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Message: 6600 - Contents - Hide Contents Date: Thu, 06 Mar 2003 16:26:29 Subject: Re: 114 temperaments in the big parade From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> If anyone can explain how to evaluate these, I'm interested. Saying > brute force doesn't help, since I don't know what I should try to >force.how about finding the ones that best approximate werckmeister III, kirnberger II, kirnberger III, and valotti, for a start. be sure to try all rotation of each. ! werck3.scl ! Andreas Werckmeister's temperament III (the most famous one, 1681) 12 ! 256/243 192.18000 32/27 390.22500 4/3 1024/729 696.09000 128/81 888.26999 16/9 1092.18000 2/1 ! kirnberger2.scl ! Kirnberger 2: 1/2 synt. comma. "Die Kunst des reinen Satzes" (1774) 12 ! 135/128 9/8 32/27 5/4 4/3 45/32 3/2 405/256 895.11186 16/9 15/8 2/1 ! kirnberger3.scl ! Kirnberger 3: 1/4 synt. comma (1744) 12 ! 135/128 193.15686 32/27 5/4 4/3 45/32 696.57843 405/256 889.73529 16/9 15/8 2/1 Vallotti & Young scale (Vallotti version) 12 ! 94.135 196.090 298.045 392.180 501.955 592.180 698.045 796.090 894.135 1000.000 1090.225 2
Message: 6601 - Contents - Hide Contents Date: Thu, 06 Mar 2003 23:22:27 Subject: Re: 114 temperaments in the big parade From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote: Thanks; I was planning to look at beat ratios for some well-known temperaments (starting from meantones wasn't getting me anywhere, for one thing.)
Message: 6602 - Contents - Hide Contents Date: Thu, 06 Mar 2003 00:53:05 Subject: Re: need definition of "brat" From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:> Gene or whomever, > > > can i please get a good, complete definition > (with examples) of "brat", for the Tuning Dictionary?For a major triad in close root position, let f be the ratio of the fifth to the root, and t the ratio of the major third to the root. Then the beat ratio, or brat is brat = (6t - 5f)/(4t - 5) If t = 5/4, so that major thirds are pure, the brat is considered to be infinity, rather than undefined, unless the minor thirds are pure also, in which case we have a JI triad and the brat is undefined. For any tuning system with uniform fifths and major thirds, the brat for the system is the brat for any one of the major triads of the system. For example, for 12-equal the brat is (6 2^(1/3) - 5 2^(7/12))/(4 2^(1/3) - 5) = 1.713299...> i've decided to create a page about Wendell > well-temperaments. this is the incipient version: > 404 Not Found * [with cont.] Search for http://sonic-arts.org/dict/wendell.htm in Wayback MachineWhy not a page about the more general concept of well temperaments with synchronized beating? Unless that's what you meant...
Message: 6603 - Contents - Hide Contents Date: Thu, 06 Mar 2003 14:22:04 Subject: Eight more well temperaments From: Gene Ward Smith These each have five pure fifths. I give the scale, followed by the brats for each scale step. These are exact; there is an odd man out which is either 8/5, 12/7 or 16/9. [1, 1215/1144, 160/143, 10213/8580, 180/143, 1145/858, 405/286, 214/143, 2729/1716, 240/143, 139/78, 270/143] [2, 2, 3/2, 2, 3/2, 12/7, 3/2, 2, 2, 3/2, 2, 3/2] [1, 96/91, 393/350, 108/91, 573/455, 243/182, 642/455, 3/2, 144/91, 153/91, 162/91, 66/35] [3/2, 3/2, 2, 3/2, 2, 16/9, 2, 2, 3/2, 2, 3/2, 2] [1, 535/506, 284/253, 300/253, 1275/1012, 675/506, 65/46, 3/2, 400/253, 425/253, 450/253, 955/506] [3/2, 2, 2, 3/2, 2, 8/5, 2, 2, 3/2, 3/2, 3/2, 2] [1, 763/720, 9/8, 571/480, 4549/3600, 427/320, 637/450, 3/2, 763/480, 27/16, 427/240, 6809/3600] [3/2, 3/2, 3/2, 2, 2, 8/5, 2, 3/2, 2, 2, 3/2, 2] [1, 17/16, 9/8, 381/320, 81/64, 171/128, 227/160, 3/2, 509/320, 27/16, 57/32, 379/200] [3/2, 2, 3/2, 2, 2, 16/9, 2, 3/2, 2, 3/2, 3/2, 2] [1, 637/600, 9/8, 763/640, 81/64, 427/320, 6809/4800, 3/2, 763/480, 27/16, 571/320, 4549/2400] [3/2, 2, 3/2, 2, 2, 8/5, 2, 3/2, 3/2, 3/2, 2, 2] [1, 18/17, 286/255, 81/68, 64/51, 3403/2550, 24/17, 382/255, 27/17, 428/255, 4547/2550, 32/17] [2, 3/2, 2, 2, 3/2, 12/7, 3/2, 2, 3/2, 2, 2, 3/2] [1, 800/759, 850/759, 300/253, 955/759, 675/506, 1070/759, 1136/759, 400/253, 425/253, 450/253, 130/69] [2, 3/2, 3/2, 3/2, 2, 8/5, 2, 2, 3/2, 2, 3/2, 2]
Message: 6605 - Contents - Hide Contents Date: Fri, 07 Mar 2003 13:26:39 Subject: Re: New file uploaded to tuning-math From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx tuning-math@xxxxxxxxxxx.xxx wrote:> > Hello, > > This email message is a notification to let you know that > a file has been uploaded to the Files area of the tuning-math > group.I uploaded plots of the major thirds of some well-temperaments, going around the circle of fifths. The three "Smith Wells", decribed in the file, are the ones with squirrel index 8 and seven pure fifths. These are, of course, far closer to 12-equal than Werckmeister III, which I also give a plot for. I was going to put my version of this temperament on the same plot, but they are so close this is impossible, as one covers the other. Does Robert read this list? I'd be interested in comments from well temperament mavens about these plots, which strike me as a useful way of looking at wells.
Message: 6606 - Contents - Hide Contents Date: Sun, 9 Mar 2003 15:07:32 Subject: good 11-limit meantones From: monz hey all, i've found some interesting meantones which give the lowest error-from-JI values thru the 11-limit: 11/48-, 8/35-, and 3/13-comma are three representative examples. can anyone (Gene?) give some data on these? if it's been done before, please supply links. thanks. -monz
Message: 6607 - Contents - Hide Contents Date: Sun, 9 Mar 2003 15:10:59 Subject: Re: good 11-limit meantones From: monz> From: "monz" <monz@xxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Sunday, March 09, 2003 3:07 PM > Subject: [tuning-math] good 11-limit meantones > > > i've found some interesting meantones which give > the lowest error-from-JI values thru the 11-limit: > 11/48-, 8/35-, and 3/13-comma are three representative > examples. > > can anyone (Gene?) give some data on these? > if it's been done before, please supply links. > thanks.2/9-comma is also quite good in 11-limit, and in fact is the one that got me started on this. -monz
Message: 6608 - Contents - Hide Contents Date: Sun, 9 Mar 2003 16:07:11 Subject: Re: good 11-limit meantones From: monz ----- Original Message ----- From: "monz" <monz@xxxxxxxxx.xxx> To: <tuning-math@xxxxxxxxxxx.xxx> Sent: Sunday, March 09, 2003 3:07 PM Subject: [tuning-math] good 11-limit meantones> hey all, > > > i've found some interesting meantones which give > the lowest error-from-JI values thru the 11-limit: > 11/48-, 8/35-, and 3/13-comma are three representative > examples.as for equal-temperaments which fall in this range, 31edo is pretty darn good for a low-cardinality EDO, generator 2^(18/31). 2^(79/136) is very good, 2^(97/167) better still, and 2^(176/303) really fantastic. feedback appreciated. -monz
Message: 6609 - Contents - Hide Contents Date: Mon, 10 Mar 2003 08:56:18 Subject: Re: good 11-limit meantones From: monz hi Graham,> From: "Graham Breed" <graham@xxxxxxxxxx.xx.xx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Monday, March 10, 2003 2:19 AM > Subject: [tuning-math] Re: good 11-limit meantones > > > monz wrote: >>> 2^(79/136) is very good, 2^(97/167) better still, >> and 2^(176/303) really fantastic. >> What mapping are you using for 136?i made the generator 2^(79/136), and the mapping of ratios to generators follows table i put at the bottom of this webpage: Definitions of tuning terms: meantone-from-JI ... * [with cont.] (Wayb.) gen. ratio -18 16/11 -17 12/11 -16 18/11 ... -10 8/7 -9 12/7 -8 14/11 ... -6 10/7 ... -4 8/5 -3 6/5 ... -1 4/3 ... +1 3/2 ... +3 5/3 +4 5/4 ... +6 7/5 ... +8 11/7 +9 7/6 +10 7/4 ... +16 11/9 +17 11/6 +18 11/8> The red line shows 3, so remember 9 is twice as far out. Why aren't > ratios of 9 included in your applet? I can see that 3/13-comma meantone > would be close to the minimax for the simpler mapping if you ignore 9:8, > 10:9 and 9:7.oops ... i'm always thinking in terms of prime-numbers, and including ratios according to prime-limit rather than odd-limit. i guess i should add ratios of 9, but that will be a lot of work as i'll have to redo every graph. -monz
Message: 6610 - Contents - Hide Contents Date: Mon, 10 Mar 2003 10:19:39 Subject: Re: good 11-limit meantones From: Graham Breed monz wrote:> 2^(79/136) is very good, 2^(97/167) better still, > and 2^(176/303) really fantastic.What mapping are you using for 136? I can find two fairly good meantones [136, 215, 316, 382, 470] [136, 215, 316, 381, 470] Both have a fifth of 79 steps, which matches your generator. And both have a worst 11-limit error of 11.7 cents. For 31-equal, this is only 11.1 cents. There are two relatively simple 11-limt meantone mappings. This one, consistent with 31- and 43-equal: 1 0 2 -1 4 -4 7 -10 11 -18 and this one 1 0 2 -1 4 -4 7 -10 -2 13 consistent with 31 and 50. The first one optimizes with a fourth of 503.3 cents and a worst 11-limit error of 11.0 cents. That's 0.2437-comma meantone, you can find rational approximations I'm sure. The other one optimizes at 502.4 cents (exactly 1/4-comma by the looks of it) and a worst error of 10.8 cents. My approximation graphs are here: Meantone temperaments * [with cont.] (Wayb.) The red line shows 3, so remember 9 is twice as far out. Why aren't ratios of 9 included in your applet? I can see that 3/13-comma meantone would be close to the minimax for the simpler mapping if you ignore 9:8, 10:9 and 9:7. Graham
Message: 6611 - Contents - Hide Contents Date: Mon, 10 Mar 2003 10:22:36 Subject: Re: good 11-limit meantones From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:> hey all, > > > i've found some interesting meantones which give > the lowest error-from-JI values thru the 11-limit: > 11/48-, 8/35-, and 3/13-comma are three representative > examples.The first order of business is to decide which of the two extensions of septimal meantone, both good, we are looking at--what I've called "meanpop" or "meantone". "Meantone" uses the map [1,4,10,18] whereas "meanpop" goes [1,4,10,-13]. 18 - (-13) = 31 and these are the same for the 31-et, which is one possible choice for the boundry between them. The 30th row of the Farey sequence for poptimal q-comma meantones goes 3/13 < 7/30 < 4/17 < 5/21 < 6/25 < 7/29. The Farey sequence for the 250th row for rational (equal temperament) poptimal meantones is 83/198 < 96/225 < 13/31 The range of poptimal brats is from 5.892 to 16.083. The 40th row of the Farey sequence for meanpop goes 10/39 < 9/35 < 8/31. The 250th row of the Farey sequence for the generator goes 73/174 < 60/143 < 47/112. The range of poptimal brats is [infinity, -10.4789]. It would seem that the 31-et is the obvious choice for meantone, and the 112-et, as I've mentioned before, works nicely for meanpop. If we want good q-comma numbers, 3/13 or 4/17 would be fine for meantone, and 10/39, 9/35 or 8/31 would have to do for meanpop. So far as brats go, choosing infinity for meanpop and 6 for meantone would work. All of these are minimal, in that the poptimal values at least include the ones given, but might include more (especially under Paul's definition.)
Message: 6612 - Contents - Hide Contents Date: Mon, 10 Mar 2003 10:44:50 Subject: Re: good 11-limit meantones From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> It would seem that the 31-et is the obvious choice for meantone, and > the 112-et, as I've mentioned before, works nicely for meanpop. If we > want good q-comma numbers, 3/13 or 4/17 would be fine for meantone, > and 10/39, 9/35 or 8/31 would have to do for meanpop.Duh. I forgot 1/4-comma for meanpop! Use that for meanpop, and 3/13-comma for meantone, and everything is kopacetic.
Message: 6613 - Contents - Hide Contents Date: Wed, 12 Mar 2003 09:23:03 Subject: Rothenberg giveaway From: Carl Lumma All; I've got a spare copy of Rothenberg's three seminal, and very mathy papers on scale theory. The first person to write off-list with a valid snail address and legible sentence saying why you're interested, gets it. -Carl
Message: 6614 - Contents - Hide Contents Date: Fri, 21 Mar 2003 02:48:02 Subject: More on MOS/temperaments From: Gene Ward Smith If T is a linear temperament, and if we tune a scale and/or temperament on n notes to a tuning of T, we may denote this by T[n]. T[n] has certain commas associated to it, in addition to the ones belonging to T itself, namely q such that q mapes to +-n generator steps in terms of the (octave based) period and generator description of T. These associated commas give us equivalences when going around the circle of generators in T[n], since they represent the n-step jumps which occur when we reach the end of the chain of generators. When these commas are "interesting", meaning such that they allow for nice equivalences, we are in a particularly nice situation. Perhaps the most interesting comma to have in this way is 36/35, which as I mentioned before has a numerator (36) which is both square and triangular and which allows for a lot of useful equivalences, since for instance 1-5/4-3/2-7/4 is converted to 1-9/7-3/2-9/5 by two 36/35's, and likewise 1-6/5-3/2-12/7 to 1-7/6-3/2-5/3. Other commas, eg 15/14, 21/20, 25/24, 49/48, 45/44, etc. have their own properties, and this could become a rather involved study. To start with, Here are some T[n] with 36/35 as an associated comma: Orwell[9], Tertiathirds[9] Supersupermajor[10] Superkleismic[11] Meantone[12], Pajara[12], Tripletone[12], Injera[12] Octafifths[13] Amity[14] Catakleismic[15] Muggles[16] Beatles[17], Squares[17] Miracle[21] Schismic[24], Diaschismic[24] At which point I'll stop. How these work out in practice depends on the details of the mapping, but in general tetrads tend to be sent to alternative versions which have more or less nice consonance properties. And recall, this is only one comma!
Message: 6615 - Contents - Hide Contents Date: Fri, 21 Mar 2003 16:26:16 Subject: Re: More on MOS/temperaments From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> If T is a linear temperament, and if we tune a scale and/or temperament > on n notes to a tuning of T, we may denote this by T[n]. T[n] has > certain commas associated to it, in addition to the ones belonging to > T itself,for example the chromatic unison vector!> namely q such that q mapes to +-n generator steps in terms > of the (octave based) period and generator description of T.that's it! i think you've finally stumbled on the chromatic unison vector, which means that from now on, you may understand what it means!> These associated commas give us equivalences when going around the > circle of generators in T[n], since they represent the n-step jumps > which occur when we reach the end of the chain of generators. When > these commas are "interesting", meaning such that they allow for nice > equivalences, we are in a particularly nice situation. Perhaps the most > interesting comma to have in this way is 36/35, which as I mentioned > before has a numerator (36) which is both square and triangular and > which allows for a lot of useful equivalences, since for instance > 1-5/4-3/2-7/4 is converted to 1-9/7-3/2-9/5 by two 36/35's, and > likewise 1-6/5-3/2-12/7 to 1-7/6-3/2-5/3. Other commas, eg 15/14, > 21/20, 25/24, 49/48, 45/44, etc. have their own properties, and this > could become a rather involved study.we already started on this in a series of posts, in particular we were looking at scales where the major tetrad and the minor tetrad arise from the same pattern of scale steps.> And recall, this is only > one comma!i'm not sure what you mean to imply by that -- and of course this chroma is only defined plus or minus any arbitrary number of commatic unison vectors.
Message: 6616 - Contents - Hide Contents Date: Fri, 21 Mar 2003 17:15:14 Subject: Re: More on MOS/temperaments From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> > wrote:>> namely q such that q maps to +-n generator steps in terms >> of the (octave based) period and generator description of T. >> that's it! i think you've finally stumbled on the chromatic unison > vector, which means that from now on, you may understand what it > means!Looks that way. Thanks for pointing this out. Communicating with me is a pain, I suppose. :)
Message: 6617 - Contents - Hide Contents Date: Fri, 21 Mar 2003 19:37:55 Subject: Re: More on MOS/temperaments From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus" > <wallyesterpaulrus@y...> wrote:>> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> >> wrote: >>>> namely q such that q maps to +-n generator steps in terms >>> of the (octave based) period and generator description of T. >>>> that's it! i think you've finally stumbled on the chromatic unison >> vector, which means that from now on, you may understand what it >> means! >> Looks that way. Thanks for pointing this out. > > Communicating with me is a pain, I suppose. :)nah, this was a rare case of something that it seemed you were not understanding, since you got it wrong a few times. but hopefully the next time the whole discussion that kalle started and that carl was interested in reviving comes around, you'll be in a position to contribute to it, and thus we will all be far better informed than if you weren't around. (hint, hint, carl, gene, etc.)
Message: 6618 - Contents - Hide Contents Date: Fri, 21 Mar 2003 22:18:30 Subject: 12 notes with 36/35 a chromatic comma From: Gene Ward Smith As I pointed out, for septimal systems we get meantone, pajara, injera and tripletone in this way. Below I give the name, the wedgie, a TM reduced scale, major and minor tetrads going around this scale. In the case of meantone I add the cicle of fifths version. Meantone [1, 4, 10, 4, 13, 12] [1, 3/2, 9/8, 5/3, 5/4, 15/8, 7/5, 21/20, 14/9, 7/6, 7/4, 21/16] [1, 15/14, 9/8, 6/5, 5/4, 4/3, 7/5, 3/2, 8/5, 5/3, 9/5, 15/8] Tetrads around circle of fifths 0 [1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3] 1 [1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3] 2 [1, 5/4, 3/2, 9/5] [1, 7/6, 3/2, 5/3] 3 [1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3] 4 [1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3] 5 [1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3] 6 [1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3] 7 [1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3] 8 [1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 5/3] 9 [1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7] 10 [1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7] 11 [1, 9/7, 32/21, 9/5] [1, 6/5, 32/21, 12/7] Tetrads around scale degrees 0 [1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3] 1 [1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3] 2 [1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3] 3 [1, 5/4, 3/2, 9/5] [1, 7/6, 3/2, 5/3] 4 [1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7] 5 [1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3] 6 [1, 9/7, 32/21, 9/5] [1, 6/5, 32/21, 12/7] 7 [1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3] 8 [1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3] 9 [1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 5/3] 10 [1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3] 11 [1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7] Pajara [2, -4, -4, -11, -12, 2] [1, 15/14, 8/7, 6/5, 5/4, 4/3, 7/5, 3/2, 8/5, 12/7, 7/4, 15/8] Tetrads 0 [1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 12/7] 1 [1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3] 2 [1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3] 3 [1, 5/4, 35/24, 7/4] [1, 7/6, 35/24, 5/3] 4 [1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7] 5 [1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7] 6 [1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 12/7] 7 [1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3] 8 [1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3] 9 [1, 5/4, 35/24, 7/4] [1, 7/6, 35/24, 5/3] 10 [1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7] 11 [1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7] Injera [2, 8, 8, 8, 7, -4] [1, 15/14, 9/8, 7/6, 9/7, 4/3, 7/5, 3/2, 14/9, 5/3, 9/5, 27/14] Tetrads 0 [1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 5/3] 1 [1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7] 2 [1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7] 3 [1, 9/7, 32/21, 9/5] [1, 6/5, 32/21, 12/7] 4 [1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3] 5 [1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3] 6 [1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 5/3] 7 [1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7] 8 [1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7] 9 [1, 9/7, 32/21, 9/5] [1, 6/5, 32/21, 12/7] 10 [1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3] 11 [1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3] Tripletone [3, 0, -6, -7, -18, -14] [1, 16/15, 10/9, 6/5, 5/4, 4/3, 7/5, 3/2, 8/5, 5/3, 7/4, 15/8] Tetrads 0 [1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 5/3] 1 [1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3] 2 [1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 12/7] 3 [1, 5/4, 35/24, 7/4] [1, 7/6, 35/24, 5/3] 4 [1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 5/3] 5 [1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3] 6 [1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 12/7] 7 [1, 5/4, 35/24, 7/4] [1, 7/6, 35/24, 5/3] 8 [1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 5/3] 9 [1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3] 10 [1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 12/7] 11 [1, 5/4, 35/24, 7/4] [1, 7/6, 35/24, 5/3]
Message: 6619 - Contents - Hide Contents Date: Fri, 21 Mar 2003 22:33:59 Subject: Re: 12 notes with 36/35 a chromatic comma From: wallyesterpaulrus keep it up, gene! i think one of the more important chromatic unison vectors kalle and carl were interested in was 49/48. the reason i don't refer to these chromatic unison vectors as "commas" is because my terminology is based on generalizing common-practice western musical theory (generalizing an old subject is usually the best way to terminologize a new one). common-practice western musical theory is based on meantone-7, and there the chromatic unison vector is 25:24 -- known classically as the "chroma"; while the commatic unison vector is 81:80 -- known classically as the "comma". --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> As I pointed out, for septimal systems we get meantone, pajara, injera > and tripletone in this way. Below I give the name, the wedgie, a TM > reduced scale, major and minor tetrads going around this scale. In the > case of meantone I add the cicle of fifths version. > > Meantone > > [1, 4, 10, 4, 13, 12] > [1, 3/2, 9/8, 5/3, 5/4, 15/8, 7/5, 21/20, 14/9, 7/6, 7/4, 21/16] > [1, 15/14, 9/8, 6/5, 5/4, 4/3, 7/5, 3/2, 8/5, 5/3, 9/5, 15/8] > > Tetrads around circle of fifths > > 0 [1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3] > 1 [1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3] > 2 [1, 5/4, 3/2, 9/5] [1, 7/6, 3/2, 5/3] > 3 [1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3] > 4 [1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3] > 5 [1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3] > 6 [1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3] > 7 [1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3] > 8 [1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 5/3] > 9 [1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7] > 10 [1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7] > 11 [1, 9/7, 32/21, 9/5] [1, 6/5, 32/21, 12/7] > > Tetrads around scale degrees > > 0 [1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3] > 1 [1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3] > 2 [1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3] > 3 [1, 5/4, 3/2, 9/5] [1, 7/6, 3/2, 5/3] > 4 [1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7] > 5 [1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3] > 6 [1, 9/7, 32/21, 9/5] [1, 6/5, 32/21, 12/7] > 7 [1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3] > 8 [1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3] > 9 [1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 5/3] > 10 [1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3] > 11 [1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7] > > > Pajara > > [2, -4, -4, -11, -12, 2] > [1, 15/14, 8/7, 6/5, 5/4, 4/3, 7/5, 3/2, 8/5, 12/7, 7/4, 15/8] > > Tetrads > > 0 [1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 12/7] > 1 [1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3] > 2 [1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3] > 3 [1, 5/4, 35/24, 7/4] [1, 7/6, 35/24, 5/3] > 4 [1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7] > 5 [1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7] > 6 [1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 12/7] > 7 [1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3] > 8 [1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3] > 9 [1, 5/4, 35/24, 7/4] [1, 7/6, 35/24, 5/3] > 10 [1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7] > 11 [1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7] > > > Injera > > [2, 8, 8, 8, 7, -4] > [1, 15/14, 9/8, 7/6, 9/7, 4/3, 7/5, 3/2, 14/9, 5/3, 9/5, 27/14] > > Tetrads > > 0 [1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 5/3] > 1 [1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7] > 2 [1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7] > 3 [1, 9/7, 32/21, 9/5] [1, 6/5, 32/21, 12/7] > 4 [1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3] > 5 [1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3] > 6 [1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 5/3] > 7 [1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7] > 8 [1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7] > 9 [1, 9/7, 32/21, 9/5] [1, 6/5, 32/21, 12/7] > 10 [1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3] > 11 [1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3] > > > Tripletone > > [3, 0, -6, -7, -18, -14] > [1, 16/15, 10/9, 6/5, 5/4, 4/3, 7/5, 3/2, 8/5, 5/3, 7/4, 15/8] > > Tetrads > > 0 [1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 5/3] > 1 [1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3] > 2 [1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 12/7] > 3 [1, 5/4, 35/24, 7/4] [1, 7/6, 35/24, 5/3] > 4 [1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 5/3] > 5 [1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3] > 6 [1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 12/7] > 7 [1, 5/4, 35/24, 7/4] [1, 7/6, 35/24, 5/3] > 8 [1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 5/3] > 9 [1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3] > 10 [1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 12/7] > 11 [1, 5/4, 35/24, 7/4] [1, 7/6, 35/24, 5/3]
Message: 6620 - Contents - Hide Contents Date: Sat, 22 Mar 2003 18:43:19 Subject: Re: More on MOS/temperaments From: Carl Lumma Hmm, something's amiss. Anybody else get this list sent to them by e-mail? I got msg. 6056 but not 6057-6067. Just got 6068 & 9.>we already started on this in a series of posts, in particular we >were looking at scales where the major tetrad and the minor tetrad >arise from the same pattern of scale steps.! Just when I was thinking, "when would I ever be interested in this incomplete-chain extra-intervals stuff"... Good looking out, Paul! More later. -C.
Message: 6621 - Contents - Hide Contents Date: Sat, 22 Mar 2003 22:24:22 Subject: Re: this T[n] business From: Carl Lumma>>> >o it seems my assertion is wrong; simple ratios don't tend >>> to be bigger. >>>> that's the great thing about tenney complexity (as opposed to >> farey, mann, etc.)! >>Ah, you've said that before, I think!I've just verified this for the Farey series. -Carl
Message: 6622 - Contents - Hide Contents Date: Sat, 22 Mar 2003 19:12:26 Subject: Re: More on MOS/temperaments From: Carl Lumma>> >nd recall, this is only one comma! >>i'm not sure what you mean to imply by that -- and of course this >chroma is only defined plus or minus any arbitrary number of >commatic unison vectors.Maybe Gene's alluding to the planar and higher cases. -Carl
Message: 6623 - Contents - Hide Contents Date: Sat, 22 Mar 2003 22:27:54 Subject: T[n] where n is small From: Carl Lumma Gene, What about turning this on scales, n < 11? I don't know how many lines of maple you do this with, but if they're few you can post them here and I can either translate to Scheme or run them in maple myself. -Carl
Message: 6624 - Contents - Hide Contents Date: Sat, 22 Mar 2003 19:14:57 Subject: Re: 12 notes with 36/35 a chromatic comma From: Carl Lumma