Tuning-Math Archive Section 6: 5000

Message: 5000 - Contents - Hide Contents Date: Tue, 11 Jun 2002 19:48:51 Subject: bye From: Carl Lumma I'm singing off of yahoo groups. Tuning-math and harmonic-entropy tomorrow, and tuning when anything I'm involved in has died down. As always, I reserve the right to change my mind at any point. :) I'll keep up the tuning-math list at freelists.org for a time, if anybody's interested in it. If folks want to switch, I'm happy to do the grunt work, or anyone else who'd like to do it is welcome to what I have so far. The list is very configurable, so there's all sorts of things to vote on, though the current config should be at least as good as anything we've had so far. It's possible to subscribe many people in one go, such as everybody on tuning-math here. Maybe Robert Walker knows how to snag archives. Freelists' web interface and archive search seem quite good. As always, feel free to mail me at carl-lumma.org, where the - is to be replaced by an @. In particular, if Gene ever gets interested in harmonic entropy, or if Paul ever runs the validation exercise, if the new notation is released, or the search of planar temperaments turns up any really good 5-10 tone generalized-diatonics... and if any of you create music; I always love to listen, so drop me a note! -Carl

Message: 5001 - Contents - Hide Contents Date: Tue, 11 Jun 2002 13:48 +0 Subject: Re: Help requested From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <ae4erq+a8rm@xxxxxxx.xxx> kalleaho wrote:

> What should I read in the Web and in the Lists to get a good > understanding of the notation and terminology used in tuning-math?

As we haven't written up the processes yet, the best place is still the list archives.

> I understand what linear temperaments are but the notation used is > not self-evident to me. I also have a basic understanding of > periodicity blocks but hmm... wedges? commatic/chromatic unison > vectors?

Wedge products are explained at <http://mathworld.wolfram.com/WedgeProduct.html * [with cont.] >. The importance here is that the wedge product of the commas defining a linear temperament family is the complement of the wedge product of two equal temperaments belonging to the same family. The chromatic unison vector is the one you don't temper out to get a linear temperament. So for 7 note meantone, this is the chromatic semitone 25:24. Graham

Message: 5003 - Contents - Hide Contents Date: Tue, 11 Jun 2002 19:49:22 Subject: A twelve-note, 11-limit scale From: genewardsmith This results from tempering a variety of Fokker blocks using the planar temperament defined by 126/125~176/175~1. I've used the 120-et for the results; since I already called the 108-et the crazy uncle of the family, I don't know where to place 120. Scale in 120-et [0, 8, 23, 31, 39, 50, 62, 70, 81, 89, 101, 112] Interval and triad count 5: 23, 12 7: 36, 36 9: 42, 58 11: 49, 82 Connectivities: 2 5 5 8 Fokker blocks which temper to this scale 1, 21/20, 8/7, 6/5, 5/4, 168/125, 10/7, 3/2, 8/5, 42/25, 25/14, 48/25 1, 21/20, 8/7, 25/21, 5/4, 4/3, 10/7, 3/2, 8/5, 5/3, 25/14, 40/21 1, 25/24, 144/125, 6/5, 5/4, 4/3, 36/25, 3/2, 8/5, 5/3, 9/5, 48/25 1, 21/20, 8/7, 6/5, 5/4, 4/3, 10/7, 3/2, 8/5, 5/3, 9/5, 40/21 1, 22/21, 63/55, 6/5, 5/4, 4/3, 63/44, 3/2, 8/5, 5/3, 9/5, 21/11

Message: 5005 - Contents - Hide Contents Date: Tue, 11 Jun 2002 21:22:41 Subject: Re: A twelve-note, 11-limit scale From: genewardsmith --- In tuning-math@y..., "emotionaljourney22" <paul@s...> wrote:

> i mention it in my paper, it's a pajara temperament.

You mentioned the temperament, or 120-et? I don't see what either has to do with pajara, since 50/49 is 4 120-et steps and 64/63 is 3.

Message: 5006 - Contents - Hide Contents Date: Wed, 12 Jun 2002 03:27:02 Subject: Re: A common notation for JI and ETs From: gdsecor --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote [#4434]:

> --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:

>> --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote: > [GS:]

>> After carefully considering your mirroring proposal, I am making a >> counter-proposal for the determination of apotome complements that >> also eliminates both of these biggest disadvantages of the rational >> complements. This will look familiar, except that it has one added >> clause (to cover wavy flags): >> >> For a symbol consisting of: >> 1) a left flag (or blank) >> 2) a single (or triple) stem, and >> 3) a right flag (or blank): >> 4) convert the single stem to a double (or triple to an X); >> 5) replace the left and right flags with their opposites according to >> the following: >> a) a straight flag is the opposite of a blank (and vice versa); >> b) a convex flag is the opposite of a concave flag (and vice versa); >> c) a wavy flag is its own opposite. >

> Wavy being its own opposite isn't new either. I think I proposed that > back when we were still deciding what the wavy's would mean.

That's correct, except that we decided that )| and (| wouldn't have any complements. In determining the 217-ET notation, each time we avoided )| and didn't use (| for any single-flag symbol smaller than a half-apotome. That enabled us to arrive at a consistent set of complements for 217.

>> This preserves most of the symbol arithmetic without encountering >> either of the two disadvantages you gave for mirrored complements. >> >> >> ... Let me know what you think about this (not-so-new) idea. >

> Now that we've exhaustively (exhaustingly?) considered the > alternatives, I think it looks absolutely brilliant!!!!

(Whew, that's a relief!)

> If you have time, could you repost your latest proposals, including > any whose first half-apotome we've agreed on, using these complements?

For the ones you haven't covered below, I'll do that in a another message. We'll also have to discuss 217 again -- you'll recall that the way we did it was slightly different before we worked on the rational complements, and I think we'll need to get that settled (again) before tackling the other divisions above 100.

> I've been working on what I call horizontal consistency in the first > half-apotome. I believe it is more important than vertical > consistency. Vertical consistency is between ETs where fifth-size is > the same but number of steps per apotome in one is a multiple of the > other, e.g. 48-ET and 96-ET. Horizontal consistency is between ETs > that have the same number of steps per apotome, but have slightly > different fifth sizes. ETs with same steps-per-apotome and adjacent > fifth sizes, always differ by 7, e.g. 41-ET, 48-ET, 55-ET.

With large-numbered ET's, we might also wish to have some commonality of symbol usage even among those with different numbers of steps per apotome. For example, the difference between the 171 and 183 notations will probably involve only the addition of the (|) symbol for the latter. And the symbols for 183 might be a subset of those for 217.

> Here are the proposals that have come from that investigation so far. > I've added complements as per the above. > > To be notated as subsets of larger ETs: > 2,3,4,5,6,7,8,9,10,11,13,14,15,16,18,20,21,23,25,28,30,33,35,40,47. Okay. > 1 step per apotome > 12,19,26: /||\ Agreed. > 2 steps per apotome > 17,24,31,38: /|\ /||\ Agreed. > 45,52: /|) /||\ [13-comma]

Okay, that works.

> 3 steps per apotome > 22,29: /| ||\ /||\

Okay. I don't think that there would be a problem having a difference between the notation for 29 and that for every other degree of 58. The latter gives |\ /|| /||\ (version 1) or |) ||) /||\ (version 2), either of which is easy enough to comprehend, e.g., in a portion of a piece in 58-ET using only every other tone.

> 36: |) ||) /||\ Agreed! > 43,50,57,64: /|) (|\ /||\ [13-commas]

Agreed! (Yikes, this is too easy!)

> 4 steps per apotome > 27: /| /|) ||\ /||\ [13-comma] Okay. > 34,41,(48?): /| /|\ ||\ /||\

For 34 & 41, agreed! In 48, the 5 factor is more than 45 percent of a degree false, so there would not be a strong reason to do 48 this way. While I would prefer this to what might be done for 55 (below), I have yet another preference (following).

> (48?),55: ~|) /|\ ~||( /||\

I think that 48 and 55 have sufficiently different properties that there would be no reason to insist on doing them alike. Since I would do 96 this way: 96: /| |) /|) /|\ (|\ ||) ||\ /||\ I wouldn't see any problem with doing 48 as a subset of 96, particularly since 7 and 11 are among the best factors represented in 48: 48: |) /|\ ||) /||\ Now 55 is a real problem, because nothing is really very good for 1deg. The only single flags that will work are |( (17'-17) or (| (as the 29 comma), and the only primes that are 1,3,5,n-consistent are 17, 23, and 29. If I wanted to minimize the number of flags, I could do it by introducing only one new flag: 55: ~|\ /|\ ~|| /||\ so that 1deg55 is represented by the larger version of the 23' comma symbol. Or doing it another way would introduce only two new flags: 55: ~|~ /|\ ~||~ /||\ The latter has for 1deg the 17+23 symbol, and its actual size (~25.3 cents) is fairly close to 1deg55 (~21.8 cents). Besides, the symbols are very easy to remember. So this would be my choice. What was your reason for choosing ~|)?

> 62: |) /|\ (|\ /||\ [13-commas]

Considering that 7 is so well represented in this division, I would hesitate to use |) in the notation if it isn't being used as the 7 comma. In fact, I don't think I would want to use |) for a symbol unless it *did* represent the 7 comma (lest the notation be misleading), although I would allow its use it in combination with other flags. So I would prefer this: 62: /|) /|\ (|\ /||\ [13-commas]

> 69,76: |) ?? (|\ /||\ [13-comma]

Again, I wouldn't use |) by itself defined as a 13-comma symbol, but would choose /|) instead: 69,76: /|) )|\ (|\ /||\ [13-comma] For 2deg of either 69 or 76, )|\ is about the right size.

> 5 steps per apotome > 32: )| /|\ (|) (||\ /||\

Very good! The 19 comma is small, but its usage is quite vailid, considering how accurately 19 is represented. (This is one division I hadn't looked at before.)

> 39,46,53: /| /|\ (|) ||\ /||\ Agreed! > 60: /| |) ||) ||\ /||\

I notice that 13 is much better represented than 7, so I would prefer this (in which the JI symbols also more closely approximate the ET intervals): 60: /| /|) (|\ ||\ /||\

> 67,74: ~|) /|) (|\ ~||( /||\

I'm certainly in agreement with the 2deg and 3deg symbols, and if you must do both ET's alike, then what you have for 1deg would be the only choice (apart from (| as the 29 comma). We both previously chose )|) for 1deg74 (see message #4412), presumably because it's the smallest symbol that will work, and I chose |( for 1deg67 (in #4346), which would give this: 67: |( /|) (|\ /||) /||\ 74: )|) /|) (|\ (||( /||\ So what do you prefer?

> 81,88: )|) /|) (|\ (||( /||\ [13-commas]

This is exactly what I have for 74, above. Should we do 67 as I did it above and do 74, 81, and 88 alike? On the other hand, why wouldn't 88 be done as a subset of 176? It is with some surprise that I find that |( is 1deg in both 67 and 81, so 81 could also be done the same way as I have for 67, above.

> 6 steps per apotome > 37,44,51: )| /| /|) ||\ (||\ /||\ [13-commas] > or > 37,44,51: |) )|) /|) (||( ||) /||\ [13-commas]

For 51 I had something a bit simpler (using lower primes): 51: |) /| /|) ||\ ||) /||\

> 58: /| |\ /|\ /|| ||\ /||\ > or > 58: /| |) /|\ ||) ||\ /||\ [13-comma]

I think I would avoid your version2 -- this is another instance where it's too easy to be misled into thinking that |) is the 7 comma. If we wanted to avoid the confusability of all straight flags, we could try: 58: /| /|) /|\ (|\ ||\ /||\ Here |) would be kept as the 7 comma and (| would be the 11'-7 comma of 2deg58. However, I think that it would be too easy to forget that /|) and (|\ aren't representing ratios of 13. So I think that the safest choice is version 1 -- all straight flags.

> 65,72,79: /| |) /|\ ||) ||\ /||\

Agreed! (After all we went through before about 72, this one is now almost a no-brainer!)

> 86,93,100: )|) |) )|\ (|\ (||( /||\ [13-commas] > or > 86,100: )|( |) )|\ (|\ (||) /||\ [13-commas] > 93: |( |) )|\ (|\ /||) /||\ [13-commas]

I would do 93-ET and 100-ET as subsets of 186-ET and 200-ET, respectively. For 86, I wouldn't use |) by itself as anything other than the 7 comma, as explained above, but would use convex flags for symbols that are actual ratios of 13. So this is how I would do it: 86: ~|~ /|) (|~ (|\ ~||~ /||\ [13-commas and 23-comma] The two best primes are 13 and 23, so there is some basis for defining |~ as the 23 flag. In any event, I believe that (|~ can be a strong candidate for half an apotome if neither /|\ nor /|) nor (|\ can be used. --George

Message: 5007 - Contents - Hide Contents Date: Wed, 12 Jun 2002 22:39:34 Subject: Two 9-note scales in the temperamentt From: Gene W Smith Commas {49/48, 21/20, 99/98, 121/120} Block [1, 12/11, 7/6, 14/11, 4/3, 3/2, 11/7, 12/7, 11/6] 22-et version [0, 3, 5, 8, 9, 13, 14, 17, 19] 5: 8, 0 7: 19, 11 9: 29, 41 11: 32, 56 31-et version [0, 4, 7, 11, 13, 18, 20, 24, 27] 5: 5, 0 7: 15, 6 9: 22, 17 11: 32, 56 53-et version [0, 7, 12, 19, 22, 31, 34, 41, 46] 5: 5, 0 7: 15, 6 9: 22, 17 11: 32, 56 Commas {128/125, 36/35, 99/98, 121/120} Block [1, 35/33, 33/28, 5/4, 175/132, 264/175, 8/5, 56/33, 66/35] 22-et version [0, 2, 5, 7, 9, 13, 15, 17, 20] 5: 12, 4 7: 22, 17 9: 27, 32 11: 30, 45 31-et version [0, 3, 7, 10, 13, 18, 21, 24, 28] 5: 12, 4 7: 21, 14 9: 25, 26 11: 30, 45 53-et version [0, 5, 12, 17, 22, 31, 36, 41, 48] 5: 12, 4 7: 21, 14 9: 25, 26 11: 30, 45

Message: 5010 - Contents - Hide Contents Date: Thu, 13 Jun 2002 05:03:16 Subject: Re: A twelve-note, 11-limit scale From: genewardsmith --- In tuning-math@y..., "emotionaljourney22" <paul@s...> wrote:

> that depends on your mapping! if you use pajara with a 710-cent > generator, you're in 120-equal!

In this case, the mapping was defined by the fact that it had to temper out 126/125 and 176/175.

Message: 5011 - Contents - Hide Contents Date: Thu, 13 Jun 2002 15:28:26 Subject: A 9-note scale in the planar temperamentt From: Gene W Smith I looked at a number of these, and this was the best I found: Block [1, 11/10, 8/7, 5/4, 11/8, 16/11, 8/5, 7/4, 20/11] 22-et version [0,3,4,7,10,12,15,18,19] 3: 2 0 5: 11 4 7: 19 12 9: 29 39 11: 33 63 11-limit connectivity 7 31-et version [0,4,6,10,14,17,21,25,27] 3: 2 0 5: 11 4 7: 19 12 9: 22 15 11: 32 58 11-limit connectivity 6 46-et version [0,6,9,15,21,25,31,37,40] 3: 2 0 5: 11 4 7: 19 12 9: 19 12 11: 32 58 11-limit connectivity 6

Message: 5012 - Contents - Hide Contents Date: Thu, 13 Jun 2002 19:27:49 Subject: Re: A common notation for JI and ETs From: gdsecor --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote [#4405]:

> > I am instead inclined to totally ignore rational complements with regard to > ETs, especially for the lower numbered ones. One reason is that I feel that > the choice of double-shaft symbols cannot in any way be allowed to > influence the choice of single-shaft. One must first choose the

best set of

> single shaft symbols (ignoring complements) since some users will have no > interest in the double-shaft symbols and should not be penalised for it. > > In fact, (and I've been making gentle noises about this possibility for a > some time now), I'm willing to throw away everything we agonised over with > regard to rational complements and instead adopt a simple system that > applies automatically to all ETs and rational tunings.

I am replying to an earlier message with a different proposal, now that I have had the experience of actually trying to do a considerable number of ET's (both small and large) following both your mirroring proposal and my counter-proposal. This experience can be summarized in the following observations: 1) I find that all of the ET's below 100 on which we agree (per message #4443) can be done, without exception, in exactly the same way using the rational complementation scheme that we just abandoned. 2) The ET's on which we did not agree fall into two categories: a) Those that are fairly simple to do (but have more than one possible way), on which we have not yet formally agreed; all of these can also be done in exactly the same way using the rational complementation scheme. b) Those that are more difficult or not so obvious; these are the less-known, little-used, and just-plain-weird ET's, for which the choice of symbols amounts to "we do the best we can." 3) I tried some of the easier ET's above 100 and found that at least half of them ended up with unavoidable inconsistencies in symbol arithmetic, and those that had a matching sequence of symbols in the half-apotomes were a rarity. I consider this a rather high price to pay for an easy-to-remember complementation scheme. 4) When I previously did these ET's above 100 within the rational complementation scheme, I was able to do all of these with completely consistent symbol arithmetic and most of them with either a matching sequence of symbols or rational complementation -- and sometimes both. 5) Our chief problem with the rational complements is that they are not very easy to remember. However, when you consider that this statement applies *only* to commas above the 13 limit, I don't think that this is a major drawback. There are only 8 pairs of rational complements to remember, and nearly half of them can be formulated into rules (represented symbolically as): [ / <=> b , | <=> || , \ <=> b ] /| <=> ||\ |\ <=> /|| nat. <=> /||\ [ b <=> b , | <=> || , ) <=> ) ] |) <=> ||) [ ( <=> ) , | <=> || , ~ <=> b ] )| <=> (||~ )|~ <=> (|| (| <=> )||~ (|~ <=> )| I would therefore recommend going back to the rational complementation system and doing the ET's that way as well. Or, if you like, we could do them both ways and then decide. I would be agreeable to doing all of the ET's (with the rational complementation scheme) using the symbols that we agreed on in message #4443. --George

Message: 5015 - Contents - Hide Contents Date: Fri, 14 Jun 2002 18:47:34 Subject: Re: Finding linear temperaments From: Gene W Smith Of course, people studying the Riemann Zeta function may in effect have used computers to find ets before anyone, without knowing it. On the last page of Titchmarsh, "The Theory of the Riemann Zeta-Function" (Oxford, 1951) he mentions the 140-et without knowing it, and one of the first things to get worked over when computers came along was Zeta(s).

Message: 5016 - Contents - Hide Contents Date: Sat, 15 Jun 2002 19:32:03 Subject: Re: figurate number expansions as scales From: genewardsmith --- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:

> Some interesting expansions and scales can be derived from figurate > numbers.

Numbers of the form n/(n-1) where n is figurate show up a lot; you could look at my discussion of "jacks", for instance. The fact that triangle and square demomenators lead to other triangle and square denomenators allows us to create series of scales.

Message: 5018 - Contents - Hide Contents Date: Sat, 15 Jun 2002 11:30:02 Subject: Re: A common notation for JI and ETs From: dkeenanuqnetau --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:

> I would therefore recommend going back to the rational > complementation system and doing the ET's that way as well.

Agreed. Provided we _always_ use rational complements, whether this results in matching half-apotomes or not.

> Or, if you like, we could do them both ways and then decide. No need. > I would be agreeable to doing all of the ET's (with the rational > complementation scheme) using the symbols that we agreed on in > message #4443. OK.

I will respond to your suggestions for the remaining ones of 6 or less steps per apotome when I get more time. Then move on to 7 steps per apotome 42,49,56,63,70,77,84,91,98,105 8 steps per apotome 54,61,68,75,82,89,96,103,110,117 9 steps per apotome 59,66,73,80,87,94,101,108,115,122,129 10 steps per apotome 71,78,85,92,99,106,113,120,127,134,141 etc. I think we can do some with 23 steps per apotome, maybe even 25.

Message: 5021 - Contents - Hide Contents Date: Mon, 17 Jun 2002 03:12:55 Subject: Seven and eleven limit comma lists From: genewardsmith Here are comma lists for the 7 and 11 limits. Each comma is less than fifty cents, and each has the property that if the comma is p/q>1 in reduced form, then ln(p-q)/ln(q) < .5 in the 7-limit, and < .3 in the 11-limit. I've found this weaking of the superparticularity condition useful in the past, and it occurred to me it would be one way of getting a finite list of temperaments a la Dave--we could simply require it to have a basis of commas passing such a condition. The lists below may be complete; at least, I haven't been able to add to them. Seven limit list, ln(p-q)/ln(q)<1/2, cents < 50 [1029/1000, 250/243, 36/35, 525/512, 128/125, 49/48, 50/49, 3125/3072, 686/675, 64/63, 875/864, 81/80, 3125/3087, 2430/2401, 2048/2025, 245/243, 126/125, 4000/3969, 1728/1715, 1029/1024, 15625/15552, 225/224, 19683/19600, 16875/16807, 10976/10935, 3136/3125, 5120/5103, 6144/6125, 65625/65536, 32805/32768, 703125/702464, 420175/419904, 2401/2400, 4375/4374, 250047/250000, 78125000/78121827] Eleven limit list ln(p-q)/ln(q) < .3, cents < 50 [36/35, 77/75, 128/125, 45/44, 49/48, 50/49, 55/54, 56/55, 64/63, 81/80, 245/242, 99/98, 100/99, 121/120, 245/243, 126/125, 1331/1323, 176/175, 896/891, 1029/1024, 225/224, 243/242, 3136/3125, 385/384, 441/440, 1375/1372, 6250/6237, 540/539, 4000/3993, 5632/5625, 43923/43904, 2401/2400, 3025/3024, 4375/4374, 9801/9800, 151263/151250, 3294225/3294172]