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Message: 4852 - Contents - Hide Contents Date: Fri, 17 May 2002 00:34 +0 Subject: Re: definitions of period, equivalence, etc. (was: Re: graham's line From: graham@xxxxxxxxxx.xx.xx emotionaljourney22 wrote:> i think the burden of proof rests on the other side. we all agree > that torsion is something that needs to be handled separately, and so > far the only method i've seen for detecting torsion uses IoE-specific > representations!Torsion can be detected with IoEE representations. The problem, if there is one, would be distinguishing torsion that tells you to divide the octave from torsion that tells you the unison vectors aren't in their simplest terms.. If you can't detect the difference, it isn't a problem. If you can detect it, you can deal with it. Why assume otherwise?>> I'm sure the torsion problem will go away when somebody looks at it >> properly. >> again, the burden would seem to "somebody" to show that. > you're "sure", i'm "sure not".Perhaps you could start by showing what the problem is.>> It doesn't matter anyway if we're only considering linear >> temperaments, because they don't have to be derived from unison >> vectors. >> they have to be derivable from unison vectors. the construction > starts with the just lattice and then establishes equivalencies. the > unison vectors simply tell you what the equivalencies were in the > original just lattice.Say what? We've agreed on a definition of linear temperament that says nothing about unison vectors. That definition is directly applicable to the octave-equivalent case. How they're derived is secondary. The unison vectors tell you more than the equivalences. They also tell you what's considered to be a unison. For example, in twintone (or whatever it's being called today) 50:49 is a unison vector. In octave equivalent terms, that is (-2 2). So it follows that half the unison, (-1 1), will also be an equivalence. That's 7:5. As 7:5 isn't half of 50:49, but much closer to half of an octave, we know the period can't be the octave. No problem at all. You don't need to consider the octave-specific case. Give a counter example. Graham

Message: 4853 - Contents - Hide Contents Date: Fri, 17 May 2002 20:26:03 Subject: Re: A common notation for JI and ETs From: gdsecor --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote:>> --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote: >> Here's what I did [for 152-ET] a couple of weeks back, and after looking > at the>> rational complements, I would still do it this way: ... >> ... I've (re-)realised that there is no need to go beyond the 19- prime-limit > for notating any ETs (that we _can_ notate). So, for notating ETs, all the > symbols must be given their 19-limit definitions. ... > > For notating ETs: > v| is always 512:513 > |v is always 288:289 > w| is always 2176:2187 > |w is always 722:729 > s| is always 80:81 > |s is always 54:55 > |x is always 63:64 > > but > > x| can be either 45056:45927 or 715:729 > > In some ETs these will be the same number of steps and the choice doesn't > need to be made. But when the choice _is_ made, the meaning of (|( (|~ (|\ > and (|) all follow automatically from it. > > I now believe that the rational complements beyond the 19-limit (i.e. if > either of the pair is outside the 19-limit in the rational conception) are > very unimportant for notating ETs, and would only be used as a tie- breaker > if all else fails. ... > > So here's the full set for 152-ET. > > 1 )| > 2 |~ > 3 /| > 4 |\ > 5 /|~ > 6 /|) > 7 /|\ > 8 (|) > 9 (|\ or )|| ? > 10 ||~ > 11 /|| > 12 ||\ > 13 /||~ > 14 /||) > 15 /||\ > > We now only disagree on the 1 step symbol. ... > > I don't see how changing 1 step from )| to |( improves ease of remembering. > In fact with )| we have that property that you admired in 217-ET, that the > flags alternate sides as you go up.I'll give my conditional assent to what you have. I want to see how all of this holds up once we do a lot more ET's. I'm going to have to take some time (a week or so) away from the list to catch up on some other things, but you can work on this, and I'll see what you have when I get back. --George

Message: 4855 - Contents - Hide Contents Date: Fri, 17 May 2002 04:50:48 Subject: Re: A common notation for JI and ETs From: genewardsmith --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:> The problem is not the fault of the notation so much as the weirdness > of the division -- I hesitate to call it a tonal system. Any > systematic notation is going to have problems with 74-ET.Sharps and flats are a systematic notation, and since 74 is a meantone system, they would suffice.

Message: 4856 - Contents - Hide Contents Date: Fri, 17 May 2002 10:48 +0 Subject: Re: definitions of period, equivalence, etc. (was: Re: graham's line From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <ac1g4r+m2vm@xxxxxxx.xxx> emotionaljourney22 wrote:> take the classic example of torsion. the fokker determinant is 24. it > seems like you're getting a system with 24 distinct pitch classes per > octave -- but you aren't. you need the octave-specific > representations to find that true number is 12. right?That's no problem at all. The temperament mapping comes out correct, usually as schismic. When you look at the 24 pitch classes you'll see they aren't distinct. Me:>> Say what? We've agreed on a definition of linear temperament that > says>> nothing about unison vectors. That definition is directly > applicable to>> the octave-equivalent case. How they're derived is secondary. Paul:> i'm referring to the definition of "temperament" that we used to > convince dave keenan that contorsion cases aren't kinds of > temperament. this would seem to be an important part of the > definition.It doesn't seem like that at all to me. You don't need to look at unison vectors to recognize contorsion. Only take the gcd of the generator mapping. Unison vectors don't do it anyway -- you have to make a deliberate attempt to remove torsion. You could equate it with contorsion instead. If things with contorsion aren't temperaments, it's because they aren't consonance connected.> 7:5 an equivalence? no way, dude. never, nunca, jamas.It's equivalent to the period in twintone. If that isn't an "equivalence" then unison vectors don't show equivalences, or we have to take torsion at face value. Graham

Message: 4857 - Contents - Hide Contents Date: Fri, 17 May 2002 10:48 +0 Subject: Re: definitions of period, equivalence, etc. (was: Re: graham's line From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <ac1mbm+b027@xxxxxxx.xxx> emotionaljourney22 wrote:> that wasn't an example of torsion, graham, but anyway . . .Torsion's easy do deal with and divisions of the octave are easy to deal with. It's only telling which is which that may be a problem. If it is, it's more with using unison vectors to define temperaments than with temperaments themselves. I still need an example that causes problems. Don't you have an algorithm for generating periodicity blocks?> there is a difference in character between an algorithm that has to > make a "much closer to" judgment, and an algorithm like gene's which > gives the answer directly. in the former, you have to stop the > mechanism, fiddle around with stuff, and get your hands dirty. in the > latter, the whole thing is sleek and axiomatic.Oh, it's Gene's algorithm now, is it? It's part of the definition of a unison vector that it should be close to a unison. I thought Gene had made a strict definition for that. If we can't differentiate a unison vector from a period-equivalence, torsion can't be differentiated for divisions of the octave. So it's GIGO. The algorithm for getting the period part of the mapping is already fairly dirty. I'm certainly assuming it won't get cleaner by ignoring the octave, which is why I don't do it that way. For the octave-equivalent case it'd probably mean trying all consonances and getting them pitch-ordered, or generating a periodicity block or something like that. In which case torsion should fall out the same way it did with the octave-specific algebraic approach. Or you could always construct the period mapping algebraically, but not *say* that it's an octave-specific process. There aren't any "much closer to" judgements anyway. Only higher/lower. If a unison vector gets bigger when you divide it, you need to divide the octave as well. Graham

Message: 4858 - Contents - Hide Contents Date: Sat, 18 May 2002 06:24:56 Subject: definitions of period, equivalence, etc. (was: Re: graham's line From: genewardsmith --- In tuning-math@y..., "emotionaljourney22" <paul@s...> wrote:> gene, any reason you're staying out of this?I'm not sure what the point of it all is. If you leave off octaves and just deal with pitch classes, you need to put the octave information back into the mix in one way or another. Why not just leave it in?

Message: 4859 - Contents - Hide Contents Date: Sat, 18 May 2002 06:35:05 Subject: definitions of period, equivalence, etc. (was: Re: graham's line From: genewardsmith --- In tuning-math@y..., "emotionaljourney22" <paul@s...> wrote:>> It's part of the definition of a unison vector that it should be > close to>> a unison. I thought Gene had made a strict definition for that. > gene?> my brain is fried.I don't recall trying to introduce something about closeness to unity. The most obvious approach is to say a unison vector is one element of a minimal generating set for the kernel, which has no such requirement.

Message: 4860 - Contents - Hide Contents Date: Sat, 18 May 2002 04:12:25 Subject: Re: A common notation for JI and ETs From: David C Keenan>--- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:>> How about "the 13-schisma" or the "tridecimal schisma". >>That sounds good. We should probably propose that term on the main >tuning list, to see if anyone knows whether it has already been used >for a different schisma.Go ahead. I'm sure enough that it hasn't, that I can't be bothered.>Since the (| flag is undoubtedly going to be used so much more often >in connection with ratios of 11 and 13 -- as (|) and (|\ -- than for >ratios of 29, I would prefer to keep its standard definition as other >than 256:261 (the 29 comma). I would also prefer the 13'-(11-5) >ratio to (11'-7) because, > >1) The numbers in the ratio are smaller (715:729 vs. 45056:45927); and > >2) The 13'-(11-5) comma (33.571 cents) is much closer in size to the >29 comma (33.487 cents) than is the (11'-7) comma (33.148 cents).I say we can totally forget the 29 comma definition of (| for notating ETs. But I think we need to decide, for every ET individually, whether x| is defined as 13'-(11-5) or (11'-7) (or both, when they are the same number of steps).>>> I suggest that 37-ET be notated as a subset of 111-ET, with the >>> latter having a symbol sequence as follows: >>>> Yes. That's also what I suggested in a later message (4188). >>>>> 111: w|, s|, |s, w|s, s|s, x|s, w||, s||, ||s, w||s, s||s. >>>> And that's almost the notation I proposed in the same message (with >> its implied complements), except that I would use x|x (|) as the >> complement of s|s /|\. Surely that is what you would want too, >since>> it represents a lower prime and is the rational complement? >>I used x|s (|\ as 6deg111 because x|x (|) calculates to 5deg111 and, >in addition, 26:27 is closer in size to 6deg111 than is 704:729. >However, if we think that there should be no problem in redefining >x|x as 6deg111 (as it would seem to make more sense), then so be it!(|\ is only 6deg111 if you define (| as 13'-(11-5), in which case you should probably also use /|) for 5deg111 instead of /|\. In this case /|) is defined as the 13 comma, not 5+7 comma. This is something else that we need to define on an ET by ET basis, whether |) is the 7 comma or the 13-5 comma. If we favout 7 over (13-5) in 111-ET then we probably shouldn't use any commas involving 13, and should therefore define (| as (11'-7). In this case we have /|\ for 5deg111 and (|) for 6deg111.>>> However, a more difficult problem is posed by 74-ET, and the idea >of>>> having redefinable symbols may be the only way to handle >situations>>> such as this. Should we do that, then there should probably be >>> standard (i.e., default) ratios for the flags, and the specific >>> conditions under which redefined ratios are to be used should be >>> identified. >>>> I think 74-ET is garbage. >>Be careful when you say something like that around here -- do you >remember my "tuning scavengers" postings?Yes, I remember. That's why I said it. So I'd get corrected as quickly as possible if it _wasn't_ garbage. :-) It isn't. See the topic "74-EDO challenge" on the main tuning list.>The problem is not the fault of the notation so much as the weirdness >of the division -- I hesitate to call it a tonal system. Any >systematic notation is going to have problems with 74-ET.Here's my proposal for notating 74-ET using its native fifth (since it's a meantone), despite the 1,3,9 inconsistency. Steps Symbol Comma ---------------------- 1 )|) 19+7 2 )|\ 19+(11-5) 3 /|\ 11 4 )||\ 5 /||\ The )| flag actually has a value of -1 steps, but it never occurs alone, so it doesn't really matter.>Would you also now prefer my selection of the /|) symbol for [6deg152] >to your choice of (|~ on the grounds that it is a more commonly used >symbol, particularly in view of the probability that you might want >to use (|\ instead of )|| or ||( for 9deg as its complement?Yes, but not on those grounds.>>> One thing that I thought should be taken into consideration is >that,>>> where appropriate, ET's that are subsets of others should make >use of>>> a subset of symbols of the larger ET. This would especially be >>> advisable for ET's under 100 that are multiples of 12 -- if you >learn>>> 48-ET, you have already learned half of 96-ET. >>>> Certainly. It's only the question of how we tell "when appropriate" >> that remains to be agreed. I've proposed two and only two reasons >in>> message 4188. You might say what you think of these. >>They sound reasonable enough. Until I thought of 7-ET, which seems >to be a "natural" for the 7 naturals. Of course, a simple way around >that is to put the modifying symbols from 56-ET into a key signature, >a solution that would keep the manuscript clean and make everybody >happy.A brilliant solution. It's a pity the same thing won't work for 37-ET as a subset of 111-ET (or will it?) because I know that some folks will prefer to notate it based on its native best fifth. The case of 74-ET has shown me that my requirement of not using the native fifth if it is 1,3,9-inconsistent, unless we don't use any flags for any prime greater than 9, may need to be relaxed in some cases.>>> I previously did symbol sets for about 20 different ET's, but >that>>> was before the latest rational complements were determined, so >I'll>>> have to review all of those to see what I would now do >differently. >>Here's what I did a couple of weeks ago for some of the ET's (in >order of increasing complexity): > >12, 19, 26: s||sAgreed. My 19 and 26 were wrong.>17, 24, 31: s|s s||s17 and 24 agreed. I guess you want (|) for 1deg31 because it is closer in cents than |), but I feel folks are more interested in its approximations of 7, than 11.>22: s| ||s s||sI agree, but how come you didn't want s|s for 1deg22? It's also arguable that it could be s| s|| s||s, making the second half-apotome follow the same pattern of flags as the first, but what you've got makes more sense to me.>36, 43: |x ||x s||sAgreed for 36. But I wanted a single-shaft symbol for 2deg43 so it is possible to notate it with monotonic letter names and without double-shaft symbols when using a notation that combines standard sharp and flat symbols with sagittals. One could use either /|\ or (|\. e.g I want to be able to notate the steps between B and C as B|), B/|\ or B(|\, C/|\ or C(|\, C!).>29: w|x w||v s||sWhy wouldn't you use the same notation as for 22-ET? There's no need to bring in primes higher than 5.>50: w|w x|s s||sFor 50-ET, {1, 3, 5, 7, 9, 13, 15, 17, 19} is the maximal consistent (19-limit) set containing 1,3,9. So I like x|s for 2 steps (as 13'), and if it's OK here, why not also in 43-ET? But w|w as 17+(19'-19) is actually -1 steps of 50-ET. The only options for 1deg50, that don't involve 11 are )|) as 19+7 or ~|) as 7+17 and /|) as 13. /|) seems the obvious choice to me.>34, 41: s| s|s ||s s||s Agreed. >27: s| x|s ||s s||sWhy do you prefer (|\ to /|)?>48: |x s|s ||x s||sIn 48-ET, {1, 3, 7, 9, 11} has only slightly lower errors than {1, 3, 5, 9, 11}, 10 cents versus 11 cents. Why prefer the above to the lower prime scheme 48: /| /|s ||\ /||\ ?>46, 53: s| s|s x|x ||s s||s Agreed. >58, 72: s| |s s|s s|| ||s s||s (version 1 -- simpler, but more >confusability) >72: s| |x s|s ||x ||s s||s (version 2 -- more complicated, but >less confusability)Of course, I prefer version 2 for 72-ET, since I started the whole confusability thing. It isn't significantly more complicated.>58: s| w|x s|s w||v ||s s||s (version 2 -- more complicated, >but less confusability)I'm inclined to go with version 1 despite the increased lateral confusability, rather than introduce 17-flags. Version 2 is a _lot_ more complicated.>96: s| |x |s s|s s|| ||x ||s s||s (version 1 -- simpler, but >more confusability) >96: s| |x w|s s|s w|| ||x ||s s||s (version 2 -- more >complicated, but less confusability)The only maximal 1,3,9-consistent 19-limit set for 96-ET is {1, 3, 5, 9, 11, 13, 15, 17}. It is not 1,3,7-consistent so the |) flag should be defined as the 13-5 comma (64:65) if it's used at all. The 17 and 19 commas vanish, so we should avoid )| |( ~| and |~. So I end up with 96: /| |) /|) /|\ /|| ||) /||) /||\ Simple _and_ non confusable.>94: w| s| w|s s|s x|x w|| ||s w||s s||sWhy do you prefer that to>94: ~| /| |) /|\ (|) ~|| ||\ ||) /||\Surely we're more interested in the 7-comma than the 17+(11-5) comma. Also, it makes sense that /| + ||\ = /||\, but it makes the second half apotome have a different sequence of flags to the first. Which should we use, /|| or ||\ ?>111 (37 as subset): w| s| |s w|s s|s x|s w|| s|| ||s w||s >s||sDealt with above. I'd prefer (|) for 6deg111.>140: |v |w s| |s s|w s|x s|s x|s ||w s|| ||s s||w s||x >s||s >152: |v |w s| |s s|w s|x s|s x|x x|s ||w s|| ||s s||w >s||x s||sDealt with elsewhere. I see no reason to use |( which is really zero steps, when )| is 1 step.>171: |v w|v s| |x |s w|s s|x s|s x|s w||v s|| ||x ||s >w||s s||x s||sWhy not ~| for 1 step?>183: |v w|v s| |x |s w|s s|x s|s x|x x|s w||v s|| ||x >||s w||s s||x s||sWhy not use w| for 1deg183, being a simpler comma than |v? 17 vs. 17'-17.>181: |v w| w|v s| |s w|x w|s s|x s|s x|x w|| w||v s|| >||s w||x w||s s||x s||sI don't see how |) can be 5deg181 or how /|\ can be 9deg181. The only symbol that can give 9deg181 with 19-limit commas is (|~. Here's my proposal. 181: |( ~| |~ /| /|( (| (|( /|) (|~ (|\ ~|| ||~ /|| /||( (|| (||( /||) /||\>217: |v w| |w s| |x |s w|x w|s s|x s|s x|x x|s w|| ||w >s|| ||x ||s w||x w||s s||x s||s Agreed.-- Dave Keenan Brisbane, Australia Dave Keenan's Home Page * [with cont.] (Wayb.)

Message: 4861 - Contents - Hide Contents Date: Sun, 19 May 2002 09:08:53 Subject: Notation for n*12-ETs (was: A common notation for JI and ETs) From: David C Keenan>--- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:>>> One thing that I thought should be taken into consideration is >that,>>> where appropriate, ET's that are subsets of others should make >use of>>> a subset of symbols of the larger ET. This would especially be >>> advisable for ET's under 100 that are multiples of 12 -- if you >learn>>> 48-ET, you have already learned half of 96-ET.Below, I notate all the most important multiples of 12-ET consistently with each other, using symbols that correspond to 17-limit commas as described previously in the "A common notation for JI and ETs" thread. The real symbols that correspond to the ASCII versions below (and a lot more), can be seen in Yahoo groups: /tuning-math/files/secor/notatio... * [with cont.] But first some observations about apotome complements. There is often a conflict between wanting to (a) make ||\ the complement of /| or make /|| the complement of |\, and (b) make the second half-apotome follow the same pattern of flags as the first. There is another problem with making the second half apotomes agree between n*12-ET and every second step of 2n*12-ET, even when there's perfect agreement in the first half apotome. This problem occurs between 36 and 72, but 24, 48 and 96 are ok. In general, complement symbols are a pain in the posterior, and I'll leave it for you to wrestle with them. I'm starting to think that the only way to make them work is to make the second half-apotome the mirror image of the first, (with the addition of a second shaft to each symbol). So here's the first half-apotome of the most important n*12-ETs notated in such a way that the same number of cents always has the same symbol. |) is always the 7 comma and (| is always the 11'-7 comma. In all but 60, 72 and 132 they are also the 13-5 and 13'-(11-5) commas respectively. 1 2 3 4 5 Cents -> 012345678901234567890123456789012345678901234567890 12 (6 4 3 2): 24 (8): /|\ 36 (18): |) 48 (16): ~|) /|\ 60 (15, 20, 30): /| |\ 72: /| |) /|\ 84 (42): /| |) /|) 96 (32): /| ~|) |\ /|\ 108 (54): /| //| |) /|) 132 (44 66): ~|( (| |) |\ /|) Cents -> 012345678901234567890123456789012345678901234567890 1 2 3 4 5 If you really don't want //| as 2deg108 (and hence 1deg54) then you could use /|( which isn't correct for 108-ET, but is correct for every 2nd degree of 216-ET. //| is correct for both. 120-ET is omitted because it does not serve to notate any lower ETs (as shown in parenthesis for those above). Also, it is not possible to notate it consistently with 60-ET while respecting the same comma values for the symbols. The 20 cent accidental is the problem. In 120-ET it must be (|, which is zero steps in 60-ET. In 60-ET it must be /| which is 1 step (10 cents) in 120-ET. Regards, -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page * [with cont.] (Wayb.)

Message: 4862 - Contents - Hide Contents Date: Sun, 19 May 2002 23:45:35 Subject: Temperaments with a 7/5 generator From: Gene W Smith Linear temperaments with a generator which is itself a consonant interval seem to me of particular interest, so I thought I would explore what is out there for generators of about 7/5. While nothing dramatic turned up, these systems might be of interest. [3, -5, -6, -1, -15, -18, -12, 0, 15, 18] <56/55, 64/63, 77/75> badness = 326 rms = 13.78 [3, -5, -6, 0, 18, -15] <64/63, 392/375> badness = 532 rms = 14.78 7/15 < 15/32 < 8/17 15/32 is a nearly exact 11-limit generator; 11/8 is closer than 7/5 to this generator, which is convenient. [3, 12, 11, -1, 12, 9, -12, -8, -44, -41] <56/55, 81/80, 540/539> badness = 404 rms = 12.62 [3, 12, 11, -8, -9, 12] <81/80, 686/675> badness = 634 rms = 9.05 8/17 < 17/36 < 9/19 17/36 is nearly exact 11-generator; again, 11/8 is closer. These two are the same in the 17-et. [5, -11, -12, -3, -29, -33, -22, 3, 31, 33] <121/120, 225/224, 441/440> badness = 355 rms = 5.15 [5, -11, -12, 3, 33, -29] <225/224, 50421/50000> badness = 376 rms = 2.68 15/31 generator [7, -15, -16, -3, -40, -45, -29, 5, 45, 47] <225/224, 441/440, 1344/1331> badness = 485 rms = 4.19 [7, -15, -16, 5, 45, -40] <225/224, 2500000/2470629> badness = 491 rms = 1.91 20/41 < 61/125 < 41/84 20/41 generator, or 61/125 if you are really picky. [7, 26, 25, -3, 25, 20, -29, -15, -97, -95] <540/539, 896/891, 1375/1372> badness = 318.5 rms = 2.58 [7, 26, 25, -15, -20, 25] <4000/3969, 10976/10935> badness = 641 rms = 1.89 39/80 < 59/121 < 79/162 < 20/41 59/121 generator; 20/41 makes this the same as the previous. [30, 13, 14, 3, -49, -62, -99, -4, -38, -40] <385/384, 2401/2400, 4000/3993> badness = 372 rms = 1.18 [30, 13, 14, -4, 62, -49] <2401/2400, 390625/387072> badness = 443 rms = 1.48 generator 35/72

Message: 4863 - Contents - Hide Contents Date: Sun, 19 May 2002 17:46:39 Subject: tempered versions of Ken Wauchope's scales From: Carl Lumma Some years ago Ken Wauchope posted to the list what he called a symmetrical scale in just intonation: 404 Not Found * [with cont.] Search for http://www.aic.nrl.navy.mil/~wauchope/audio/tuning/symscale.html in Wayback Machine It is a superpostion of two 10:12:15:18 chords rooted a 7:5 apart. It seems that Gene's "star" scale is a tempered version of this. Comments? You can do the same thing with a pair of 8:10:12:15 chords. If you temper the result in 22-tet, you get a subset of Paul Erlich's Symmetrical Decatonic. -Carl

Message: 4864 - Contents - Hide Contents Date: Mon, 20 May 2002 01:24:31 Subject: latest generalized diatonic review From: Carl Lumma At Graham's suggestion I've tried to make my gd rules more simple and objective. Resulting in the following spec: * [with cont.] (Wayb.) I applied it to the usual suspects, and the results are shown in this excel spreadsheet: * [with cont.] (Wayb.) The scala files I used, a text-file version of the results with notes: * [with cont.] (Wayb.) There are 28 scales in all. There are only two scales which are in the top 14 of all four areas: the usual diatonic in 12-tet and Balzano's nonatonic in 20-tet. There are 7 scales common to the top-14 of the 3rd and 4th areas: octatonic blackwood symmaj diatonic balzano hahn-433 pentmaj -Carl

Message: 4865 - Contents - Hide Contents Date: Mon, 20 May 2002 10:11:58 Subject: Re: latest generalized diatonic review From: genewardsmith --- In tuning-math@y..., Carl Lumma <carl@l...> wrote: This spec does not pinpoint the features of a scale which make it a good one, IMHO. Part of the problem is that you assume a sort of rough linearity ("higher values are better.") Are they really? For number of scale steps, you restrict to 5-10, which is reasonable, but claim that in that range lower is better; I don't agree. The numbers given by "modal transposition" reflect scale regularity to some extent, but seem to suggest an equal division is melodically perfect--in fact, like number of scale steps, this is highly nonlinear, and you want to do a Goldilocks and come out somewhere in the sweet spot. The numbers for the Miracle-10 MOS (0.75) and Porcupine-7 (0.73) reflect the bland and pudding-like quality which makes them less interesting than they might be, but the Orwell-9 (0.71), despite its comparitive regularity, is melodically wonderful, like Meantone-7 (ie, diatonic) at 0.61. You need a number to reflect the difference, which is crucial to the sound of these scales!

Message: 4866 - Contents - Hide Contents Date: Mon, 20 May 2002 10:28:49 Subject: Re: tempered versions of Ken Wauchope's scales From: genewardsmith --- In tuning-math@y..., Carl Lumma <carl@l...> wrote:> Some years ago Ken Wauchope posted to the list what he called a > symmetrical scale in just intonation: > > 404 Not Found * [with cont.] Search for http://www.aic.nrl.navy.mil/~wauchope/audio/tuning/symscale.html in Wayback Machine > > It is a superpostion of two 10:12:15:18 chords rooted a 7:5 apart. > > It seems that Gene's "star" scale is a tempered version of this. > Comments?Star is also a 126/125 tempering of two chains of three minor thirds a fifth apart, giving a parallogram Fokker block in the 5-limit, which is a convenient thing to know if you are writing music in it: 1--27/25--6/5--5/4--36/25--3/2--5/3--9/5

Message: 4867 - Contents - Hide Contents Date: Mon, 20 May 2002 14:02 +0 Subject: Re: tempered versions of Ken Wauchope's scales From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <4.3.2.20020519173652.02471620@xxxxx.xxx> Carl Lumma wrote:> It is a superpostion of two 10:12:15:18 chords rooted a 7:5 apart. > > It seems that Gene's "star" scale is a tempered version of this. > Comments?That's Ken Wauchope's minor from your list. 1/1 21/20 7/6 5/4 7/5 3/2 5/3 7/4 2/1 I couldn't work out a temperament for it. What's this star scale?> You can do the same thing with a pair of 8:10:12:15 chords. If > you temper the result in 22-tet, you get a subset of Paul Erlich's > Symmetrical Decatonic.Ken Wauchope's major 1/1 21/20 5/4 21/16 7/5 3/2 7/4 15/8 2/1 does that. And the tempered equivalent is also a subset of the pentachordal decatonic. Graham

Message: 4868 - Contents - Hide Contents Date: Mon, 20 May 2002 08:57:49 Subject: Re: latest generalized diatonic review From: Carl Lumma>> >ttp://lumma.org/spec.txt >>This spec does not pinpoint the features of a scale which make it a good >one, IMHO.It isn't supposed to do that -- just the features that make scales like the diatonic scale, in its application in Western music.>Part of the problem is that you assume a sort of rough linearity >("higher values are better.") Are they really? Sure. >For number of scale steps, you restrict to 5-10, which is reasonable, >but claim that in that range lower is better; I don't agree.Me either. Lower values make pitch tracking easier. I didn't say this desirable. I do cut it off at 5, which is enough to prevent the scale from sounding like a chord in most cases.>The numbers given by "modal transposition" reflect scale regularity to >some extent, but seem to suggest an equal division is melodically >perfect--in fact, like number of scale steps, this is highly nonlinear, >and you want to do a Goldilocks and come out somewhere in the sweet spot.An equal division is supposedly perfect with respect to modal transposition, but will be poor with respect to mode autonomy.>The numbers for the Miracle-10 MOS (0.75) and Porcupine-7 (0.73) reflect >the bland and pudding-like quality which makes them less interesting than >they might be, but the Orwell-9 (0.71), despite its comparitive >regularity, is melodically wonderful, like Meantone-7 (ie, diatonic) at >0.61. You need a number to reflect the difference, which is crucial to >the sound of these scales!So what do you think is going on here? -Carl

Message: 4869 - Contents - Hide Contents Date: Mon, 20 May 2002 09:02:56 Subject: Re: tempered versions of Ken Wauchope's scales From: Carl Lumma>That's Ken Wauchope's minor from your list. > >1/1 21/20 7/6 5/4 7/5 3/2 5/3 7/4 2/1Right -- I pulled it this time around in favor of star.>I couldn't work out a temperament for it. What's this star scale?Gene just posted it, or see the Scala files in gd.zip.>> You can do the same thing with a pair of 8:10:12:15 chords. If >> you temper the result in 22-tet, you get a subset of Paul Erlich's >> Symmetrical Decatonic. >>Ken Wauchope's major > >1/1 21/20 5/4 21/16 7/5 3/2 7/4 15/8 2/1 > >does that. And the tempered equivalent is also a subset of the >pentachordal decatonic.I dropped it too, in favor of the decatonics. I should point out that this one wasn't by Ken, but by me in the spirit of the other one. -Carl

Message: 4870 - Contents - Hide Contents Date: Tue, 21 May 2002 17:34:46 Subject: Central-African rhythms From: manuel.op.de.coul@xxxxxxxxxxx.xxx Here's an interesting and nicely made website that explains why Pygmee rhythms are asymmetrical: Ethnomusicologie, ethnomathématique (Colloque ... * [with cont.] (Wayb.) In terms of mode, they are MOS and tritoneless, if you'll allow the analogy. Take a look at the other two pages too. Manuel

Message: 4872 - Contents - Hide Contents Date: Wed, 22 May 2002 16:45:22 Subject: Re: latest generalized diatonic review From: manuel.op.de.coul@xxxxxxxxxxx.xxx Carl, maybe there's an error in 08-star.scl? When you take the 46-tET version of the scale Gene posted you get 5 7 3 9 3 7 5 7 which is different and has a higher stability. 07-graham.scl is a mode of harmonic major in 31-tET. Manuel

Message: 4873 - Contents - Hide Contents Date: Wed, 22 May 2002 15:13:03 Subject: Re: latest generalized diatonic review From: Carl Lumma>Carl, maybe there's an error in 08-star.scl? >When you take the 46-tET version of the scale >Gene posted you get 5 7 3 9 3 7 5 7 which is >different and has a higher stability. Gene?0 3 12 15 22 27 34 [37] 46>07-graham.scl is a mode of harmonic major in 31-tET.Yeah, I saw that. Graham asked me to put it in. -Carl

Message: 4874 - Contents - Hide Contents Date: Thu, 23 May 2002 05:03:47 Subject: Re: latest generalized diatonic review From: genewardsmith --- In tuning-math@y..., Carl Lumma <carl@l...> wrote:>> Carl, maybe there's an error in 08-star.scl? >> When you take the 46-tET version of the scale >> Gene posted you get 5 7 3 9 3 7 5 7 which is >> different and has a higher stability. > > Gene?I should have written [1,25/24,6/5,5/4,36/25,3/2,5/3,9/5], the 46-et version of which is [0, 3, 12, 15, 24, 27, 34, 39]. However, the alternative with second degree being approximately 27/25 is very much worthy of notice also--like star, it is a 126/125-tempered version of a Fokker block, consisting of two parallel chains of minor thirds, with a lot of nice harmonic properties. Being a new star, maybe it's "nova" :)

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