Tuning-Math Archive Section 1: 600

Message: 600 - Contents - Hide Contents Date: Mon, 30 Jul 2001 00:44:14 Subject: Re: Hey Carl From: Paul Erlich --- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:

> Hi Paul, > > Lots of stuff to chew on there, but here's some quick thoughts... > > <<Still can't see what a 2D generator could mean.>> > > When I say it I mean it in a very simple way -- as I see it it's > exactly the generalized difference between the 7-tone Pythagorean > single 1D plane built from 2:3s, and the classic JI, 5-limit diatonic > 2D plane built from 2:3s and 4:5s.

So wouldn't any connected, convex chunk of an N-dimensional JI lattice be "generatable" by this definition?

> > > <<Here's why the hypothesis should work [SNIP] QED -- right?>> > > Well it might make sense right off to some folks but it's a lot for me > to follow right now without some examples... do you think you could > flesh out the steps with some?

I need to draw some diagrams for you . . .

> > > <<another thought is that trivalency is a very special case>> > > Certainly when compared to "bivalency" it's not as omnipresent (as > I've said before all M-out-of-N sets are bivalent within a period), > but I could come up with several trivalent examples pretty easily, so > I don't think it's exactly "very special".

By "very special" here I didn't mean to imply that it would be hard for you to find examples . . . just that in general, a 2D hyper-MOS would have up to four specific sizes for each generic interval.

Message: 602 - Contents - Hide Contents Date: Sun, 29 Jul 2001 19:20:01 Subject: Re: TGs, Unison Vectors, and Octaves From: monz

> From: Paul Erlich <paul@s...> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Sunday, July 29, 2001 5:46 PM > Subject: [tuning-math] Re: TGs, Unison Vectors, and Octaves > >> [monz:]

>> It seems to me like you're alluding to my idea of "finity", >> in that the composers make use of unison-vectors and the >> listeners pick that up, without anyone really being very >> conscious of it all. Am on I the right track? >

> Sure -- I've been implying something to that effect specifically on > the Tuning List, rather than here. And of course, your inquiries into > finity are what led me to study PBs in the first place.

Right... duh! on my part... of course I remember that -- I was *with* you when it all started! But as far as the *unconscious* aspect of it... you haven't really made that explicit, and I suppose that's what I was doing here. That's probably the thing I find most interesting about your avenue of exploration. It seems that there are mathematical ways of modeling our unconscious tonal habits after all. love / peace / harmony ... -monz Yahoo! GeoCities * [with cont.] (Wayb.) "All roads lead to n^0" _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 604 - Contents - Hide Contents Date: Mon, 30 Jul 2001 11:23 +0 Subject: Re: Magic lattices From: graham@m... In-Reply-To: <3.0.6.32.20010729153925.00a89630@u...> Dave Keenan wrote:

>> I've discovered that "Multiple Approximations Generated Iteratively > and

>> Consistently" is an acronym for "MAGIC". What a coincidence! >

> Tee hee! Yes I _had_ noticed that. > > By the way, you can delete the second ocurrence of it in your catalog. > The > 5-limit one. That was my fault. Oops.

I've also updated the Bohlen-Pierce entry.

>> Now Dave Keenan's found an alternative simplified Miracle lattice, > let's

>> see if he can make anything of this. >

> The lattice I gave for Miracle works just as well for this temperament > (without the 11s of course), because the 224:225 is distributed in this > temperament too.

Yes, it does! I also ended up with my own lattice that removes the 224:225. Graham

Message: 606 - Contents - Hide Contents Date: Mon, 30 Jul 2001 18:48:46 Subject: Re: BP linear temperament From: Paul Erlich --- In tuning-math@y..., <manuel.op.de.coul@e...> wrote:

> > Paul 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? >

> Yes, with APPROXIMATE while using one of the attribute functions > for your definition of "simple". >

>>> 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! >

> In case of the Triple BP scale I don't see a generator for it.

Are you including the full 39 pitches per tritave?

Message: 608 - Contents - Hide Contents Date: Mon, 30 Jul 2001 22:45:05 Subject: Another BP linear temperament? (was: Magic lattices) From: Dave Keenan --- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:

> Hi Graham, > > <<I've also updated the Bohlen-Pierce entry.>> > > The BP linear temperament was so obvious that I could hardly call it > much of a contribution... it's an observation at best! > > Personally I'm much more satisfied with the 2D and especially the 1/6 > comma generalized meantone contributions, as I think they're more than > simple observations of the obvious. > > The 1/6 comma generalization would be accomplished by using 1/6 of the > 118098/117649 2D comma to make the 177147/117649 in a chain of 9/7s > pure: > > 1468 434 > 1034 868 > 600 1302 > 166 1736 > 1634 268 > 1200 702 > 766 1136 > 332 1570 > 1800 102 > 1366 536 > 932 970 > 498 1404 > > Here's the generalized or remapped 1/6 comma Bohlen-Pierce meantone > rotations: > > 0 268 434 600 868 1034 1302 1468 1736 1902 > 0 166 332 600 766 1034 1200 1468 1634 1902 > 0 166 434 600 868 1034 1302 1468 1736 1902 > 0 268 434 702 868 1136 1302 1570 1736 1902 > 0 166 434 600 868 1034 1302 1468 1634 1902 > 0 268 434 702 868 1136 1302 1468 1736 1902 > 0 166 434 600 868 1034 1200 1468 1634 1902 > 0 268 434 702 868 1034 1302 1468 1736 1902 > 0 166 434 600 766 1034 1200 1468 1634 1902 > > Note the pure 1:2s, 2:3s and the perfect 1:2^(1/2)s. > > (Come to think of it, even the n*n figurate number lattice based on > the 3:5:7 seems like more of a contribution to me...) Dan,

Can you give us the mapping from primes to numbers of generators for this temperament? Also the mapping between primes (or whatever) that takes us from meantone to this temperament? -- Dave Keenan

Message: 611 - Contents - Hide Contents Date: Tue, 31 Jul 2001 15:31 +0 Subject: Re: Another BP linear temperament? From: graham@m... In-Reply-To: <9k4o1h+jqn@e...> Dan Stearns wrote:

>> The BP linear temperament was so obvious that I could hardly call it >> much of a contribution... it's an observation at best!

Well, that's all I'm cataloguing. I've written a little script to go with my temperament finder for this. Your BP mapping is the best in the "7-limit" with a tritave as the interval of equivalence. Here's the printout: 7/30, 279.0 cent generator basis: (1.0, 0.23246075055125442) mapping by period and generator: [(1, 0), (-1, 7), (1, 2), (2, -1)] mapping by steps: [(17, 13), (11, 8), (25, 19), (30, 23)] unison vectors: [[9, 2, -7, 0], [-13, 1, 0, 7]] highest interval width: 8 complexity measure: 8 (9 for smallest MOS) highest error: 0.003704 (4.445 cents) unique The top 10 list should be at <2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 ... * [with cont.] (Wayb.)> and the script at <import temper, string * [with cont.] (Wayb.)>. Graham

Message: 612 - Contents - Hide Contents Date: Tue, 31 Jul 2001 18:37:33 Subject: Re: BP linear temperament From: Paul Erlich --- In tuning-math@y..., <manuel.op.de.coul@e...> wrote:

> > Paul wrote:

>> Are you including the full 39 pitches per tritave? >

> No, I wasn't. But trying that, after filling the gaps I got the scale: > 1/1 65/63 35/33 27/25 55/49 15/13 25/21 11/9 49/39 9/7 65/49 15/11 7/5 > 13/9 49/33 75/49 11/7 21/13 5/3 77/45 135/77 9/5 13/7 21/11 49/25 > 99/49 27/13 15/7 11/5 147/65 7/3 117/49 27/11 63/25 13/5 147/55 25/9 > 99/35 189/65 3/1 > I saw there's no L+S approximation to it that is Myhill. > If you give me the set of unison vectors

You can find the unison vectors by calculating the ratios of (differences between) various pairs of generic seconds, pairs of generic thirds, etc.

> I can try to find some other > versions using the periodicity block shaping of Scala and see if any > of those can be approximated by a MOS, but the chance looks small that > it can be found like this by hand.

Well at least you're not trying to do it entirely by hand!

Message: 613 - Contents - Hide Contents Date: Tue, 31 Jul 2001 18:39:30 Subject: Re: BP linear temperament From: Paul Erlich --- In tuning-math@y..., <manuel.op.de.coul@e...> wrote:

> > > If you give me the set of unison vectors I can try to find some other > versions using the periodicity block shaping of Scala and see if any > of those can be approximated by a MOS, but the chance looks small that > it can be found like this by hand.

P.S. if you temper out all but one of the unison vectors, you're guaranteed to get an MOS, by my Hypothesis . . . right? I'd look for a set of unison vectors that are all super-super-particular (n=d+2), then temper out all but the largest one (the one with the smallest numbers) . . .

Message: 614 - Contents - Hide Contents Date: Tue, 31 Jul 2001 19:47:03 Subject: Re: BP linear temperament From: Paul Erlich --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

> > You can find the unison vectors by calculating the ratios of > (differences between) various pairs of generic seconds, pairs of > generic thirds, etc.

Here are some I found, just by looking at the generic seconds: 243:245 273:275 1323:1331 1617:1625 That should be enough . . . in 4-D, (5,7,11,13) space, with 3 as the interval of equivalence, these UVs become (-1 -2 0 0) (-2 1 -1 1) ( 0 2 -3 0) (-3 2 1 -1) and the determinant of this matrix is -39. Whew! Now what's the generator of the MOS we get by tempering out all of the above UVs, except for, say, 243:245?

Message: 615 - Contents - Hide Contents Date: Wed, 1 Aug 2001 03:10:28 Subject: Re: finding n-limit ratios From: Robert Walker Hi there, I've been working recently on section of FTS that finds close approximations to n-limit ratios You can find it in current beta preview, but it is very slow for large quotients. It does it just by testing all quotients, and working out what the denominator would be for that quotient, then if denom / denum is within the desired tolerance, checks if they are both n-limit. If denom is n-limit, and denum isn't, does an increment / decrement of denum until it finds one that is okay, or until it goes outside the desired tolerance. However, even that fairly crude technique is useful for this work, where small values of denom. and denum. are interesting to find. It's not too bad for max quotient of 1000. I'm interested to extend the range. I've been trying a very simple technique: suppose r = ratio, and suppose you want it 5 limit, then write r~= 2^a*3^b*5^c as log r ~= a log 2 + b log 3 + c log 5 Now set a max quotient to find and tolerance in cents in advance. From that, one can calculate a max possible power for each of the exponents. E.g. b < log(N)/log 3 where N is the max quotient. Now just loop over all the possible values for b and c. Leave a free, since as a power of 2, it will be the largest. For each value of b, c, you can then work out a value for a as a = (log r - b log 3 - c log 5)/ log 2 and since you want an integer, use the floor and ceil of a i.e. nearest integers above and below. This is very fast for 5 limit ratios. E.g. for max quotient of 1000000, you have integer part of log(1000000)/log(5)= 10 and of log(1000000)/log(3)= 12 So, taking negative numbers into account you have 24*16 = 384 loops to do for each ratio. That's tiny, even if you are testing maybe 20 numbers or more, and doing a fair bit of processing in each loop. Even up to max quotient of 1000000000000 works fine. Doubling the number of digits will only double the range for each of the powers, and since there are two involved, that quadruples the time. Add in fourth factor though 2 3 5 7 and you begin to get into seriously large numbers. Each time you double the number of digits you multiply the time needed by 8. That's a siginificant difference. Add a couple more factors and it is getting on for as slow as the method of checking all the quotients in succession, for small quotients (of course it will always be faster for large quotients if you are willing to wait for the answer). I can refine this a little by just doing things more efficiently. However, I wondered if there is any way to go more directly to the solution rather than search all the points of the lattice. I've been puzzling it over but haven't come up with anything yet. I've heard of the LLL and PLSQ algorithms. However, seems to me if you look for an integer relation between (r, log 2, log 3, log 5, log 7) then you are as likely to find a relation with 0 coeff. for r as one involving r and the remaining ones. Also if you find a relation involving r, then the coeff. for r could be greater than 1, and that would be no good either. Am I missing something? I've had a look for free source code for PLSQ but only found gpld code which I can't use in FTS. However, if it is likely to solve this I'll be interested to look more thoroughly, or if nec, try and see if I can code it myself, which should be poss, if I can find a good clear description somewhere of how it works. Any thoughts? Or hints of an idea even that might lead to something? Robert

Message: 616 - Contents - Hide Contents Date: Wed, 1 Aug 2001 03:21:15 Subject: Re: finding n-limit ratios From: Robert Walker Hi there,

> However, seems to me if you look for an integer > relation between (r, log 2, log 3, log 5, log 7)

that of course should be

> relation between (log r, log 2, log 3, log 5, log 7)

so if coeff of log r > 1 you end up with a nice expression of, say, r^3 as an n-limit ratio. Robert

Message: 618 - Contents - Hide Contents Date: Wed, 1 Aug 2001 00:07:52 Subject: Sumerian 12-EDO? (was: [tuning] fractional exponents of prime-factors) From: monz I wrote:

> From: monz <joemonz@y...> > To: <tuning@xxxxxxxxxxx.xxx> > Sent: Wednesday, July 18, 2001 3:09 AM > Subject: [tuning] fractional exponents of prime-factors > > > For a very simple example, if we use vector subtraction to > calculate what note results from tempering the Pythagorean > "perfect 5th" by 1/12 of a Pythagorean comma, we get: > > 2^x 3^y > > |-12/12 12/12| = 2^-1 * 3^1 = 3/2 = "perfect 5th" > - |-19/12 12/12| = ((2^-19)*(3^12))^(1/12) = 1/12 Pythagorean comma > ----------------- > | 7/12 0 | = 2^(7/12) = 12-EDO "5th"

Upon re-reading this, it just struck me that perhaps this was the reasoning which may have led the Sumerians to 12-EDO (if I'm correct that they did "go there"). I'm not suggesting that the Sumerians conceived of numbers in exponential terms, especially not numbers which describe ratios of *frequencies* (well, OK, maybe I am), but if they did, this calculation would look like this in base-60: |-1 1| - |-1,35 1| ------------ | 35 0| So the "cycle of 5ths" in 12-EDO would be: exponent in base-60 2^(6/12) 30 2^(11/12) 55 2^(4/12) 20 2^(9/12) 45 2^(2/12) 10 2^(7/12) 35 2^(0/12) 0 2^(5/12) 25 2^(10/12) 50 2^(3/12) 15 2^(8/12) 40 2^(1/12) 5 2^(6/12) 30 which form an exponent-cycle of +35 (or -25) mod 60. They certainly knew about the so-called "Pythagorean" cycle, and my research leads me to believe that they knew how to calculate 12-EDO. If they realized that the 12-EDO "5th" is tempered by 1/12 of a Pythagorean comma, as is shown by my vector subtraction, then there ought to be some evidence of their reasoning. On firmer ground, let's take a look at string-length calculations. The string-length which produces 2^(7/12), which is quite close to 1772/2655, can be described extremely accurately in base-60 as 40,2,42,42,15. Do any of you know of any "ancient mathematics" groups whose subscribers might know about ancient references to these numbers, if there are any? (either groups or references, that is) love / peace / harmony ... -monz Yahoo! GeoCities * [with cont.] (Wayb.) "All roads lead to n^0" _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 619 - Contents - Hide Contents Date: Wed, 1 Aug 2001 00:31:16 Subject: Re: Sumerian 12-EDO? (was: [tuning] fractional exponents of prime-factors) From: monz

> From: monz <joemonz@y...> > To: <tuning-math@xxxxxxxxxxx.xxx>; <celestial-tuning@xxxxxxxxxxx.xxx> > Sent: Wednesday, August 01, 2001 12:07 AM > Subject: [tuning-math] Sumerian 12-EDO? (was: [tuning] fractional

exponents of prime-factors)

> > > > I wrote: >

>> ... if we use vector subtraction to >> calculate what note results from tempering the Pythagorean >> "perfect 5th" by 1/12 of a Pythagorean comma, we get: >> >> 2^x 3^y >> >> |-12/12 12/12| = 2^-1 * 3^1 = 3/2 = "perfect 5th" >> - |-19/12 12/12| = ((2^-19)*(3^12))^(1/12) = 1/12 Pythagorean comma >> ----------------- >> | 7/12 0 | = 2^(7/12) = 12-EDO "5th" > > >

> Upon re-reading this, it just struck me that perhaps this > was the reasoning which may have led the Sumerians to 12-EDO > (if I'm correct that they did "go there").

A few more quite accurate base-60 string-length ratios which are important in pursuing this idea: Pythagorean comma: 59,11,32,43,11 1/12 Pythagorean comma: 59,55,56,13,9 love / peace / harmony ... -monz Yahoo! GeoCities * [with cont.] (Wayb.) "All roads lead to n^0" _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 622 - Contents - Hide Contents Date: Wed, 01 Aug 2001 20:03:41 Subject: Re: BP linear temperament From: Paul Erlich --- In tuning-math@y..., <manuel.op.de.coul@e...> wrote:

> > Paul wrote: >

>> Here are some I found, just by looking at the generic seconds: >> >> 243:245 >> 273:275 >> 1323:1331 >> 1617:1625 >

> Creating a periodicity block with these unison intervals was an > immediate result! This is the scale: > 1/1 77/75 35/33 27/25 55/49 63/55 25/21 11/9 1701/1375 9/7 33/25 > 15/11 7/5 275/189 81/55 75/49 11/7 441/275 5/3 77/45 135/77 9/5 > 275/147 21/11 49/25 99/49 567/275 15/7 11/5 25/11 7/3 825/343 27/11 > 63/25 55/21 147/55 135/49 99/35 225/77 3/1 > It is 11-limit. The 56th root of 3 quantisation is Myhill and the > generator is 11/7 (782.49 c.) or 21/11 (1119.46 c.). > The least squares best generator for the above scale is 780.8095 > cents, s=39.924 cents (22) and L=60.213 cents (17).

Your least squares optimization uses what mapping from generators to primes? And it considers the errors for intervals 3:5, 3:7, 3:11, 3:13, 5:7, 5:11, 5:13, 7:11, 7:13, 11:13 . . . right?

> So it's very > close to the 95th root of 3. > 1/1 39.9241 100.1373 140.0614 200.2747 240.1987 300.4120 340.3361 > 380.2601 440.4734 480.3975 540.6107 580.5348 640.7481 680.6721 > 740.8854 780.8095 820.7335 880.9468 920.8709 981.0841 1021.0082 > 1081.2215 1121.1455 1161.0696 1221.2829 1261.2069 1321.4202 > 1361.3443 1421.5575 1461.4816 1521.6949 1561.6189 1601.5430 > 1661.7563 1701.6803 1761.8936 1801.8177 1862.0309 3/1 > > Manuel

Message: 623 - Contents - Hide Contents Date: Thu, 02 Aug 2001 01:51:52 Subject: Re: Another BP linear temperament? From: Dave Keenan --- In tuning-math@y..., graham@m... wrote:

> Here's the printout: > > 7/30, 279.0 cent generator

Oops! Your generator should be 442.1 cents. You've multiplied by cents per octave when it should have been cents per tritave. I get 442.6 cents as the 7-limit optimum. I notice you put "7 limit" in scare quotes, and rightly so. I'd call what you've given "7-limit without 2s". By analogy: For an octave repeating scale, "7 limit" means ratios of all (whole number) non-multiples of 2 up to 7, i.e {1,3,5,7} (and their octave equivalents). So for a tritave repeating scale I think "7 limit" should mean ratios of all non-multiples of 3 up to 7, i.e. {1,2,4,5,6,7}. -- Dave Keenan

Message: 624 - Contents - Hide Contents Date: Thu, 02 Aug 2001 02:55:24 Subject: Re: Another BP linear temperament? From: Paul Erlich --- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:

> I get 442.6 cents as the 7-limit optimum. I notice you put "7 limit" > in scare quotes, and rightly so. I'd call what you've given "7-limit > without 2s". > > By analogy: > For an octave repeating scale, "7 limit" means ratios of all (whole > number) non-multiples of 2 up to 7, i.e {1,3,5,7} (and their octave > equivalents). > > So for a tritave repeating scale I think "7 limit" should mean ratios > of all non-multiples of 3 up to 7, i.e. {1,2,4,5,6,7}.

Since the "orthodox" BP consonances are ratios of {3,5,7}, and the triple-BP consonances are ratios of {3,5,7,11,13}, and their tritave-equivalents, and these have been referred to as "7-limit" and "13-limit", respectively, I think this could lead to confusion . . . let's abandon the term "limit" when leaving the octave-equivalent world and adopt more explicit mathematical terminology. (Remember, Pierce's idea for the scale that became known as BP was that if the even partials are absent, the 3:1 would take on the role of the octave, and ratios involving even numbers would no longer be "consonances" . . . whether or not the idea holds up in practice, it's conceptually neat enough that it makes sense to at least honor it on equal footing with other conceptions . . . Graham, or anyone else who can "hear" 3:1 equivalence . . . does it help to eliminate even partials from the timbres?)