--George

I've spent the past couple of weeks working on a brand new tuning that I'm presenting here in my second reply to Igs' message #102777, which follows in its entirety.

Those who haven't been following this thread should first read these two messages, to get the objective of this discussion in its full context:
/tuning/topicId_102422.html#102422
/tuning/topicId_102746.html#102748

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:
>
> Hi George! Great to hear from you, as always, and happy to have broached a subject that inspired you to make a come-back here!
>
> --- In tuning@yahoogroups.com, "gdsecor" <gdsecor@> wrote:
> > I suspect that you wouldn't be happy with just a single tuning, because I'll guarantee that
> > you'll be frustrated by its shortcomings. Instead, I would recommend that you settle on
> > two or three different tunings that collectively possess all of the properties you're seeking.
> > If you work with one for a while and then work with another, I think you'll find the change
> > refreshing.
>
> Oh, of course--I highly doubt I'll ever be a one-tuning kinda guy, but I would like to settle enough on a flagship tuning (or rather, family of related tunings that could all share a common music theory, despite a host of intonational differences) that I can promote and teach. I have been pruning my stable of microtonal guitars--31, 22, and 19 are all gone (much to Carl and Gene's chagrin), as well as 15 and 18, while 13, 16, 17, 20, and 23 remain, with the smallest three holding the strongest gravity for me as an instrumentalist. But I've been doing a lot of work with subgroup temperaments, and might trade out one of these for a different tuning depending on my findings. In any case, though, 17 is a tough number to beat, both among the ETs and among the unequal temperaments. Good ETs make good unequal temperaments, after all....
>
> > I would say that we should not look for a *successor* to meantone so much as an
> > *alternative* that could compete with it.
>
> Yeah, I was using Carl's words. I doubt meantone and 12-TET will ever be replaced, and I think the key to finding an alternative is not trying to find a tuning that does what meantone does *AND* does something different, but just focus on doing really well in the areas that meantone does poorly.
>
> > Bingo! You've just unlocked the door to the solution you're seeking. Reasonable-number
> > EDO's (for my purposes, that would be <24) won't deliver the intonational accuracy you're
> > seeking, whereas >5-limit JI with a reasonable number of tones/octave doesn't offer
> > unlimited (or even generous) ability to transpose. When I consider the number of 12-tone
> > circulating temperaments that have been devised over the last several centuries, I'm
> > amazed that hardly anyone else other than myself (I know only of Margo Schulter & Joe
> > Monzo) has worked with any sort of non-12 circulating temperament.
>
> It's really a very elegant solution, and I anticipate that having a spectrum of 17-note tunings that range from equal-but-inaccurate (allowing for the invocation of a variety of rank-2 temperaments, just like how 12-TET allows the invocation of meantone, diminished, augmented, pajara, 6-EDO, etc.) to optimized rank-2, to circulating well-temperaments that offer a large variety of near-Just harmonies, will make an overall **17-tone** meta-system more broadly appealing. Rather than focusing on finding a single tuning that tries to fit everyone's desiderata, it strikes me that finding a meta-system with a spectrum of instantiations that share a common musical logic is a much more sensible approach.
>
> > I would consider 17 to be the ideal number of tones for a meantone alternative.
>
> Yay! I have to say I agree, if for no other reason than my 17-tone guitar is by far the easiest to play. Easier even than 19 was, and oddly easier than 15 and 16
>
> > However, I'd be looking for alternatives to my 17-WT, because I think the omission of
> > prime 5 is going to be a deal-breaker for some folks. (BTW, I would insist on having an
> > alternative 17-tone tuning with a full 15-odd limit, with less error than 31-equal, as one
> > of my alternatives, and it would also have to offer free modulation. Impossible? Not if you > think outside the box. Wait & see!)
>
> The full 15-odd-limit, less error than 31, free modulation, and only 17 notes? I'm fairly certain there's not one of us here who wouldn't be interested in such a tuning. I can't wait to see what you're about to roll out! Sleep well, take care of yourself, make a full recovery, and then blow all our minds with a new 17-tone WT.
>
> -Igs

After working on a couple of new ideas for the past several weeks, here's the new tuning that I completed last Friday. I spent a couple of hours over the past weekend trying it out on my generalized-keyboard Scalatron, and I'm very happy with the result:

! Secor17-ZRT.scl
!
George Secor's 17-tone Zany Rational Temperament, 27 Jan 2012
17
!
555/536
289/268
75/67
157/134
163/134
2015/1608
802/603
369/268
5075/3618
401/268
835/536
869/536
1010/603
1064/603
981/536
15/8
2/1

If you're trying this out with Scala's chromatic clavier, enter the commands "set nota sa24" and "set sagi mixed" to get the tonal layout to display properly. (This tuning departs significantly from 17-equal, and notations 17E or SA17 don't get it quite right.)

Let's check this "zany rational temperament"against the four requirements ("the full 15-odd-limit, less error than 31, free modulation, and only 17 notes?").

1) Only 17 notes?

Yep!

2) The full 15-odd-limit?

Obviously it would be impossible to have a full 15-odd limit rooted in all 17 keys, but here's what I came up with.

In JI it would take 18 tones to build three 15-limit otonal ogdoads (8:9:10:11:12:13:14:15), rooted on 1/1, 3/2, and 4/3. To get this in 17 tones, something must be tempered out. 17-ZRT is the offspring of a marriage of my 1/5-comma temperament extraordinaire:
/tuning-math/files/secor/scl/Secor1_5TX.scl
with my 17-tone well-temperament:
/tuning-math/files/secor/scl/secor17wt.scl
such that both 80:81 and 63:64 are tempered out. There are six (rationally approximated) 1/5-comma fifths in a chain of fifths from C to F# and a (rational) chain of ten fifths averaging around 5 cents wide from Ebb (=C#) to C. (There is a wide wolf fifth between F# and C#, which I'll discuss below.)

Taking C=1/1, in 17-ZRT there are 15-limit otonal ogdoads rooted on F, C, and G. If you're willing to cut some slack with the intonation, you also get a complete one on D (7 and 11 suffer most) and a partial one (lacking 13) on Bb (5 suffers most; however, the 6:7:9:11 approximation, with F=6, is excellent!).

3) less error than 31?

Taken overall, I'd say that 17-ZRT is competitive with 31-equal. Some odd harmonics (3, 9, 11, 13) are better, the others (5, 7, 15) not as good. At any rate, it's a lot better than 12-equal.

In 31-equal there are 5 sizes of 3rds: subminor, minor, neutral (or middle), major, and supermajor. 17-ZRT also has all of these (and more!), but there are only 3 sizes (which I'll refer to, relatively, as small, medium, and large) in any given key. Here's the "zany" part: the small thirds approximate 6:7 in 9 keys, 5:6 in 5 keys, and 13:15 in 3 keys; the medium thirds approximate neutral (or middle) 3rds (9:11 or 13:16) in 10 keys, 5:6 in 3 keys, and 4:5 in 4 keys; the large thirds approximate 7:9 in 10 keys, 4:5 in 5 keys, and 10:13 in 2 keys. As you go around the circle of fifths, the intonation changes drastically. Observe that the (5-limit) major and minor 3rds (and 6ths) each occur in more than one relative-size-class. (As you'll see below, this can be a rather useful property of the tuning.)

4) free modulation?

There's a complete circle of 17 fifths, so you can transpose anything from one key to any other. However, no two intervals of the same number of degrees of 17 are tuned exactly alike, and some of these intervals sound quite different (e.g., major becomes supermajor, subminor becomes minor, neutral becomes major or minor, etc.). Note, however, that in conventional harmony, a modulation from a major key to a minor key is still a modulation, even if the sizes of the intervals in the scale change. Thus, in 17-ZRT it is possible to make the claim that there is free modulation.

And there are still other features in addition to the four above.

5) conventional 5-limit harmony?

Yep, it's there (as 1/5-comma meantone, which has significantly less error than 12-equal) in 3 major keys (F, C, & G) and 3 minor keys (D, A, & E).

In the minor keys, you have the option of using dominant 7th chords with either middle 3rds (approximating 4:5) or large 3rds (wider than 7:9). The large 3rds are quite dissonant relative to 5-limit harmony and resolve very effectively (both melodically and harmonically) to the tonic minor triad (or to a final major triad with picardy 3rd). I discussed this at some length in my 17-tone well-temperament paper (beginning on p. 73, alias p. 19 in the pdf file):
http://anaphoria.com/Secor17puzzle.pdf
A lot of other things in that paper regarding 17-WT will also apply to 17-ZRT; e.g., virtually all of the chord progressions in Figures 5 and 6 will work, so this should be a good tuning for Margo Schulter's neo-Medieval music.

I mentioned above that there's a wide wolf 5th between F# and C# in the circle of 17 fifths. If you omit the tones in the circle from C# to E# (note: E#=Gb, also E#=F^), you'll get a circle of 12 fifths with a narrow wolf 5th between F# and Db (Db=C^). You'll get an E-major scale if you use Db, Ab, and Eb as the 3rds of A, E, and B, respectively, but there will be a false 5th between F# and Db (the narrow wolf in the 12-circle), as occurs in the supertonic triad in strict JI. Also note that in the 12-circle all of the neutral 3rds and 6ths are gone.

I attempted to eliminate the wide wolf 5th in the 17-circle by distributing the error among several consecutive 5ths, but I was not happy with the result. The prime 13 tones were too low in pitch, with the 8:13 approximations sounding more like minor 6ths than neutral 6ths; also, the 3rd of the D-major triad was a full pythagorean comma wide, and I still ended up with a couple of 5ths about 16 cents wide, which is not too good. I decided that it would be better to keep the two wolf fifths (the wide one in the 17-circle and the narrow now in 12, each with F# as the root), inasmuch as either of these could be used for a shock effect. With proper marketing, a defect can be turned into a feature!

6) incorporating Eastern scales?

Yep! There are plenty of neutral (or middle) intervals in 17-ZRT. All of the middle 3rds on Bb, F, C, G, and D are neutral 3rds, and the small 3rds on Bb, F, C, and G are all subminor 3rds, so there will be an ample number of 6:7:9:11 tetrads that can be used for non-12 harmony. (That's just in the "common" keys. You can also go to the far side of the 17-circle, where there are no major or minor 3rds.)

So that's my Meantone Alternative #1. (In case this one doesn't get rave reviews, I have some other alternatives in the pipeline.)

--George

--- In tuning@yahoogroups.com, "gdsecor" <gdsecor@...> wrote:
> After working on a couple of new ideas for the past several weeks, here's the new tuning that I completed last Friday. I spent a couple of hours over the past weekend trying it out on my generalized-keyboard Scalatron, and I'm very happy with the result:
>
> ! Secor17-ZRT.scl
> !
> George Secor's 17-tone Zany Rational Temperament, 27 Jan 2012
> 17
> !
> 555/536
> 289/268
> 75/67
> 157/134
> 163/134
> 2015/1608
> 802/603
> 369/268
> 5075/3618
> 401/268
> 835/536
> 869/536
> 1010/603
> 1064/603
> 981/536
> 15/8
> 2/1

Here's my personal opinion of this:

1. The wolf fifth is a deal-breaker for me. I wouldn't even call this a circulating temperament because of it.

2. I thought we were talking about meantone alternatives, but the part of this temperament that has good 5-limit harmony is meantone-based.

17-tone circulating temperaments are very attractive, and I can see room for at least four different categories of them:

I) Temperaments that don't even attempt to approximate the prime 5 at all. These would be much closer to 17edo than the other alternatives, because 17edo is such a good no-5s temperament to begin with.

II) Temperaments with a circle of fifths, one side of which is like meantone (using the 17c val), and the other side of which therefore has much wider fifths, like superpyth (using the 17p val). Such scales would probably be Rothenberg improper, and 5/4 would be represented inconsistently as sometimes 6 steps and sometimes 5 steps (similarly 6/5 would be sometimes 5 steps and sometimes 4 steps).

III) Temperaments with a circle of fifths, one side of which is like schismatic (using the 17p val), and the other side superpyth. These temperaments would consistently represent both 5/4 and 6/5 as 5 steps, leading to the interesting phenomenon that as you modulated through different keys, the thirds would gradually change from major to neutral to minor, and then back again. Supermajor and subminor thirds would always be available as 6-step and 4-step intervals respectively.

IV) Temperaments with a circle of neutral thirds, one side of which has flat fifths and tempers out 81/80 (like "mohaha" temperament), and the other side of which has sharp fifths and more resembles "beatles" temperament. In these temperaments 5/4 and 6/5 would consistently be represented as 6 steps and 4 steps respectively (i.e. the 17c val).

I wrote these category descriptions before really looking at your temperament's interval inventory in detail, and it appears that it falls squarely into category II (despite the wolf fifth making it not-really-circulating, or circulating only in a loose sense, if you prefer). Your widest 5-step interval is 912/725, or 397.263 cents, and your narrowest 6-step interval is 203/162, or 390.583 cents. This means it's Rothenberg improper and has an inconsistent mapping of 5/4 as sometimes 5 steps and sometimes 6 steps, as I suspect all good meantone/superpyth combinations will.

Keenan

--- In tuning@yahoogroups.com, "Keenan Pepper" <keenanpepper@...> wrote:
>
> --- In tuning@yahoogroups.com, "gdsecor" <gdsecor@> wrote:
> > After working on a couple of new ideas for the past several weeks, here's the new tuning that I completed last Friday. I spent a couple of hours over the past weekend trying it out on my generalized-keyboard Scalatron, and I'm very happy with the result:
> >
> > ! Secor17-ZRT.scl
> > !
> > George Secor's 17-tone Zany Rational Temperament, 27 Jan 2012
> > 17
> > !
> > 555/536
> > 289/268
> > 75/67
> > 157/134
> > 163/134
> > 2015/1608
> > 802/603
> > 369/268
> > 5075/3618
> > 401/268
> > 835/536
> > 869/536
> > 1010/603
> > 1064/603
> > 981/536
> > 15/8
> > 2/1
>
> Here's my personal opinion of this:
>
> 1. The wolf fifth is a deal-breaker for me. I wouldn't even call this a circulating temperament because of it.

F#-C# is almost exactly 13:20, which is pretty dissonant by itself. But if you add another tone a small (A) or medium-sized 3rd (A^) above the F#, the resulting triads actually sound halfway decent. (But adding the large 3rd, A#, 461 cents, retains the tension!) Given that there's so much variation in the sizes of the 3rds, I don't think it inappropriate to have a wolf 5th in the mix.

Anyway, I'll go back to the drawing board and see if I can come up with an 11-limit circulating temperament of 17 fifths.

> 2. I thought we were talking about meantone alternatives, but the part of this temperament that has good 5-limit harmony is meantone-based.

A portion of it would have to be meantone-based in order to have the A function as both prime 5 of root F and odd 9 of root G. But that doesn't disqualify it from being a meantone alternative, if only for the fact that the meantone feature is restricted to only a few keys. I think that lack of at least some meantone capability and/or prime 5 would be a deal-breaker for some folks.

> 17-tone circulating temperaments are very attractive, and I can see room for at least four different categories of them:
>
> I) Temperaments that don't even attempt to approximate the prime 5 at all. These would be much closer to 17edo than the other alternatives, because 17edo is such a good no-5s temperament to begin with.

But of course! I designed something I call a 17-tone well-temperament way back in 1978:
/tuning-math/files/secor/scl/secor17wt.scl
It's fully explained in this paper:
http://anaphoria.com/Secor17puzzle.pdf

One way to get prime 5 is to add another 17-WT circle of fifths 600 cents apart from the first one. That gets you a 17-limit tuning that's near-optimal for pajara (with optional expanded harmonic limit) in 10 different keys:
/tuning-math/files/secor/scl/secor34wt.scl

However, 34 tones is considerably more than Igs wanted for a meantone alternative. But, as an alternative to 31-equal, it's a viable option, and since it's 17-limit, it's also a nice alternative to 46-equal (Mike B. take note!).

While I'm on the subject of circulating alternatives in the 30-tone neighborhood, here's a 29-tone "HTT" 15-odd-limit circulating temperament (no wolf fifths!) that has error so low (in 6 different keys) that it sounds like JI:
/tuning-math/files/secor/scl/secor29htt.scl
And to drive home the point, here's a recording of a live performance from 1975 using this tuning on a generalized-keyboard Scalatron:
/tuning-math/files/secor/improv29.mp3
If you're trying it out with Scala's chromatic clavier, enter the command "set nota sahtt".

I also have a 17-tone (non-circulating) version of this "HTT" tuning (15-odd-limit in 3 different keys), in case anyone is interested.

> II) Temperaments with a circle of fifths, one side of which is like meantone (using the 17c val), and the other side of which therefore has much wider fifths, like superpyth (using the 17p val). Such scales would probably be Rothenberg improper, and 5/4 would be represented inconsistently as sometimes 6 steps and sometimes 5 steps (similarly 6/5 would be sometimes 5 steps and sometimes 4 steps).

My response is below.

> III) Temperaments with a circle of fifths, one side of which is like schismatic (using the 17p val), and the other side superpyth. These temperaments would consistently represent both 5/4 and 6/5 as 5 steps, leading to the interesting phenomenon that as you modulated through different keys, the thirds would gradually change from major to neutral to minor, and then back again. Supermajor and subminor thirds would always be available as 6-step and 4-step intervals respectively.

Okay, this is worth looking into. I'll check it out.

> IV) Temperaments with a circle of neutral thirds, one side of which has flat fifths and tempers out 81/80 (like "mohaha" temperament), and the other side of which has sharp fifths and more resembles "beatles" temperament. In these temperaments 5/4 and 6/5 would consistently be represented as 6 steps and 4 steps respectively (i.e. the 17c val).

That's also worth a try.

> I wrote these category descriptions before really looking at your temperament's interval inventory in detail, and it appears that it falls squarely into category II (despite the wolf fifth making it not-really-circulating, or circulating only in a loose sense, if you prefer).

Yes, that's part of the zaniness (as is the not-really-circulating circle of 12 fifths without neutral 3rds).

> Your widest 5-step interval is 912/725, or 397.263 cents, and your narrowest 6-step interval is 203/162, or 390.583 cents. This means it's Rothenberg improper and has an inconsistent mapping of 5/4 as sometimes 5 steps and sometimes 6 steps, as I suspect all good meantone/superpyth combinations will.

Yes, another zany "feature" that drives the theorist up the wall, but which could be very useful to a composer. The ability to do one or two special things in a given key that can't be done in most of the other keys can make this a very versatile tuning, full of surprises.

I was interested in some feedback about whether I was successfully representing all of the 15-odd-limit intervals, particularly in the 3 main keys (F, C, G), but also elsewhere. Looking at the numbers to check the amount of error is one thing, but a listening test may tell a different story. For example, plus-error is more tolerable for major 3rds and harmonic 7ths than minus-error.

--George