Tuning-Math Archive Section 1: 550

Message: 550 - Contents - Hide Contents Date: Tue, 24 Jul 2001 20:11:34 Subject: Re: Hey Carl From: Paul Erlich --- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:

> Paul, > > If you look at the old BP posts I linked yesterday and the recent > comma-chroma posts, I'm using a generalization that would find the > simplest 1 or 2D "generators" as identities that fall within a given > range (i.e., 2:3 for the 1D 7-tone Pythagorean or 4:5:6 for the 2D > syntonic diatonic). > > This is of course one way to generate periodicity blocks that'll > accomplish what you've outlined here to two dimensions -- only it does > it by starting with some arbitrary MOS index (i.e., x small steps and > y large steps) and not a set of UVs. >

Interesting . . . so what would be the 2D "generator" for Dave Keenan's 31-tone planar microtemperament?

Message: 551 - Contents - Hide Contents Date: Tue, 24 Jul 2001 20:36:29 Subject: Re: Hey Carl From: carl@l...

> Hey Carl, just wanted to let you know that my "Hypothesis" is > dedicated to you.

Neato! It's quite an honor to have such a cool hypothesis dedicated to you.

> You asked what a 2D and higher-dimensional generalization of an > MOS might be. There was a lot of talk on the Tuning list about > trivalent scales (scales where each generic interval has exactly > three specific step sizes) for a while but those seem too rarefied > to be the "right" answer.

Yeah, "trihill" never seemed right.

> Keenan's 31-tone 11-limit planar microtemperament, where 2 of the 4 > unison vectors are tempered out.

Is that the pre-Canasta one (Canasta having three unison vectors tempered out... 'zthat right? And 31-tet all four?) -Carl

Message: 553 - Contents - Hide Contents Date: Tue, 24 Jul 2001 21:51:43 Subject: Re: Hey Carl From: Paul Erlich --- In tuning-math@y..., carl@l... wrote:

>> Keenan's 31-tone 11-limit planar microtemperament, where 2 of the 4 >> unison vectors are tempered out. >

> Is that the pre-Canasta one (Canasta having three unison vectors > tempered out... 'zthat right? And 31-tet all four?) > > -Carl

You got it!

Message: 554 - Contents - Hide Contents Date: Tue, 24 Jul 2001 20:54:45 Subject: Re: BP linear temperament From: David C Keenan --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

> Perhaps my view was too severe, but it definitely seems to contradict > the _spirit_ of the near-just approximations to all simple ratios of > odd numbers, since the approximations to simple ratios involving even > numbers are so poor.

Oh no they're not. For sure they are bad in 13-ED3, but the 7-limit (1,2,4,5,7) optimum version of the Bohlen-Pierce-Stearns temperament (1:3 period, 442.60 cent generator) has no error greater than 8.4 cents. The octave itself is only 3.8c narrow. A chain of 15 generators (16 notes) is required for a complete 7-limit pentad (e.g. 3:4:5:6:7 or 4:5:6:7:9). Remember that we have tritave equivalence instead of octave equivalence. So for example 1:5, 3:5, 5:9 are all "tritave equivalent" so they all have the same error magnitude in this temperament, and they do not have the same error as 4:5, 5:6, or 9:10. This 7-limit MA (max-absolute) optimum generator is a 7:9 widened by 2/15 of the following "BPS-comma". 1647086 2^1 * 7^7 ------- = ------- ~= 56.37 c 1594323 3^13 And in fact the following 10-limit ratios come in under 8.4 cents error as well. 1:10 (9:10), 7:8, 7:10. But not 1:8 (8:9) or 5:8, (11.3c and 12.1c errors). The 10-limit (1,2,4,5,7,8,10) MA optimum generator is 442.77 cents with max error of 8.9c. This is 3/22 BPS-comma. So we get denominators of convergents (and semiconvergents) for this temperament being: 4 (5) (9) 13 (17) 30 43 (73) 116 These can be read as MOS cardinalities and the larger ones as possibly useful ED3s. -------- Keyboard -------- I've attached a .gif) of the keyboard mapping (showing cents) for 30 notes of this temperament, which also makes it clear that the "natural" notation for it would have only 4 nominals to the tritave. To see a 3:4:5:6:7 pattern, look at notes marked 0 491 885 1196 1459 (cents). ------ Guitar ------ This 30-note per tritave proper MOS will work incredibly well on a guitar! Simply tune adjacent open strings one generator (a supermajor third) apart. The smallest step is 48 cents. This is no worse than a 24-EDO guitar, but notice that it has 19 steps per octave. 4:5:6:7:9 barre chords should be playable. Here's the optimum rotation of the scale for fretting such a guitar. The zero of the rotation shown on the keyboard map, corresponds to the 16th fret (1017 cents) below. The keyboard scale is then playable everywhere on the top string, and as far from the nut as possible on the other strings. Fret Step posn size (to the nearest cent) ----------- 0 48 48 84 132 48 179 84 263 48 311 84 395 48 443 84 526 48 574 48 622 84 706 48 754 84 837 48 885 84 969 48 1017 48 1065 84 1148 48 1196 84 1280 48 1328 84 1411 48 1459 48 1507 84 1591 48 1639 84 1722 48 1770 84 1854 48 1902 Good work Dan!

Message: 555 - Contents - Hide Contents Date: Wed, 25 Jul 2001 22:14:20 Subject: Re: Hey Carl From: Dave Keenan --- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:

> Hi Paul, > > What is Dave Keenan's 31-tone planar microtemperament exactly... I > either missed it or have forgotten it -- any links? > > --Dan Stearns

It's keenan5.scl in the Scala archive.

Message: 556 - Contents - Hide Contents Date: Wed, 25 Jul 2001 22:32:46 Subject: Re: Hey Carl From: Paul Erlich --- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:

> --- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote: >> Hi Paul, >>

>> What is Dave Keenan's 31-tone planar microtemperament exactly... I >> either missed it or have forgotten it -- any links? >> >> --Dan Stearns >

> It's keenan5.scl in the Scala archive.

Dave, where's the tuning list post where you describe it, lattice it, and tell us which unison vectors are tempered out and which aren't? I was just looking at it a few days ago, but now I can't find it.

Message: 557 - Contents - Hide Contents Date: Wed, 25 Jul 2001 22:57:36 Subject: Re: BP linear temperament From: Paul Erlich --- In tuning-math@y..., David C Keenan <D.KEENAN@U...> wrote:

> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

>> Perhaps my view was too severe, but it definitely seems to contradict >> the _spirit_ of the near-just approximations to all simple ratios of >> odd numbers, since the approximations to simple ratios involving even >> numbers are so poor. >

> Oh no they're not.

I meant those producable by a small number of generators. In any case, I think it's worth pursuing this from _both_ angles. How about the triple-BP scale I discovered (The Bohlen-Pierce Site: BP Scale Structures * [with cont.] (Wayb.))? Is there a particular linear temperament that this cries out for?

Message: 558 - Contents - Hide Contents Date: Thu, 26 Jul 2001 00:15:11 Subject: Re: Hey Carl From: Dave Keenan --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

> Dave, where's the tuning list post where you describe it, lattice it, > and tell us which unison vectors are tempered out and which aren't? I > was just looking at it a few days ago, but now I can't find it. Yahoo groups: /tuning/message/7202 * [with cont.] Yahoo groups: /tuning/message/7279 * [with cont.] Yahoo groups: /tuning/message/7341 * [with cont.]

I don't say anything about commas _not_ tempered out. Those tempered out are 224:225 and 384:385. -- Dave Keenan

Message: 559 - Contents - Hide Contents Date: Thu, 26 Jul 2001 00:41:18 Subject: Re: Hey Carl From: Paul Erlich --- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:

> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

>> Dave, where's the tuning list post where you describe it, lattice > it,

>> and tell us which unison vectors are tempered out and which aren't? > I

>> was just looking at it a few days ago, but now I can't find it. > > Yahoo groups: /tuning/message/7202 * [with cont.] > Yahoo groups: /tuning/message/7279 * [with cont.] > Yahoo groups: /tuning/message/7341 * [with cont.] >

> I don't say anything about commas _not_ tempered out. Those tempered > out are 224:225 and 384:385.

I'm almost positive it was a different message I was looking at. Dang . . . should have bookmarked it. Anyway, the first two above may just confuse people.

Message: 560 - Contents - Hide Contents Date: Thu, 26 Jul 2001 00:53:33 Subject: Re: BP linear temperament From: Dave Keenan --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

> --- In tuning-math@y..., David C Keenan <D.KEENAN@U...> wrote:

>> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

>>> Perhaps my view was too severe, but it definitely seems to > contradict

>>> the _spirit_ of the near-just approximations to all simple ratios > of

>>> odd numbers, since the approximations to simple ratios involving > even

>>> numbers are so poor. >>

>> Oh no they're not. >

> I meant those producable by a small number of generators.

I think 7 is a small number. That's how many it needs for the octave (and hence the fifth) with 3.8c errors. Prime No. Generators ----- -------------- 2 7 3 0 5 2 7 -1

> In any > case, I think it's worth pursuing this from _both_ angles.

Sure. If 2s are omitted, the MA optimum generator is 439.82c and the max error is 4.8c.

> How about the triple-BP scale I discovered > (The Bohlen-Pierce Site: BP Scale Structures * [with cont.] (Wayb.))? Is there a > particular linear temperament that this cries out for?

Probably, but I don't have time. A suggested approach: Give the simplest frequency ratio you can for each scale degree. Then list the step sizes between these, classify them into two sizes, L and s, then figure out what the generator is from that. -- Dave Keenan

Message: 561 - Contents - Hide Contents Date: Thu, 26 Jul 2001 01:32:46 Subject: Re: BP linear temperament From: Paul Erlich --- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:

> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote: >>

>> I meant those producable by a small number of generators. >

> I think 7 is a small number. That's how many it needs for the octave > (and hence the fifth) with 3.8c errors. > > Prime No. Generators > ----- -------------- > 2 7 > 3 0 > 5 2 > 7 -1

By these standards I think 7 is a large number! It's much larger in magnitude than 0, 2, or -1. And to get some of the simplest ratios involving 2 (such as 4:3, 5:4, and 7:4) you need around 14 generators. Quite a qualitative difference, in my opinion, compared with what you need for the basic BP consonances.

Message: 563 - Contents - Hide Contents Date: Thu, 26 Jul 2001 21:07:53 Subject: Re: BP linear temperament From: Paul Erlich --- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:

> > Probably, but I don't have time. A suggested approach: Give the > simplest frequency ratio you can for each scale degree.

Hmm . . . this would be the Kees van Prooijen definition of a periodicity block. Have you studied this page? Manuel, can we do this in SCALA?

> Then list the > step sizes between these, classify them into two sizes, L and s, then > figure out what the generator is from that.

The Hypothesis in action!

Message: 564 - Contents - Hide Contents Date: Fri, 27 Jul 2001 01:20:01 Subject: 152-tET From: Paul Erlich Graham, did you come across anything relating to 152-tET in your searches? I notice it's near-just, and also contains 76-tET, which contains many linear temperaments (meantone, paultone, double- negative-7-limit-tetradecatonic-thingy . . .)?

Message: 565 - Contents - Hide Contents Date: Fri, 27 Jul 2001 04:58:20 Subject: Re: Hey Carl From: Dave Keenan --- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:

> Hi Dave, > > I don't use Scala, could you either give a link to an old post or just > repost the scale in cents? > > thanks, > > --Dan Stearns Hi Dan,

As it turns out, the scale that appears as keenan5.scl doesn't have only 3 step sizes. It's the second scale I give in the same post, (as a "near miss") that has only 3 step-sizes. Although I give a lattice for it in that post, I didn't give the cents. Yahoo groups: /tuning/message/7341 * [with cont.] So here they are: 0.0 s 30.40953308 L 79.61111475 m 115.8026469 m 151.994179 L 201.1957607 s 231.6052938 m 267.7968259 L 316.9984076 s 347.4079406 m 383.5994728 L 432.8010544 m 468.9925866 s 499.4021197 L 548.6037013 m 584.7952335 s 615.2047665 m 651.3962987 L 700.5978803 s 731.0074134 m 767.1989456 L 816.4005272 m 852.5920594 s 883.0015924 L 932.2031741 m 968.3947062 s 998.8042393 L 1048.005821 m 1084.197353 m 1120.388885 L 1169.590467 s 1200.0 What we want to know is: What are the generators for this particular 31 note hyper-MOS of a planar temperament? How you do you find them? Are they unique? What is the mapping from generators to primes? How do you construct this tuning from the generators? How do you construct other examples of the same planar temperament from them? How do you find linear temperaments that cover them? How do you make only those examples having exactly 3 step sizes (hyper-MOS)? Of those, how do you make only strictly proper ones? Is there a smaller strictly-proper 3-step-size scale in this planar temperament? Are there different 3-step-size scales in this planar temperament, having 31 notes? We can answer all the corresponding questions for linear temperaments. I think I know the answer to some of these questions for this particular planar temperament, but not for planar temperaments in general. Any light you can shed will be much appreciated. -- Dave Keenan

Message: 566 - Contents - Hide Contents Date: Fri, 27 Jul 2001 05:43:01 Subject: Re: Hey Carl From: Dave Keenan Here's a set of unison vectors for the tuning in the previous post. 3 5 7 11 -------------------------- 3 7 0 0 4 -1 0 0 2 2 -1 0 -1 1 1 1 It has a determinant of 31, the first two vectors are chromatic (not tempered out), the last two are commatic (tempered out). 3 7 4 -1 has a determinant of -31 by itself. Is it possible that although the scale has 3 step sizes and is symmetrical, it is not a hyper-MOS? -- Dave Keenan

Message: 567 - Contents - Hide Contents Date: Fri, 27 Jul 2001 09:48 +0 Subject: Re: 152-tET From: graham@m... In-Reply-To: <9jqfk1+ifbe@e...> Paul wrote:

> Graham, did you come across anything relating to 152-tET in your > searches? I notice it's near-just, and also contains 76-tET, which > contains many linear temperaments (meantone, paultone, double- > negative-7-limit-tetradecatonic-thingy . . .)?

No. I was originally only considering ETs with less than 100 notes. I've expanded that now, but it doesn't seem to be 13-limit consistent, and is too high to be considered for the other limits. I did notice 72 came up a lot in the higher limits. Here's what you get by combining the two: 3/28, 16.2 cent generator basis: (0.125, 0.013529015588479165) mapping by period and generator: ([8, 0], ([13, 19, 23, 28, 30], [-3, -4, -5, -3, -4])) mapping by steps: [(152, 72), (241, 114), (353, 167), (427, 202), (526, 249), (562, 266)] unison vectors: [[-3, -1, 0, -1, 0, 2], [1, 0, 6, -4, 0, -1], [-7, 1, 0, -3, 4, 0], [1, 10, 0, - 6, 0, 0]] highest interval width: 6 complexity measure: 48 (56 for smallest MOS) highest error: 0.004556 (5.467 cents) 11-limit: highest interval width: 6 complexity measure: 48 (56 for smallest MOS) highest error: 0.001072 (1.286 cents) unique 7-limit: highest interval width: 5 complexity measure: 40 (48 for smallest MOS) highest error: 0.001044 (1.253 cents) unique Graham

Message: 570 - Contents - Hide Contents Date: Fri, 27 Jul 2001 19:58:54 Subject: Re: Hey Carl From: Paul Erlich --- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:

> Here's a set of unison vectors for the tuning in the previous post. > > 3 5 7 11 > -------------------------- > 3 7 0 0 > 4 -1 0 0 > 2 2 -1 0 > -1 1 1 1

FWIW, the Fokker parallelopiped PB corresponding to these four unison vectors is entirely within the 3-5 plane: cents numerator denominator 0 1 1 41.059 128 125 70.672 25 24 111.73 16 15 162.85 1125 1024 203.91 9 8 223.46 256 225 274.58 75 64 315.64 6 5 335.19 4096 3375 386.31 5 4 427.37 32 25 478.49 675 512 498.04 4 3 539.1 512 375 590.22 45 32 609.78 64 45 660.9 375 256 701.96 3 2 721.51 1024 675 772.63 25 16 813.69 8 5 864.81 3375 2048 884.36 5 3 925.42 128 75 976.54 225 128 996.09 16 9 1037.1 2048 1125 1088.3 15 8 1129.3 48 25 1158.9 125 64 and contains the following step sizes: 41.059 29.614 41.059 51.12 41.059 19.553 51.12 41.059 19.553 51.12 41.059 51.12 19.553 41.059 51.12 19.553 51.12 41.059 19.553 51.12 41.059 51.12 19.553 41.059 51.12 19.553 41.059 51.12 41.059 29.614 41.059

Message: 571 - Contents - Hide Contents Date: Fri, 27 Jul 2001 20:00:20 Subject: Re: 152-tET From: Paul Erlich --- In tuning-math@y..., graham@m... wrote:

> > I did notice 72 came up a lot in the higher limits. Here's what you get > by combining the two:

Combining the two what? How?

Message: 572 - Contents - Hide Contents Date: Fri, 27 Jul 2001 20:03:15 Subject: Re: Hey Carl From: Paul Erlich --- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:

> Hi Dave, > > <<Is it possible that although the scale has 3 step sizes and is > symmetrical, it is not a hyper-MOS?>> > > Right, that's sort of what I just posted, but then again I'm not sure > of exactly what definition of "hyper-MOS" we're going by!

Perhaps Dave is trying to proceed by analogy from, say, the situation where a scale like 2 2 1 2 1 2 2 in 12-tET has 2 step sizes and is symmetrical, but is not an MOS?