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Message: 7100 Date: Fri, 25 Jul 2003 23:05:22 Subject: Re: Creating a Temperment /Comma From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "paulhjelmstad" <paul.hjelmstad@u...> wrote:

> > Thanks. (3)^8/(5)^5*(2) Hmm. 84 cents? I christen it "sloppyjoe"

> Rats. It's been taken. Can't name it. It's "Diaschizoid"

Taken by what?

Message: 7101 Date: Fri, 25 Jul 2003 16:14:00 Subject: Re: Creating a Temperment /Comma From: Carl Lumma

>> >I would like to know which temperment/comma connects 12, 47, 35, and >> >23 ets on zoomr.gif. How could I calculate these for myself? Thanks! >> >If this is a new one, could I name it? What is the 5-limit vector?

>> >> The comma is 6561/6250, according to Gene's maple, if I did it right.

> >I get the same. You get a temperament with a generator the size of a >semitone, five of which give a fourth, and eight of which give a minor >sixth, which obviously is compatible with 12-et.

So what I'm missing is... the heuristic says temperaments based on this comma will be bad. And indeed, 23, 47 don't seem to be very good. But 12 is good. So what gives? -Carl

Message: 7103 Date: Fri, 25 Jul 2003 16:23:20 Subject: Re: Creating a Temperment /Comma From: Carl Lumma

>Well, I can't find it so I doubt it's much good. Thanks. If I bought >Maple and installed it, could I possibly run these calculations >myself?

I'm sure Gene will give you the code. Maybe he'll even post it on his web site. By the Way, Paaauuul (Erlich), I'm told Matlab has the Maple engine built into it!! -Carl

Message: 7105 Date: Fri, 25 Jul 2003 23:32:18 Subject: Re: Creating a Temperment /Comma From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> >I get the same. You get a temperament with a generator the size of a > >semitone, five of which give a fourth, and eight of which give a minor > >sixth, which obviously is compatible with 12-et.

> > So what I'm missing is... the heuristic says temperaments based on > this comma will be bad. And indeed, 23, 47 don't seem to be very > good. But 12 is good. So what gives?

As a linear temperament, it isn't very interesting, because its complexity is too high. As a means of tempering 12 notes, it becomes much more interesting. Looking at temperaments dominated (as this one is) by 12 strikes me as worthwhile from that point of view. I think I'll check out this idea of semitones of around 101.5 cents for 12 notes.

Message: 7106 Date: Fri, 25 Jul 2003 23:33:52 Subject: Re: Creating a Temperment /Comma From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> >Well, I can't find it so I doubt it's much good. Thanks. If I bought > >Maple and installed it, could I possibly run these calculations > >myself?

> > I'm sure Gene will give you the code. Maybe he'll even post it > on his web site.

I'm planning too. Hope no one laughs.

Message: 7107 Date: Fri, 25 Jul 2003 23:41:53 Subject: Re: Creating a Temperment /Comma From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "paulhjelmstad" <paul.hjelmstad@u...> wrote: (But I can't find

> anything for 12, 25, 37, 49 ets etc)

This has comma 262144/253125. Once again the generator is about a semitone and five of them give a fourth. This time, however, the semitone is a little smaller than 12-et (about 98.3 cents) and four of them give you a major third. Evidently these two belong together and should get cute matching names.

Message: 7108 Date: Fri, 25 Jul 2003 16:44:23 Subject: Re: Creating a Temperment /Comma From: Carl Lumma

>As a linear temperament, it isn't very interesting, because its >complexity is too high. As a means of tempering 12 notes, it becomes >much more interesting.

Is this because there's a T[n] comma associated with stopping at 12? Or...? By the way, what wound up being your favorite complexity measure and why? The last 5-limit list I have from you uses "geometric complexity"... which is... Oh, that's the really hard-to-understand Euclidean distance-on-the-lattice-to-the-comma thing, isn't it? Meanwhile, Dave and Graham, are you guys still using map-based complexity? Doesn't this run into problems because there are many different ways to write the map for a given temperament? Me, I like the idea of number of notes. But then we have to consider the effects of the comma formed between the ends of the chain. Has anyone ever looked at error as a function of chain length for each temperament? -Carl

Message: 7109 Date: Fri, 25 Jul 2003 19:34:19 Subject: Re: Creating a Temperment /Comma From: Carl Lumma

>>>As a linear temperament, it isn't very interesting, because its >>>complexity is too high. As a means of tempering 12 notes, it >>>becomes much more interesting.

>> >> Is this because there's a T[n] comma associated with stopping at >> 12? Or...?

//

>Just as with meantone, we can temper 12 notes, or we can allow >ourselves to take something written in 12 notes and extend it to >more using the temperament;

How does one "take something written in 12 notes and extend it with the temperament"?

>>By the way, what wound up being your favorite complexity >>measure and why?

> >I like geometric complexity because it works for all regular >temperaments.

Such as planar and higher temperaments. But the 'how many notes' complexity obviously generalizes to these as well (in the linear case it's equivalent to Dave/Graham complexity).

>> Me, I like the idea of number of notes. But then we have to >> consider the effects of the comma formed between the ends of >> the chain. Has anyone ever looked at error as a function of >> chain length for each temperament?

> >Aren't you conflating temperaments and scales?

I understand that temperaments are abstract Things, but they ultimately must be expressed as scales to be used. If geometric complexity is so good, the T[n] stuff wouldn't have been such a surprise (unless, I suppose, consonances formed by way of the extra "wolf" comma necessarily break consistency -- do they?). In my view, musically complexity has to boil down to 'how many notes get me how many consonances'. Say n/i where n is notes and i is the number of consonant dyads. Dave/Graham get rid of the denominator by standardizing i to 'a complete n-ad' (Dave once showed that for linear temperaments, otonal and utonal n-ads always come in pairs -- I wonder if this is true for planar temperaments...). I suggest we plot n/i for temperament T for all n up to some large number, and report the minima. If this is a bad idea I'd like to know why. -Carl

Message: 7110 Date: Sat, 26 Jul 2003 13:14:41 Subject: Re: Creating a Temperment /Comma From: Carl Lumma

>> Such as planar and higher temperaments. But the 'how many notes' >> complexity obviously generalizes to these as well (in the linear >> case it's equivalent to Dave/Graham complexity).

> >What about equal temperaments?

For an n-et, one can always think of the generators as 1200 and 1200/n. Then it works the same as for linear temperaments. Or isn't that what you do?

>> In my view, musically complexity has to boil down to 'how many >> notes get me how many consonances'. Say n/i where n is notes >> and i is the number of consonant dyads. Dave/Graham get rid of >> the denominator by standardizing i to 'a complete n-ad' (Dave >> once showed that for linear temperaments, otonal and utonal n-ads >> always come in pairs -- I wonder if this is true for planar >> temperaments...). I suggest we plot n/i for temperament T for >> all n up to some large number, and report the minima. If this >> is a bad idea I'd like to know why.

> >For a linear temperament, the number of consonances is (roughly) >the number of notes minus the complexity. So n-i is a constant. >If you then plot n/i, you can replace n with i+c to get >(i+c)/i = 1+c/i. As i tends to infinity, this tends to 1, which >is as small as it gets. So your measure would depend on how >large n is allowed to get.

Ok, I follow your reasoning but I'm not sure what your conclusion is. I think we'd limit n to the number of notes in the Fokker block. At that n, the complexity would be expected to match the heuristic (for linear temperaments) and geometric complexity (for everything). But when n gets smaller we need a way to quantify what happens. It was probably wrong of me to suggest this be some sort of temperament complexity. And I'm wondering if intervals achieved through use of the "wolf" break consistency, or whatever Gene calls it for regular temperaments.

>For an equal temperament, n can only ever take one value, and >n/i=1.

How do you figure the number of consonant intervals equals the number of notes in the et?

>For a planar temperament, the number of consonance depends on what >generators you choose, and how you construct the scale from them. >You can get more otonal than utonal if you like.

Aha.

>Taking 5-limit JI as the canonical planar scale, C-E-G gives 1 >more otonality than utonality. And C-D-E-G-G#-B has 2 more.

Do such blocks necessarily involve unison vectors that are larger than their smallest 2nd?

>But you should get the same value for complexity whether you use >otonal or utonal.

Excactly -- which is why they apparently both must be counted for planar temperaments. -Carl

Message: 7111 Date: Sat, 26 Jul 2003 01:19:58 Subject: Re: Creating a Temperment /Comma From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> >As a linear temperament, it isn't very interesting, because its > >complexity is too high. As a means of tempering 12 notes, it becomes > >much more interesting.

> > Is this because there's a T[n] comma associated with stopping at > 12?

Only if you think a comma would be interesting. Or...? There's a historical importance to 12 which is, of course, obvious. There are just two ways to generate a cyclic group of order 12--one gives the fourth/fifth generators we usually focus on, the other is the semitone. Just as with meantone, we can temper 12 notes, or we can allow ourselves to take something written in 12 notes and extend it to more using the temperament; it's an idea worth exploring, at any rate.

> By the way, what wound up being your favorite complexity measure > and why?

I like geometric complexity because it works for all regular temperaments. The last 5-limit list I have from you uses "geometric

> complexity"... which is... Oh, that's the really hard-to-understand > Euclidean distance-on-the-lattice-to-the-comma thing, isn't it?

Fraid so.

> Me, I like the idea of number of notes. But then we have to > consider the effects of the comma formed between the ends of > the chain. Has anyone ever looked at error as a function of > chain length for each temperament?

Aren't you conflating temperaments and scales?

Message: 7112 Date: Sat, 26 Jul 2003 06:01:46 Subject: Re: Creating a Temperment /Comma From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> How does one "take something written in 12 notes and extend it with > the temperament"?

These are 5-limit temperaments, so you should concentrate on getting good triads.

Message: 7113 Date: Sat, 26 Jul 2003 12:26:36 Subject: Re: Creating a Temperment /Comma From: Graham Breed Carl Lumma wrote:

> Such as planar and higher temperaments. But the 'how many notes' > complexity obviously generalizes to these as well (in the linear > case it's equivalent to Dave/Graham complexity).

What about equal temperaments?

> In my view, musically complexity has to boil down to 'how many > notes get me how many consonances'. Say n/i where n is notes > and i is the number of consonant dyads. Dave/Graham get rid of > the denominator by standardizing i to 'a complete n-ad' (Dave > once showed that for linear temperaments, otonal and utonal n-ads > always come in pairs -- I wonder if this is true for planar > temperaments...). I suggest we plot n/i for temperament T for > all n up to some large number, and report the minima. If this > is a bad idea I'd like to know why.

For a linear temperament, the number of consonances is (roughly) the number of notes minus the complexity. So n-i is a constant. If you then plot n/i, you can replace n with i+c to get (i+c)/i = 1+c/i. As i tends to infinity, this tends to 1, which is as small as it gets. So your measure would depend on how large n is allowed to get. For an equal temperament, n can only ever take one value, and n/i=1. For a planar temperament, the number of consonance depends on what generators you choose, and how you construct the scale from them. You can get more otonal than utonal if you like. Taking 5-limit JI as the canonical planar scale, C-E-G gives 1 more otonality than utonality. And C-D-E-G-G#-B has 2 more. But you should get the same value for complexity whether you use otonal or utonal. If you standardize on a parallelogram, like C-E-G-B, than you get equal numbers of each chord. That's assuming you take the fifth and third as generators. With a tone and semitone, it'd take 5 notes to get 1 triad, and 6 notes to get 1 of each triad (which gets you 2 of 1 kind). I haven't looked into planar temperaments yet, so I don't know if there's an obvious way to get the optimum generators. I think you still get n/i tends to 1 as n tends to infinity. Graham

Message: 7114 Date: Sat, 26 Jul 2003 12:52:03 Subject: Re: Creating a Temperment /Comma From: Graham Breed paulhjelmstad wrote:

> I would like to know which temperment/comma connects 12, 47, 35, and > 23 ets on zoomr.gif. How could I calculate these for myself? Thanks! > If this is a new one, could I name it? What is the 5-limit vector?

Go to Temperament Finder * [with cont.] (Wayb.) and enter 12 and 47 for the ets, and 5 for the limit. That gives you the mapping [(1, 0), (2, -5), (3, -8)] It doesn't show the comma because I don't see the point, and they aren't that easy to calculate for some temperaments. But for the 5-limit case you can work it out from the mapping. It must involve 3**8 and 5**5 with opposite signs, from the octave equivalent mapping (-5, -8). You then multiply (x, 8, -5) by the period part of the mapping to get x + 2*8 - 3*5. For a unison vector, this should be zero, so x = -2*8 + 3*5 = 15-16 = -1. So the comma is (-1 8 -5) or 2**(-1) * 3**8 * 5**(-5) = 6561:6250. There are different tools runnable online at Linear Temperament Finding Home * [with cont.] (Wayb.) and you can get the library for Python, which is free, and play with it as you like. To get commas for a linear temperament as ratios,

>>> map(temper.getRatio, temper.Temperament(12,47,

temper.limit5).getUnisonVectors()) [(6561, 6250)] Graham

Message: 7115 Date: Sun, 27 Jul 2003 17:54:25 Subject: Re: Creating a Temperment /Comma From: Graham Breed Carl Lumma wrote:

> For an n-et, one can always think of the generators as 1200 and > 1200/n. Then it works the same as for linear temperaments. Or > isn't that what you do?

In that case, you're turning an et into a linear temperament in an arbitrary way. If you're consistent between ets, it should be the same as taking a fixed proportion of the number of notes to the octave. For comparing equal and linear temperaments it's still arbitrary, and will make ets look more complicated then they are.

>>For a linear temperament, the number of consonances is (roughly) >>the number of notes minus the complexity. So n-i is a constant. >>If you then plot n/i, you can replace n with i+c to get >>(i+c)/i = 1+c/i. As i tends to infinity, this tends to 1, which >>is as small as it gets. So your measure would depend on how >>large n is allowed to get.

> > Ok, I follow your reasoning but I'm not sure what your conclusion > is. I think we'd limit n to the number of notes in the Fokker > block. At that n, the complexity would be expected to match the > heuristic (for linear temperaments) and geometric complexity (for > everything). But when n gets smaller we need a way to quantify > what happens. It was probably wrong of me to suggest this be some > sort of temperament complexity. And I'm wondering if intervals > achieved through use of the "wolf" break consistency, or whatever > Gene calls it for regular temperaments.

You can conclude whatever you like. What Fokker block? Why bring the heuristic into it? Geometric complexity might work, but I don't understand it.

>>For an equal temperament, n can only ever take one value, and >>n/i=1.

> > How do you figure the number of consonant intervals equals the > number of notes in the et?

I thought consonances were chords. Otherwise, why distinguish otonal and utonal? It makes more sense for planar temperaments anyway. A-B-C-G has all the 5-limit consonances, but no triads. This can't happen with linear temperaments.

>>Taking 5-limit JI as the canonical planar scale, C-E-G gives 1 >>more otonality than utonality. And C-D-E-G-G#-B has 2 more.

> > Do such blocks necessarily involve unison vectors that are larger > than their smallest 2nd?

I don't know. Would it always work out for a well formed periodicity block? I'd prefer the complexity didn't depend on the tuning.

>>But you should get the same value for complexity whether you use >>otonal or utonal.

> > Excactly -- which is why they apparently both must be counted > for planar temperaments.

I thought it was why it didn't matter. Graham

Message: 7116 Date: Sun, 27 Jul 2003 13:50:11 Subject: Re: Creating a Temperment /Comma From: Carl Lumma

>> For an n-et, one can always think of the generators as 1200 and >> 1200/n. Then it works the same as for linear temperaments. Or >> isn't that what you do?

> >In that case, you're turning an et into a linear temperament in an >arbitrary way. If you're consistent between ets, it should be the >same as taking a fixed proportion of the number of notes to the >octave. For comparing equal and linear temperaments it's still >arbitrary, and will make ets look more complicated then they are.

So you suggest taking a fixed proportion of the number of notes to the octave? What does that mean, and how does one do it?

>>>For a linear temperament, the number of consonances is (roughly) >>>the number of notes minus the complexity. So n-i is a constant. >>>If you then plot n/i, you can replace n with i+c to get >>>(i+c)/i = 1+c/i. As i tends to infinity, this tends to 1, which >>>is as small as it gets. So your measure would depend on how >>>large n is allowed to get.

>> >>Ok, I follow your reasoning but I'm not sure what your conclusion >>is. I think we'd limit n to the number of notes in the Fokker >>block. At that n, the complexity would be expected to match the >>heuristic (for linear temperaments) and geometric complexity (for >>everything). But when n gets smaller we need a way to quantify >>what happens. It was probably wrong of me to suggest this be some >>sort of temperament complexity. And I'm wondering if intervals >>achieved through use of the "wolf" break consistency, or whatever >>Gene calls it for regular temperaments.

> >You can conclude whatever you like. What Fokker block?

Any temperament can be viewed in terms of Fokker blocks.

>Why bring the heuristic into it?

Because it tells you the complexity of the temperament.

>Geometric complexity might work, but I don't understand it.

That makes two of us.

>>>For an equal temperament, n can only ever take one value, and >>>n/i=1.

>> >> How do you figure the number of consonant intervals equals the >> number of notes in the et?

> >I thought consonances were chords. Otherwise, why distinguish otonal >and utonal?

I was asking about the o/utonal distinction re. your complexity. I'm suggesting looking at dyads.

>>>Taking 5-limit JI as the canonical planar scale, C-E-G gives 1 >>>more otonality than utonality. And C-D-E-G-G#-B has 2 more.

>> >> Do such blocks necessarily involve unison vectors that are larger >> than their smallest 2nd?

> >I don't know. Would it always work out for a well formed periodicity >block? I'd prefer the complexity didn't depend on the tuning.

But tempering adds intervals/notes, which is how you lower complexity. So the tuning has to matter.

>>>But you should get the same value for complexity whether you use >>>otonal or utonal.

>> >> Excactly -- which is why they apparently both must be counted >> for planar temperaments.

So that's important result number 1 from this little exchange. -Carl

Message: 7117 Date: Mon, 28 Jul 2003 17:59:32 Subject: Re: Creating a Temperment /Comma From: Graham Breed Gene Ward Smith wrote:

> Two words: geometric complextity.

Yes, and if you follow back throught the thread you'll see that neither Carl nor I understand it. And explanation would be more useful than repeated invocation of the name. Graham

Message: 7118 Date: Mon, 28 Jul 2003 10:01:52 Subject: Re: Creating a Temperment /Comma From: Carl Lumma

>> Two words: geometric complextity.

> >Yes, and if you follow back throught the thread you'll see that neither >Carl nor I understand it. And explanation would be more useful than >repeated invocation of the name.

In fairness Gene has tried to explain it. Paul was working on adapting the heuristic. You use the heuristic of both commas, plus the angle between them ("straightness")... -Carl

Message: 7119 Date: Mon, 28 Jul 2003 10:39:20 Subject: Re: Creating a Temperment /Comma From: Graham Breed Carl Lumma wrote:

> So you suggest taking a fixed proportion of the number of notes to > the octave? What does that mean, and how does one do it?

I don't suggest anything, I only know the problems. If you're counting steps upwards from the 1/1, then a 4:5:6 chord will be the same size as a 2:3 interval. For a consistent et of n notes to the octave, that means the number of steps you need for a complete triad is the nearest integer to 0.585*n. For an inconsistent et it won't be far off. So for comparing one et to another, you get the same result as comparing the number of steps. For comparing equal with linear temperaments, you get 7-equal having the same complexity as meantone, which is underestimating 7-equal.

> Any temperament can be viewed in terms of Fokker blocks.

Usually an infinite number of them!

>>Why bring the heuristic into it?

> > Because it tells you the complexity of the temperament.

It gives you an estimate of something you can calculate exactly. At least, the heuristic I'm thinking of does, but that gives accuracy not complexity.

> I was asking about the o/utonal distinction re. your complexity. > I'm suggesting looking at dyads.

For a linear temperament, the complexity of a chord is always the same as its most complex interval. So it doesn't matter which you look at. For planar temperaments that needn't be the case, but perhaps it will be for sensibly constructed blocks.

> But tempering adds intervals/notes, which is how you lower complexity. > So the tuning has to matter.

I noticed some mention of wolves before, so perhaps this is it. It's assumed we're dealing with regular temperaments, where the tempering of each interval is always the same. So even if the tuning happens to give you some alternative approximations you ignore them. Hence the 5:1 in meantone is always four fifths, even with a Pythagorean tuning where a schismic approximation would be closer. If you wanted that, you should have asked for schismic. In meantone, a fifth is 1 generator step, a major third is 4 generators and a minor third is 3 generators. If that's all you need to calculate the complexity, you don't need the tuning. Most of the complexity measures for linear temperaments work like this. The exception is the "smallest MOS" one, because you need some idea of the size of the generator to know what MOSs are valid. That's a practical problem for a search algorithm, because it means you have to optimize the tuning for any given temperament before you can reject it for being too complex. (Then again, for temperaments defined by a pair of ets, you can get a list of MOS sizes by working backwards on the scale tree, and this will always be sufficient if the seed ets are consistent.) Because a search for planar temperaments is going to be harder, it would be nice if we could still calculate the complexity independent of tuning. But perhaps planar temperaments generated from equal temperaments can always be assigned well formed Fokker blocks without worrying about the tuning. Graham

Message: 7121 Date: Mon, 28 Jul 2003 13:24:59 Subject: Re: Creating a Temperment /Comma From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote: Because a search for planar temperaments is going to be harder, it would

> be nice if we could still calculate the complexity independent of > tuning.

Two words: geometric complextity.

Message: 7123 Date: Mon, 28 Jul 2003 10:57:06 Subject: Re: Creating a Temperment /Comma From: Carl Lumma

>> Any temperament can be viewed in terms of Fokker blocks.

> >Usually an infinite number of them!

That's why blocks was plural!

>>>Why bring the heuristic into it?

>> >> Because it tells you the complexity of the temperament.

> >It gives you an estimate of something you can calculate exactly. At >least, the heuristic I'm thinking of does, but that gives accuracy >not complexity.

There's an error heuristic |n-d|/d*log(d), and a complexity heuristic, log(d).

>> I was asking about the o/utonal distinction re. your complexity. >> I'm suggesting looking at dyads.

> >For a linear temperament, the complexity of a chord is always the same >as its most complex interval. So it doesn't matter which you look at.

But I'm suggesting counting the number of available dyads per number of notes.

>>But tempering adds intervals/notes, which is how you lower complexity. >>So the tuning has to matter.

> >I noticed some mention of wolves before, so perhaps this is it. It's >assumed we're dealing with regular temperaments, where the tempering >of each interval is always the same. So even if the tuning happens to >give you some alternative approximations you ignore them. Hence the >5:1 in meantone is always four fifths, even with a Pythagorean tuning >where a schismic approximation would be closer. If you wanted that, >you should have asked for schismic.

Right. Gene did a search for scales where the wolf was considered to be another comma. Or something. I'm trying to understand how this fits into the temperament stuff. His results looked like:

>>Blackwood[10] >>[0, 5, 0, 8, 0, -14] [[5, 8, 12, 14], [0, 0, -1, 0]] >> >>bad 1662.988586 comp 10.25428060 rms 15.81535241 >>graham 5 scale size 10 ratio 2.000000

>In meantone, a fifth is 1 generator step, a major third is 4 generators >and a minor third is 3 generators. If that's all you need to calculate >the complexity, you don't need the tuning.

Ok, but it implies some kind of non-JI tuning. Maybe I should have said "ignoring commas increases intervals/notes".

>Most of the complexity measures for linear temperaments work like >this. The exception is the "smallest MOS" one,

That doesn't sound like a very good one. Maybe Gene can prod us to figure out geometric complexity. Yahoo groups: /tuning-math/message/5546 * [with cont.] Yahoo groups: /tuning-math/message/5598 * [with cont.] Yahoo groups: /tuning-math/message/5671 * [with cont.] Yahoo groups: /tuning-math/message/5692 * [with cont.] -Carl

Message: 7124 Date: Mon, 28 Jul 2003 19:27:23 Subject: Re: Creating a Temperment /Comma From: Graham Breed paulhjelmstad wrote:

> This site is cool. I've been playing with it. However, what do I > enter (as an example) in "Temperments from Unison Vectors"?

A list of commas that you want tempered out. For example, 81:80, 225:224, 1025:1024, 126:125, 50:49 and 64:63 should give some 7-limit temperaments. It's liberal about the format, so you can paste this whole message in. Graham

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