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Message: 11125 - Contents - Hide Contents Date: Sat, 26 Jun 2004 10:12:01 Subject: Re: Paul's nifty fifty From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Herman Miller <hmiller@I...> wrote:> Paul Erlich wrote:>> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> >> wrote: >> >>>>> [1, -8, -14, -15, -25, -10] septischismic? >> >>>> Continuing on our kick of naming after people, I might call this >> Garibaldi, since Eduardo Sabat-Garibaldi gave names to 5120/5103 >> ("Beta 5") and 33554432/33480783 ("Beta 2") in his study which >> yielded the 1/9-schisma, pure-octave version of this temperament. >> 4000/3969, though, may have escaped his attention (at least Manuel >> doesn't list any other "Beta"s, or any name for 4000/3969 at all). >> >> That also fits the trend of animal names, since a garibaldi is a kind of > fish (Hypsypops rubicundus). But is anything wrong with plain "schismic" > for this one?The adjustment of it via +53 gives what I once called counterschismic, but the tuning of that is like 5-limit schismic, so I'm suggesting it should get the name schismic, and similarly "catakleismic" the name hanson. This leaves kleismic free to be a purely 7-limit name, unless we like armadillo better for that. ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ Yahoo! Groups Links <*> To visit your group on the web, go to: Yahoo groups: /tuning-math/ * [with cont.] <*> To unsubscribe from this group, send an email to: tuning-math-unsubscribe@xxxxxxxxxxx.xxx <*> Your use of Yahoo! Groups is subject to: Yahoo! Terms of Service * [with cont.] (Wayb.)

Message: 11126 - Contents - Hide Contents Date: Sun, 27 Jun 2004 18:11:56 Subject: Re: NOT tuning From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Kalle Aho" <kalleaho@m...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> > wrote:>> NOT being an acronym for No Octave Tempering. NOT tuning is TOP > tuning>> with the added constraint that octaves must be pure. For example, > the>> 7-limit NOT tuning for meantone is very close to 1/5-comma; this > makes>> the error for 3, weighted by log(3), equal to with opposite sign > from>> the error for 7 weighted by log(7). >> >> More anon, I think. > > Why?Why not? This is the tuning math list, after all. Just as TOP tuning bounds the ratio of absolute error over Tenney height, NOT does the same for odd Tenney height, defined as the Tenney height of the odd part of a positive rational number. In other words, take out the even factor, so that the numerator and denominator are two odd integers with GCD 1, and take the log of the product. For example, the 5-limit NOT meantone tuning has fifths of size 697.0197, about 2/11 comma flat, and close to many people's favored 55-equal tuning. The error in the fifth, divided by log2(3), is 2.4829 cents, the error in the major third is sharp rather than flat, but divided by log2(5) is again 2.4829. The error in the minor third is in the flat direction by about 9.7 cents; dividing this by log2(15) again gives 2.4829. The error in any 5-limit interval, divided by the log base two of the product of the numerator and denominator of the odd part, is bounded by 2.4829. It seems to me this is interesting enough to justify posting about it.

Message: 11127 - Contents - Hide Contents Date: Sun, 27 Jun 2004 18:29:09 Subject: Re: NOT tuning From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Kalle Aho" <kalleaho@m...> wrote:>> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> >> wrote:>>> NOT being an acronym for No Octave Tempering. NOT tuning is TOP >> tuning>>> with the added constraint that octaves must be pure. For example, >> the>>> 7-limit NOT tuning for meantone is very close to 1/5-comma; this >> makes>>> the error for 3, weighted by log(3), equal to with opposite sign >> from>>> the error for 7 weighted by log(7). >>> >>> More anon, I think. >> >> Why? >> Why not? This is the tuning math list, after all. Just as TOP tuning > bounds the ratio of absolute error over Tenney height, NOT does the > same for odd Tenney height, defined as the Tenney height of the odd > part of a positive rational number. In other words, take out the even > factor, so that the numerator and denominator are two odd integers > with GCD 1, and take the log of the product. > > For example, the 5-limit NOT meantone tuning has fifths of size > 697.0197, about 2/11 comma flat, and close to many people's favored > 55-equal tuning. The error in the fifth, divided by log2(3), is 2.4829 > cents, the error in the major third is sharp rather than flat, but > divided by log2(5) is again 2.4829. The error in the minor third is > in the flat direction by about 9.7 cents; dividing this by log2(15) > again gives 2.4829. The error in any 5-limit interval, divided by the > log base two of the product of the numerator and denominator of the > odd part, is bounded by 2.4829. It seems to me this is interesting > enough to justify posting about it.I'm not here. The odd Tenney height should truly be 5 for both the major third and the minor third. They're both ratios of 5 -- members of the 5-odd- limit. I'm not here.

Message: 11128 - Contents - Hide Contents Date: Sun, 27 Jun 2004 18:33:46 Subject: Re: NOT tuning From: Kalle Aho --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:>> Why?> Why not? This is the tuning math list, after all. Just as TOP tuning > bounds the ratio of absolute error over Tenney height, NOT does the > same for odd Tenney height, defined as the Tenney height of the odd > part of a positive rational number. In other words, take out the even > factor, so that the numerator and denominator are two odd integers > with GCD 1, and take the log of the product. > For example, the 5-limit NOT meantone tuning has fifths of size > 697.0197, about 2/11 comma flat, and close to many people's favored > 55-equal tuning. The error in the fifth, divided by log2(3), is 2.4829 > cents, the error in the major third is sharp rather than flat, but > divided by log2(5) is again 2.4829. The error in the minor third is > in the flat direction by about 9.7 cents; dividing this by log2(15) > again gives 2.4829. The error in any 5-limit interval, divided by the > log base two of the product of the numerator and denominator of the > odd part, is bounded by 2.4829. It seems to me this is interesting > enough to justify posting about it.I didn't mean to be rude. You can post anything you like. I just wanted to understand the point of NOT. Kalle

Message: 11129 - Contents - Hide Contents Date: Sun, 27 Jun 2004 11:57:36 Subject: Re: NOT tuning From: Carl Lumma>NOT being an acronym for No Octave Tempering. NOT tuning is TOP tuning >with the added constraint that octaves must be pure. For example, the >7-limit NOT tuning for meantone is very close to 1/5-comma; this makes >the error for 3, weighted by log(3), equal to with opposite sign from >the error for 7 weighted by log(7). > >More anon, I think.Cool; I've been waiting for this. -C.

Message: 11130 - Contents - Hide Contents Date: Sun, 27 Jun 2004 20:08:54 Subject: Re: NOT tuning From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>> NOT being an acronym for No Octave Tempering. NOT tuning is TOP tuning >> with the added constraint that octaves must be pure. For example, the >> 7-limit NOT tuning for meantone is very close to 1/5-comma; this makes >> the error for 3, weighted by log(3), equal to with opposite sign from >> the error for 7 weighted by log(7). >> >> More anon, I think. >> Cool; I've been waiting for this.Cool; let's look at some results. Meantone 5-limit: 698.0187 (43, 55, 98, 153, 251, 404) 7-limit: 697.6458 (31, 43, 934, 977; 0.0286 cents flatter than 43) 11-limit huygens: 697.6458 (same as 7-limit) 11-limit meantone (meanpop): 696.9010 (close to 31; the next convergent is 169/291) dominant sevenths: 702.1396 (12, 41, 94) flattone: 693.9317 (19, 64, 83, 313) Miracle 5-limit: 116.6593 (very close to 72; 31, 72, 2191, 2263) 7 and 11 limits are the same. Diaschismic 5-limit: 104.7764 (126 is close) 7-limit: 104.5806 (46, then 218) 7-limit pajara: 109.1845 (22 is very close; 10, 22, 2230) 7-limit shrutar: 58.3882 (46, then 252) 11-limit pajara same as 7-limit pajara 11-limit shrutar same as 7-limit shrutar Magic 5-limit: 381.1024 (continued fraction gives 19, 22, 63, 85, 148, 233, 381 with 2^(121/381) being very close. 41 isn't in there!) 7-limit: same as 5-limit 11-limit gives a family of temperaments <<5 1 12 -8 -10 5 -30 25 -22 -64|| same as 5 and 7 limit magic <<5 1 12 33 -10 5 35 25 73 51|| 380.6009 (19, 22, 41, 227, 268) <<5 1 12 14 -10 5 5 25 29 -2|| 381.4284 (19, 22, 129, 280; 280 is very close) Orwell 5-limit: 271.5994 (22, 31, 53, 190, 243) 7-limit: 271.4707 (22, 31, 53, 84, 305) 11-limit: 271.8716 (22, 53, 128, 2741)

Message: 11131 - Contents - Hide Contents Date: Sun, 27 Jun 2004 20:45:22 Subject: Absolute TOP error From: Gene Ward Smith We've been doing the weighted TOP error as the error in cents divided by the log base two of the product of the numerator and the denominator. This is good for most purposes, but if we stick to the same log for the ratio (both cents, or both log base two, etc.) then we get something with a meaning independent of unit/log base choice. It can be described as the logarithm, base N = the product of numerator and denominator, of the error; Log_N(E). The reciprocal is Log_E(N); it is how many steps of size E (the error) are required to get to N (product of numerator and denominator.) In TOP tuning, there is a minimum value for this which defines the relative error. The same remarks apply for NOT tuning and the product of the numerator and denominator of the odd part of the interval. If T is the TOP error by the definition we've been using, then 1200/T is the minimum number of error-sized steps needed to get to the product of numerator and denominator. For (5, 7, 11-limit) meantone that would be 706.497 steps, for instance; miracle would be 1901.701 and ennealimmal 32987.408.

Message: 11132 - Contents - Hide Contents Date: Sun, 27 Jun 2004 01:08:16 Subject: Re: Some 11-limit temperaments supported by 152 From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> <<1 33 27 -18 50 40 -32 -30 -156 -144|| has a generator of a fifth, > like meantone and schismic, with TM basis {540/539, 1375/1372, > 5120/5103}. The poptimal range runs from about 0.01 to 0.67 cents > sharp.This range stuck me as way too big, so I recalculated and got 0.658 to 0.670, much more reasonable for a temperament of this accuracy. The 152 fifth, 0.6766 cents sharp, is just beyond this range, but works well enough; 291/497, 0.6607 cents sharp, is poptimal. This thing has an interesting {7,11}-limit comma, 65536/65219; which is one of the five {7,11} commas of size less than 50 cents and epimericity less than 0.6: 352/343 44.840223 .376383 14641/14336 36.445863 .597701 65536/65219 8.394360 .519498 117649/117128 7.683669 .536007 5767168/5764801 .710691 .499084 Here I've listed the comma, size in cents, and epimericity; the most striking is clearly the last one, 2^18 7^(-8) 11, which could be a gold mine for people seeking the exotic harmonies of 7 and 11 in conjunction.

Message: 11133 - Contents - Hide Contents Date: Sun, 27 Jun 2004 02:17:07 Subject: A no-fives/planar 11-limit microtemperament From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> Here I've listed the comma, size in cents, and epimericity; the most > striking is clearly the last one, 2^18 7^(-8) 11, which could be a > gold mine for people seeking the exotic harmonies of 7 and 11 in > conjunction.If we take this comma and add to it 43923/43904, a {2,3,7,11} comma, we get what we can consider to be either an 11-limit planar temperament (with 2, 5, and 7 as generators) or {2,3,7,11} linear temperament; ie, a no-fives temperament. This has period an octave and generator a very slightly narrow 8/7. (If we use (48)^(1/29) as a generator, 0.0722 cents flat; if we use 2^(161/836), 0.0736 cents flat.) In terms of this generator, the mapping is 3 ~ 2^(-4) (8/7)^29 11 ~ 2^5 (8/7)^(-8) This is a strong no-fives microtemperament but trying to add five to it doesn't work so well. The Graham complexity is 66, or 37 if you are willing to forgo 9 as well as 5; we have DE of size 57 and 83. A bit too much for most of us. Of course if we are willing to settle for a {7, 11} temperament, the Graham complexity is now 9, and it's reminiscent of schismic, but of course more accurate. DE of size 11, 16, 21, 26 and 31 become relevant.

Message: 11134 - Contents - Hide Contents Date: Sun, 27 Jun 2004 11:33:25 Subject: NOT tuning From: Gene Ward Smith NOT being an acronym for No Octave Tempering. NOT tuning is TOP tuning with the added constraint that octaves must be pure. For example, the 7-limit NOT tuning for meantone is very close to 1/5-comma; this makes the error for 3, weighted by log(3), equal to with opposite sign from the error for 7 weighted by log(7). More anon, I think.

Message: 11135 - Contents - Hide Contents Date: Sun, 27 Jun 2004 15:20:07 Subject: Re: NOT tuning From: Kalle Aho --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> NOT being an acronym for No Octave Tempering. NOT tuning is TOP tuning > with the added constraint that octaves must be pure. For example, the > 7-limit NOT tuning for meantone is very close to 1/5-comma; this makes > the error for 3, weighted by log(3), equal to with opposite sign from > the error for 7 weighted by log(7). > > More anon, I think. Why? ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------Yahoo! Groups Links <*> To visit your group on the web, go to: Yahoo groups: /tuning-math/ * [with cont.] <*> To unsubscribe from this group, send an email to: tuning-math-unsubscribe@xxxxxxxxxxx.xxx <*> Your use of Yahoo! Groups is subject to: Yahoo! Terms of Service * [with cont.] (Wayb.)

Message: 11136 - Contents - Hide Contents Date: Mon, 28 Jun 2004 21:41:57 Subject: Re: HTT temperament From: George D. Secor --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> On Yahoo! Groups : MakeMicroMusic Messages * [with cont.] (Wayb.) > George Secor wrote: > > Gene, for your enlightenment: 29-HTT consists (except for one filler > tone) of 3 chains of fifths of ~703.5787c, or exactly (504/13)^ > (1/9). The 3 chains of fifths contain tones 1/1, 5/4, and 7/4, > respectively, and the tones in each chain are taken to as many places > as are required to result in otonal ogdoads on roots Bb, F, C, G, D, > and A. This also gives very-near-just diatonic (5-limit) scales in 5 > different keys. > > Since the fifth is (504/13)^(1/9), we immediately have that > > (504/13)/(3/2)^9 = 28672/28431 > > is a comma of the temperament, which must go up to the 13 limit at > least. It seems clear also that three of these slightly sharp fifths > are intended to represent 44/13, which means > > (44/13)/(3/2)^3 = 352/351 > > is another comma of the system. It does not appear any more commas are > intended, since the 5/4 and 7/4 are introduced as independent > generators.Yes. The essence of high-tolerance temperament (or HTT) is that the ratios of 7, 11, and 13 are all in a single chain of fifths, but the size of the fifth has been set so that 8:9 has the same error as 8:13, making 9:13 exact. This gives 8:11 and 7:11 almost the same error, which in turn makes 11:13 almost exact.> This means that the HTT temperament is a two-comma > temperament in the 13-limit; the TM basis for which turns out to be > 352/351 and 364/363. This is a spacial temperament, meaning one with > four generators, counting octaves. In this case we can take the > generators to be the approximation to 2,3,5,and 7, and the commas then > give us > > 11 ~ 896/81 > 13 ~ 28672/2187 > > Five is not a factor of the commas, so we can make this into a > no-fives system very easily.Yes. Margo does this by omitting the chain of fifths containing 5/4, which reduces the total number of tones to a number that she can put on two conventional keyboards, while maintaining the conventional octave distance.> The reduction to the 11-limit is > 896/891-spacial in the 11-limit, again of course a no-fives comma. > Aside from George's tuning, we have all the usual rms, minimax, TOP > etc. tunings if we want them. An interesting question is what 7- limit > JI scales would be good ones to temper using HTT;I don't know where you might go with that idea, since HTT was intended to bridge 7 with 11 and 11 with 13. --George

Message: 11137 - Contents - Hide Contents Date: Mon, 28 Jun 2004 23:51:13 Subject: Re: HTT temperament From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor" <gdsecor@y...> wrote:> Yes. The essence of high-tolerance temperament (or HTT) is that the > ratios of 7, 11, and 13 are all in a single chain of fifths, but the > size of the fifth has been set so that 8:9 has the same error as > 8:13, making 9:13 exact. This gives 8:11 and 7:11 almost the same > error, which in turn makes 11:13 almost exact.If 7 is an independent generator it isn't in the same chain of fifths. One way to put it in is via hemififths, where the no-fives commas have a basis 144/143, 243/242 and 364/363. Adding 196/195 to the mix gives you fives, or using it as a generator gives you a planar temperament. This, however, will not give you something exactly like HTT as you have described it, which seems to need 7 *not* to be in the chain of fifths.> Yes. Margo does this by omitting the chain of fifths containing 5/4, > which reduces the total number of tones to a number that she can put > on two conventional keyboards, while maintaining the conventional > octave distance.Hemififths has a 24-note MOS, which she could put in her two conventional keyboards very easily. Five would then be too complex to play much of a role, though there are plenty of 7/5s to play with. TOP tuning for it makes the error of five, over log 5, the same as the error in 13, over log 13. If we ignore 5 the TOP tuning now wants to make the error of 11, over log 11, the same as the error in 13, over log 13, and so 13/11 becomes nearly exact, which corresponds to your remark about the tuning of this interval in HTT. Making the octaves and 13/11 both pure gives a fifth of (44/13)^(1/3), and so a hemififths generator of (44/13)^(1/6). This is actually a little sharper even than the 58-et fifth, and this whole approach may not give you as much accuracy as you require. However, the 24 note MOS of 58-et with a 17 step generator is interesting; it has step sizes of 3 and 1: 3,3,3,1,3,3,1,3,3,3,1,3,3,1,3,3,3,1,3,3,1,3,3,1 if anyone feels inspired to try it.

Message: 11138 - Contents - Hide Contents Date: Mon, 28 Jun 2004 20:49:07 Subject: Re: Linear temperaments compatible with HTT From: Herman Miller Gene Ward Smith wrote:> I took a look at 13-limit linear temperaments which killed the commas > 352/351 and 364/363, and which had a TOP error less than 2.0. Below I > list some of these; the funny order is because they are listed in > order of a badness figure for the 11-limit reduction. As I expected, > mystery, Graham's discovery, put in an appearance, but the 13-limit > version of hemififths looks like a good choice also, allowing for a 41 > note DE as well as a 58 note one (or 99, if you really get worked up.) > Magic, shrutar and supersuper also show up. "Triton" is really better > in the 11-limit, an 896/891 temperament.The 13-limit hemififths looks pretty nice: the 41-note version is comparable to Erv Wilson's Cassandra 1 ( <<1, -8, -14, 23, 20, -15, -25, 33, 28, -10, 81, 76, 113, 108, -16]] ), but has a greater range of 13-limit chords available (15 complete hexads compared to only 4 for Cassandra 1[41]) Supersupermajor also has a good 41-note version, but is too complex to work very well with only 41. With only a few more notes (46), <<2, -4, 30, 22, 16, -11, 42, 28, 18, 81, 65, 52, -42, -66, -26]] has a better mapping. What really looks interesting is <<1, 21, 15, 11, 8, 31, 21, 14, 9, -24, -47, -59, -21, -33, -13]]. It actually looks a little better than nonkleismic, and the 29-note MOS could potentially use Graham Breed's fourth-based keyboard mapping.

Message: 11139 - Contents - Hide Contents Date: Mon, 28 Jun 2004 21:24:44 Subject: Some 13-limit temperaments From: Herman Miller Here are a few temperaments found by the brute-force approach (try all possible generators and periods until something interesting comes up). No. 1: Coendou? (a South American porcupine) 7L + 15s (period=1201.0c, generator=166.2c) [[1, 2, 3, 1, 4, 3], [0, -3, -5, 13, -4, 5]] <<3, 5, -13, 4, -5, 1, -29, -4, -19, -44, -8, -30, 56, 34, -32]] No. 2: Roman? (26 letters in the Roman alphabet) 3L + 23s (period=1200.6c, generator=414.5c) [[1, 4, 3, -1, 0, 3], [0, -7, -2, 11, 10, 2]] <<7, 2, -11, -10, -2, -13, -37, -40, -29, -31, -30, -12, 10, 35, 30]] No. 3: Leapday? (Feb. 29) 17L + 12s (period=1200.0c, generator=495.7c) [[1, 2, 11, 9, 8, 7], [0, -1, -21, -15, -11, -8]] <<1, 21, 15, 11, 8, 31, 21, 14, 9, -24, -47, -59, -21, -33, -13]] No. 4: Myna / Pangolin (formerly Nonkleismic) 27L + 4s (period=1198.5c, generator=310.0c) [[1, -1, 0, 1, -3, 5], [0, 10, 9, 7, 25, -5]] <<10, 9, 7, 25, -5, -9, -17, 5, -45, -9, 27, -45, 46, -40, -110]] No. 5: 26L + 6s (period=599.6c, generator=185.8c) [[2, 1, 0, 5, 6, 4], [0, 7, 15, 2, 3, 11]] <<14, 30, 4, 6, 22, 15, -33, -39, -17, -75, -90, -60, 3, 47, 54]] No. 6: 29L + 5s (period=1199.9c, generator=247.8c) [[1, 2, 5, 9, 8, 7], [0, -2, -13, -30, -22, -16]] <<2, 13, 30, 22, 16, 16, 42, 28, 18, 33, 6, -11, -42, -66, -26]] No. 7: Hitchcock? (39 steps) 7L + 32s (period=1201.1c, generator=339.7c) [[1, 3, 6, -2, 6, 2], [0, -5, -13, 17, -9, 6]] <<5, 13, -17, 9, -6, 9, -41, -3, -28, -76, -24, -62, 84, 46, -54]] No. 8: Supersupermajor 5L + 36s (period=1200.1c, generator=234.5c) [[1, 1, -1, 3, 6, 8], [0, 3, 17, -1, -13, -22]] <<3, 17, -1, -13, -22, 20, -10, -31, -46, -50, -89, -114, -33, -58, -28]] No. 9: Cassandra 1 12L + 29s (period=1200.2c, generator=498.0c) [[1, 2, -1, -3, 13, 12], [0, -1, 8, 14, -23, -20]] <<1, -8, -14, 23, 20, -15, -25, 33, 28, -10, 81, 76, 113, 108, -16]] No. 10: Hemififths 17L + 24s (period=1198.9c, generator=351.2c) [[1, 1, -5, -1, 2, 4], [0, 2, 25, 13, 5, -1]] <<2, 25, 13, 5, -1, 35, 15, 1, -9, -40, -75, -95, -31, -51, -22]] No. 11: 17L + 26s (period=1201.0c, generator=140.3c) [[1, 1, -2, 0, 1, 3], [0, 5, 37, 24, 21, 6]] <<5, 37, 24, 21, 6, 47, 24, 16, -9, -48, -79, -123, -24, -72, -57]] No. 12: 4L + 42s (period=600.5c, generator=287.8c) [[2, 7, 8, 8, 5, 5], [0, -8, -7, -5, 4, 5]] <<16, 14, 10, -8, -10, -15, -29, -68, -75, -16, -67, -75, -57, -65, -5]] No. 13: Diaschismic 12L + 34s (period=599.6c, generator=103.6c) [[2, 3, 5, 7, 9, 10], [0, 1, -2, -8, -12, -15]] <<2, -4, -16, -24, -30, -11, -31, -45, -55, -26, -42, -55, -12, -25, -15]] No. 14: (Number 33 from the 7-limit list) 26L + 20s (period=600.5c, generator=183.3c) [[2, 5, 8, 5, 6, 8], [0, -6, -11, 2, 3, -2]] <<12, 22, -4, -6, 4, 7, -40, -51, -38, -71, -90, -72, -3, 26, 36]] No. 15: 34L + 12s (period=599.7c, generator=104.9c) [[2, 3, 5, 3, 5, 6], [0, 1, -2, 15, 11, 8]] <<2, -4, 30, 22, 16, -11, 42, 28, 18, 81, 65, 52, -42, -66, -26]] No. 16: Vulture 5L + 43s (period=1199.3c, generator=475.4c) [[1, 0, -6, 4, -12, -7], [0, 4, 21, -3, 39, 27]] <<4, 21, -3, 39, 27, 24, -16, 48, 28, -66, 18, -15, 120, 87, -51]] No: 17: (Number 76 from the 7-limit list) 3L + 47s (period=1200.8c, generator=408.0c) [[1, 6, 3, 13, -3, 2], [0, -13, -2, -30, 19, 5]] <<13, 2, 30, -19, -5, -27, 11, -75, -56, 64, -51, -19, -157, -125, 53]] No. 18: 26L + 25s (period=1201.8c, generator=46.7c) [[1, 1, 1, 3, 4, 3], [0, 15, 34, -5, -14, 18]] <<15, 34, -5, -14, 18, 19, -50, -74, -27, -107, -150, -84, -22, 69, 114]] No. 19: 29L + 22s (period=1199.9c, generator=165.0c) [[1, 2, 7, 9, 8, 7], [0, -3, -34, -45, -33, -24]] <<3, 34, 45, 33, 24, 47, 63, 42, 27, 9, -41, -70, -63, -99, -39]] No. 20: 5L + 48s (period=1199.4c, generator=475.8c) [[1, 0, 17, 4, 11, 16], [0, 4, -37, -3, -19, -31]] <<4, -37, -3, -19, -31, -68, -16, -44, -64, 97, 84, 65, -43, -76, -37]] No. 21: Catakleismic (Hanson) 19L + 34s (period=1200.9c, generator=317.0c) [[1, 0, 1, -3, 9, 0], [0, 6, 5, 22, -21, 14]] <<6, 5, 22, -21, 14, -6, 18, -54, 0, 37, -66, 14, -135, -42, 126]]

Message: 11140 - Contents - Hide Contents Date: Mon, 28 Jun 2004 08:10:59 Subject: Linear temperaments compatible with HTT From: Gene Ward Smith I took a look at 13-limit linear temperaments which killed the commas 352/351 and 364/363, and which had a TOP error less than 2.0. Below I list some of these; the funny order is because they are listed in order of a badness figure for the 11-limit reduction. As I expected, mystery, Graham's discovery, put in an appearance, but the 13-limit version of hemififths looks like a good choice also, allowing for a 41 note DE as well as a 58 note one (or 99, if you really get worked up.) Magic, shrutar and supersuper also show up. "Triton" is really better in the 11-limit, an 896/891 temperament. Triton {325/324, 352/351, 364/363, 540/539} [7, 26, 25, -3, -24, 25, 20, -29, -64, -15, -97, -152, -95, -160, -72] [[1, 5, 15, 15, 2, -8], [0, -7, -26, -25, 3, 24]] 50 .647154 439.161631 Mystery {196/195, 352/351, 364/363, 676/675} [0, 29, 29, 29, 29, 46, 46, 46, 46, -14, -33, -40, -19, -26, -7] [[29, 46, 67, 81, 100, 107], [0, 0, 1, 1, 1, 1]] 29 .651558 178.354113 Hemififths {144/143, 196/195, 243/242, 352/351} [2, 25, 13, 5, -1, 35, 15, 1, -9, -40, -75, -95, -31, -51, -22] [[1, 1, -5, -1, 2, 4], [0, 2, 25, 13, 5, -1]] 26 1.056165 241.002003 "Number 80" [6, -12, 10, -14, -32, -33, -1, -43, -73, 57, 9, -30, -74, -127, -59] [[2, 4, 3, 7, 5, 3], [0, -3, 6, -5, 7, 16]] 44 .974054 534.159090 Shrutar [4, -8, 14, -2, -14, -22, 11, -17, -37, 55, 23, -3, -54, -91, -41] [[2, 3, 5, 5, 7, 8], [0, 2, -4, 7, -1, -7]] 28 1.845166 476.393162 Magic [5, 1, 12, -8, -23, -10, 5, -30, -55, 25, -22, -57, -64, -109, -50] [[1, 0, 2, -1, 6, 11], [0, 5, 1, 12, -8, -23]] 35 1.688049 632.167104 [1, 21, 15, 11, 8, 31, 21, 14, 9, -24, -47, -59, -21, -33, -13] [[1, 2, 11, 9, 8, 7], [0, -1, -21, -15, -11, -8]] 21 1.589486 254.071241 Supersupermajor [3, 17, -1, -13, -22, 20, -10, -31, -46, -50, -89, -114, -33, -58, -28] [[1, 1, -1, 3, 6, 8], [0, 3, 17, -1, -13, -22]] 39 .958363 429.837463 [2, -4, 30, 22, 16, -11, 42, 28, 18, 81, 65, 52, -42, -66, -26] [[2, 3, 5, 3, 5, 6], [0, 1, -2, 15, 11, 8]] 34 1.076337 384.072477 [2, -4, -16, -24, -30, -11, -31, -45, -55, -26, -42, -55, -12, -25, -15] [[2, 3, 5, 7, 9, 10], [0, 1, -2, -8, -12, -15]] 34 1.267597 452.320425

Message: 11141 - Contents - Hide Contents Date: Mon, 28 Jun 2004 01:59:45 Subject: Re: Absolute TOP error From: Carl Lumma>We've been doing the weighted TOP error as the error in cents divided >by the log base two of the product of the numerator and the >denominator. This is good for most purposes, but if we stick to the >same log for the ratio (both cents, or both log base two, etc.) then >we get something with a meaning independent of unit/log base choice. >It can be described as the logarithm, base N = the product of >numerator and denominator, of the error; Log_N(E). The reciprocal is >Log_E(N); it is how many steps of size E (the error) are required to >get to N (product of numerator and denominator.) In TOP tuning, there >is a minimum value for this which defines the relative error. The same >remarks apply for NOT tuning and the product of the numerator and >denominator of the odd part of the interval. > >If T is the TOP error by the definition we've been using, then 1200/T >is the minimum number of error-sized steps needed to get to the >product of numerator and denominator. For (5, 7, 11-limit) meantone >that would be 706.497 steps, for instance; miracle would be 1901.701 >and ennealimmal 32987.408.I think I understand some of this. How is it absolute? It still sounds weighted to me. -Carl

Message: 11142 - Contents - Hide Contents Date: Mon, 28 Jun 2004 02:01:18 Subject: Re: NOT tuning From: Carl Lumma>>> >OT being an acronym for No Octave Tempering. NOT tuning is TOP tuning >>> with the added constraint that octaves must be pure. For example, the >>> 7-limit NOT tuning for meantone is very close to 1/5-comma; this makes >>> the error for 3, weighted by log(3), equal to with opposite sign from >>> the error for 7 weighted by log(7). >>> >>> More anon, I think. >>>> Cool; I've been waiting for this. >>Cool; let's look at some results.Drat! I've lost Paul's comment on this. Did you see it? IIRC he accused you of measuring reciprocals differently.>Meantone > >5-limit: 698.0187 (43, 55, 98, 153, 251, 404) > >7-limit: 697.6458 (31, 43, 934, 977; 0.0286 cents flatter than 43)Hmm... I dunno, this seems a bit far from the old-style rms optimum. -Carl

Message: 11143 - Contents - Hide Contents Date: Mon, 28 Jun 2004 09:32:10 Subject: Re: Linear temperaments compatible with HTT From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> Hemififths {144/143, 196/195, 243/242, 352/351} > [2, 25, 13, 5, -1, 35, 15, 1, -9, -40, -75, -95, -31, -51, -22] [[1, > 1, -5, -1, 2, 4], [0, 2, 25, 13, 5, -1]] > 26 1.056165 241.002003The five mapping is the most complex, and 352/351 and 364/363 are no-fives commas. We can get a no-fives linear temperament which is the no-fives reduction of 13-limit hemififths from the commas 144/143, 243/242, 364/363. Margo might possibly find this of interest--the fifths are of low complexity (two generator steps), with 512/507 a comma of 13-limit hemififths, 16/13 being the generator and (16/13)^2/(3/2) = 512/507. It's a no-fives system with a sharp fifth, which is what she seems to favor. The Graham complexity is 14, which allows a lot of no-fives 13-limit chords in the 24-note DE, and even the 17 note DE isn't bad.

Message: 11144 - Contents - Hide Contents Date: Mon, 28 Jun 2004 09:33:51 Subject: Re: Absolute TOP error From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:> I think I understand some of this. How is it absolute? It still > sounds weighted to me.It's a pure number; no units of cents or whatever log base you use are used.

Message: 11145 - Contents - Hide Contents Date: Mon, 28 Jun 2004 09:39:10 Subject: Re: NOT tuning From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:> Drat! I've lost Paul's comment on this. Did you see it? IIRC he > accused you of measuring reciprocals differently.He didn't like my definition of odd Tenney height. Too bad, it's the only way this works, so we are stuck with it.>> Meantone >> >> 5-limit: 698.0187 (43, 55, 98, 153, 251, 404) >> >> 7-limit: 697.6458 (31, 43, 934, 977; 0.0286 cents flatter than 43) >> Hmm... I dunno, this seems a bit far from the old-style rms > optimum.It is in many cases. NOT meantone sneers at 6/5 and 5/3; and this is how it works in general, it just plain likes odd integer numerators better than ratios of two odd integers, which probably doesn't make a lot of sense and has to do with why Paul is complaining. ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ Yahoo! Groups Links <*> To visit your group on the web, go to: Yahoo groups: /tuning-math/ * [with cont.] <*> To unsubscribe from this group, send an email to: tuning-math-unsubscribe@xxxxxxxxxxx.xxx <*> Your use of Yahoo! Groups is subject to: Yahoo! Terms of Service * [with cont.] (Wayb.)

Message: 11146 - Contents - Hide Contents Date: Tue, 29 Jun 2004 22:57:49 Subject: Kernel intersections and unions From: Gene Ward Smith I just wrote some programs for finding kernel intersections and unions for two 7-limit linear temperaments, and may up the ante to the 11-limit, where things get more complicated, next. In the 7-limit, the Pfaffian of the two wedgies is either not zero, in which case the temperaments are unrelated and we get nothing in the intersection and a rank 4 union, or the Pfaffian is zero. In that case, the temperaments are related, and we get a single comma generating the intersection, and a single val definiting, by the commas it annihilates, the union. I took a list of 112 7-limit linear temperaments and got the usual suspects, for the most part, with temperaments related to meantone. By this I mean for the union I got the expected 5, 7, 12, 19 or 31. However, I also found three 50's and a 43. Meantone related via 50-et union: <<12 -2 20 -31 -2 52|| wizard 225/224 intersection <<13 2 30 -27 11 64|| 225/224 intersection <<11 -6 10 -35 -15 40|| 225/224 intersection Meantone related via 43-et union: <<7 -15 -16 -40 -45 -5|| 225/224 intersection Of course, 225/224 is one of the usual suspects when it comes to kernel intersections with meantone, but I got some oddballs there: Meantone related via 5103/5000 <<5 8 2 1 -11 -1|| 12 union Meantone related via 703125/702464 <<23 -1 13 -55 -44 33|| 31 union Meantone related via 3645/3584 <<1 -8 -2 -15 -6 18|| 12 union <<0 12 12 19 19 -6|| 12 union (duh)

Message: 11147 - Contents - Hide Contents Date: Tue, 29 Jun 2004 23:39:40 Subject: Mobs From: Gene Ward Smith Let us say that a set of three or more 7-limit linear temperaments forms a mob if each pair in the set is related by the same union and intersection of their kernels. For example, ennealimmal, hemiwuerscmidt and hemififths form a mob, with et 99 and comma 2401/2400. If we switch to et 72 and 2401/2400, we get another mob from ennealimmal, miracle and harry (the 2401/2400 and 19683/19600 temperament.) A mob can have many members; meantone, miracle, orwell, wuerschmidt, tritonic (meaning the <<5 -11 -12 -29 -33 3|| temperament) and slender are all mobbed up around 225/224 and 31 taken together, a mob obviously supplied with some serious muscle.

Message: 11148 - Contents - Hide Contents Date: Tue, 29 Jun 2004 03:30:25 Subject: Re: Linear temperaments compatible with HTT From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Herman Miller <hmiller@I...> wrote:> What really looks interesting is <<1, 21, 15, 11, 8, 31, 21, 14, 9, -24, > -47, -59, -21, -33, -13]]. It actually looks a little better than > nonkleismic, and the 29-note MOS could potentially use Graham Breed's > fourth-based keyboard mapping.It's also got a fifth as a generator, a big plus for Margo and George I suspect. Here's another possibility: 352/351 is a {2,3,11,13} comma; if you put it together with another such, you get a {2,3,11,13} temperament, the mapping of which you can then extend. 144/143 is one choice for this, leading to hemififths. It's a nice comma but it lowers the accuracy 352/351 gives some. Another possibility is 2197/2187 = 13^3/3^7. This starts the mapping game off with a generator which is approximately 13/9, about two cents narrow, and we now have 3 ~ (13/9)^3 11 ~ 2^(-5) (13/9)^16 13 ~ (13/9)^7 This is a {2,3,11,13} linear temperament with a Graham complexity of 16. If we now add 364/363 we will get a no-fives linear temperament compatible with HTT, and mapping 7 ~ 2^(-12) (13/9)^28 The 36 note MOS would seem like a good choice with this. If you absolutely must, you can add 5 by including 245/243 among the commas, giving 5 ~ 2^24 (13/9)^(-41) At this point you should probably just switch to 87-equal and call it a day. George's preferred 3 approximation for HTT is 2 (504/13)^(1/9) = (258048/13)^(1/9), so one possible tuning would have a generator of size (258048/13)^(1/27), which works fine, giving something whose tuning works better as an HTT extension (much more accurate for 11 and 13 in particular) than 13-limit hemififths.

Message: 11149 - Contents - Hide Contents Date: Tue, 29 Jun 2004 00:30:36 Subject: An early reference to 13-lemba From: Herman Miller I've been going through old tuning-math posts looking for 13-limit stuff. Check out the last temperament in the list at: Yahoo groups: /tuning-math/message/4916 * [with cont.]>[[2, 2, 5, 6, 5, 7], [0, 3, -1, -1, 5, 1]] [600.0000000, 231.2498543] > >rms 12.34827552 comp 8.160882306 graham 14 > >bad 287.8803242 grabad 646.8422283 > >mos [10, 16, 26, 36, 62, 88]This is the same mapping for 13-lemba that I posted to the tuning list. It only took a year and a half for me to notice it!

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