This is an Opt In Archive . We would like to hear from you if you want your posts included. For the contact address see About this archive. All posts are copyright (c).
- Contents - Hide Contents - Home - Section 76000 6050 6100 6150 6200 6250 6300 6350 6400 6450 6500 6550 6600 6650 6700 6750 6800 6850 6900 6950
6650 - 6675 -
Message: 6650 - Contents - Hide Contents Date: Sun, 23 Mar 2003 07:15:04 Subject: Re: T[n] where n is small From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:> Gene, > > What about turning this on scales, n < 11?I can, but I'm not sure how much, if any, of that will be new.> 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.I could send you some Maple code I've written for various things if you have access to Maple.
Message: 6651 - Contents - Hide Contents Date: Sun, 23 Mar 2003 08:22:11 Subject: T[7] with 25/24 chroma From: Gene Ward Smith Here's something for Carl: Meantone[7] (Diatonic) As they say about Kellogg's Corn Flakes, "The Original and Best" 81/80 [1, 9/8, 6/5, 4/3, 3/2, 5/3, 9/5] [6/5, 9/5, 4/3, 1, 3/2, 9/8, 5/3] [1, 5/4, 3/2] [1, 5/4, 3/2] [1, 5/4, 3/2] [1, 6/5, 3/2] [1, 6/5, 3/2] [1, 6/5, 3/2] [1, 6/5, 36/25] Pelogic[7] 135/128 [1, 9/8, 5/4, 4/3, 3/2, 8/5, 15/8] [5/4, 15/8, 4/3, 1, 3/2, 9/8, 8/5] Generator = 3/2 [1, 6/5, 3/2] [1, 6/5, 3/2] [1, 6/5, 3/2] [1, 5/4, 3/2] [1, 5/4, 3/2] [1, 5/4, 3/2] [1, 5/4, 25/16] Porcupine[7] 250/243 [1, 10/9, 6/5, 4/3, 3/2, 5/3, 9/5] [3/2, 5/3, 9/5, 1, 10/9, 6/5, 4/3] Generator = 10/9 [1, 6/5, 36/25] [1, 6/5, 36/25] [1, 6/5, 36/25] [1, 6/5, 3/2] [1, 6/5, 3/2] [1, 5/4, 3/2] [1, 5/4, 3/2]
Message: 6652 - Contents - Hide Contents Date: Sun, 23 Mar 2003 00:50:10 Subject: Re: T[7] with 25/24 chroma From: Carl Lumma>Here's something for Carl:Cool. That means it's working. ;) What happens when n is not a MOS or NMOS point? -Carl
Message: 6653 - Contents - Hide Contents Date: Sun, 23 Mar 2003 23:33:16 Subject: Re: T[7] with 25/24 chroma From: Carl Lumma>i'm proud to say i've created music in all three!Do you have finished pieces in Porcupine or Pelogic? -Ca.
Message: 6654 - Contents - Hide Contents Date: Sun, 23 Mar 2003 23:45:03 Subject: Re: this T[n] business From: Carl Lumma>that's the great thing about tenney complexity (as opposed to >farey, mann, etc.)! // > >something that *does* work like tenney in this way is the odd limit.Hmm, this doesn't seem right; the diamond pitches aren't very regular at all. IIRC they get thicker and thicker as they approach the identity, but then there's a gap before hitting it. -Carl
Message: 6655 - Contents - Hide Contents Date: Sun, 23 Mar 2003 00:07:03 Subject: Re: Beatles[17] and squares[17] From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus" > <wallyesterpaulrus@y...> wrote: >>> oh, the uniformity is because you put the tetrads in generator circle >> order, isn't it? >> I was for a while, but it turns out it's better to stick with > generator circles. >>> what does 36/35 have to do with it? >> 36/35 exchanges 7/6<-->6/5, 5/4<-->9/7, 5/3<-->12/7, 7/4<-->9/5 as we > move from one kind of tetrad to another.right, but the uniformity would happen no matter what you used. another appealing possibility, important in the whole version of this discussion that we were having before with kalle, is 49:48, which exchanges 7/4 <-> 12/7, 7/6 <-> 8/7, and in particular those systems where 50:49 is one of the commatic unison vectors, such that the chromatic unison vector can also be expressed as 25:24, leading to 6/5 <-> 5/4 and 8/5 <-> 5/3.
Message: 6656 - Contents - Hide Contents Date: Sun, 23 Mar 2003 00:16:17 Subject: Re: Duodecimal[36] From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> I could repeat this twice more, but you get the idea--all tetrads are > alike.Duh! Of course, they are not--the top circle and bottom circle each have their own sort of tetrad: Circles of fifths Middle [1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3] [1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3] [1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3] [1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3] [1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3] [1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3] [1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3] [1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3] [1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3] [1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3] [1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3] [1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3] Top [1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3] [1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3] [1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3] [1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3] [1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3] [1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3] [1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3] [1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3] [1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3] [1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3] [1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3] [1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3] Bottom [1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7] [1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7] [1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7] [1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7] [1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7] [1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7] [1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7] [1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7] [1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7] [1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7] [1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7] [1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
Message: 6657 - Contents - Hide Contents Date: Sun, 23 Mar 2003 00:29:47 Subject: Re: Beatles[17] and squares[17] From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:> right, but the uniformity would happen no matter what you used. > another appealing possibility, important in the whole version of this > discussion that we were having before with kalle, is 49:48, which > exchanges 7/4 <-> 12/7, 7/6 <-> 8/7, and in particular those systems > where 50:49 is one of the commatic unison vectors, such that the > chromatic unison vector can also be expressed as 25:24, leading to > 6/5 <-> 5/4 and 8/5 <-> 5/3.What strikes me as more interesting than that is 21/20, leading to 10/7<-->3/2, 8/7<-->6/5, 5/3<-->7/4, 12/7<-->9/5, 10/9<-->7/6 and which can covert a tetrad into an ass. I think I'll tackle it next.
Message: 6658 - Contents - Hide Contents Date: Sun, 23 Mar 2003 01:36:59 Subject: Magic[16] From: Gene Ward Smith This is the first 21/20 chroma scale I've looked at, and it's a beaut, fully justifying my interest in 21/20. While Rothenberg might not approve of it, check out the masses of asses, and note that the single "wolf" third hardly deserves the name. Magic[16] [5, 1, 12, -10, 5, 25] [1, 25/24, 15/14, 7/6, 6/5, 5/4, 9/7, 4/3, 35/24, 3/2, 14/9, 8/5, 5/3, 12/7, 15/8, 27/14] [12/7, 15/14, 4/3, 5/3, 25/24, 9/7, 8/5, 1, 5/4, 14/9, 27/14, 6/5, 3/2, 15/8, 7/6, 35/24] Circle of 5/4's [1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 9/5] [1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 9/5] [1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 9/5] [1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 9/5] [1, 5/4, 3/2, 5/3] [1, 6/5, 3/2, 9/5] [1, 5/4, 3/2, 5/3] [1, 6/5, 3/2, 9/5] [1, 5/4, 3/2, 5/3] [1, 6/5, 3/2, 9/5] [1, 5/4, 3/2, 5/3] [1, 6/5, 3/2, 12/7] [1, 5/4, 3/2, 5/3] [1, 6/5, 3/2, 12/7] [1, 5/4, 3/2, 5/3] [1, 6/5, 3/2, 12/7] [1, 5/4, 3/2, 5/3] [1, 6/5, 3/2, 12/7] [1, 5/4, 10/7, 5/3] [1, 6/5, 10/7, 12/7] [1, 5/4, 10/7, 5/3] [1, 8/7, 10/7, 12/7] [1, 5/4, 10/7, 5/3] [1, 8/7, 10/7, 12/7] [1, 5/4, 10/7, 5/3] [1, 8/7, 10/7, 12/7] [1, 25/21, 10/7, 5/3] [1, 8/7, 10/7, 12/7]
Message: 6659 - Contents - Hide Contents Date: Sun, 23 Mar 2003 02:17:10 Subject: Orwell[18] From: Gene Ward Smith You may recall this being discussed on the nmos thread. I've added two more circles of tetrads which may be of interest. Orwell[18] [7, -3, 8, -21, -7, 27] [1, 36/35, 15/14, 35/32, 8/7, 7/6, 5/4, 9/7, 4/3, 48/35, 35/24, 3/2, 14/9, 8/5, 12/7, 7/4, 49/27, 15/8] [8/7, 4/3, 14/9, 49/27, 15/14, 5/4, 35/24, 12/7, 1, 7/6, 48/35, 8/5, 15/8, 35/32, 9/7, 3/2, 7/4, 36/35] Circles of tetrads, generator 7/6 [1, 21/16, 3/2, 7/4] [1, 6/5, 3/2, 9/5] [1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 7/4] [1, 21/16, 3/2, 7/4] [1, 6/5, 3/2, 12/7] [1, 9/7, 3/2, 12/7] [1, 7/6, 3/2, 7/4] [1, 21/16, 3/2, 7/4] [1, 6/5, 3/2, 12/7] [1, 9/7, 3/2, 12/7] [1, 7/6, 3/2, 7/4] [1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 12/7] [1, 9/7, 3/2, 12/7] [1, 7/6, 3/2, 7/4] [1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 12/7] [1, 9/7, 3/2, 12/7] [1, 7/6, 3/2, 7/4] [1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 12/7] [1, 9/7, 3/2, 12/7] [1, 7/6, 3/2, 7/4] [1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 12/7] [1, 9/7, 3/2, 12/7] [1, 7/6, 3/2, 7/4] [1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 12/7] [1, 9/7, 3/2, 12/7] [1, 7/6, 3/2, 7/4] [1, 5/4, 3/2, 7/4] [1, 8/7, 3/2, 12/7] [1, 9/7, 3/2, 12/7] [1, 7/6, 3/2, 7/4] [1, 5/4, 3/2, 7/4] [1, 8/7, 3/2, 12/7] [1, 9/7, 3/2, 12/7] [1, 7/6, 3/2, 7/4] [1, 5/4, 3/2, 5/3] [1, 8/7, 3/2, 12/7] [1, 9/7, 3/2, 12/7] [1, 7/6, 3/2, 5/3] [1, 5/4, 10/7, 5/3] [1, 8/7, 10/7, 12/7] [1, 9/7, 10/7, 12/7] [1, 7/6, 10/7, 5/3] [1, 5/4, 10/7, 5/3] [1, 8/7, 10/7, 12/7] [1, 60/49, 10/7, 12/7] [1, 7/6, 10/7, 5/3] [1, 5/4, 10/7, 5/3] [1, 8/7, 10/7, 12/7] [1, 60/49, 10/7, 12/7] [1, 7/6, 10/7, 5/3] [1, 5/4, 10/7, 5/3] [1, 8/7, 10/7, 12/7] [1, 60/49, 10/7, 12/7] [1, 7/6, 10/7, 5/3] [1, 5/4, 10/7, 5/3] [1, 8/7, 10/7, 12/7] [1, 60/49, 10/7, 12/7] [1, 7/6, 10/7, 5/3] [1, 5/4, 10/7, 5/3] [1, 8/7, 10/7, 12/7] [1, 60/49, 10/7, 12/7] [1, 7/6, 10/7, 5/3] [1, 5/4, 10/7, 5/3] [1, 8/7, 10/7, 12/7] [1, 60/49, 10/7, 12/7] [1, 10/9, 10/7, 5/3]
Message: 6660 - Contents - Hide Contents Date: Sun, 23 Mar 2003 03:30:13 Subject: An example of 25/24 chroma From: Gene Ward Smith If we were to be completely systematic, the place to have begun this investigation would have been the 5-limit and 25/24 as a chroma; as Paul pointed out, the locus classicus for all of this is 81/80 as a comma, 25/24 as a chroma, leading to 81/80[7], better known as the diatonic scale. There are other possibilities, some known and some completely unknown. I give an example in the latter category. Name?[89] This may have been given one, but I couldn't find it Comma 381520424476945831628649898809/381469726562500000000000000000 Scale [1, 31381059609/31250000000, 1594323/1562500, 7812500000000/7625597484987, 250/243, 129140163/125000000, 6561/6250, 205891132094649/195312500000000, 62500/59049, 27/25, 847288609443/781250000000, 15625000/14348907, 312500000000/282429536481, 10/9, 3486784401/3125000000, 3906250000/3486784401, 78125000000000/68630377364883, 2500/2187, 14348907/12500000, 729/625, 22876792454961/19531250000000, 625000/531441, 59049/50000, 6/5, 94143178827/78125000000, 156250000/129140163, 3125000000000/2541865828329, 100/81, 387420489/312500000, 39062500000/31381059609, 617673396283947/488281250000000, 25000/19683, 1594323/1250000, 2541865828329/1953125000000, 6250000/4782969, 6561/5000, 4/3, 10460353203/7812500000, 1562500000/1162261467, 27/20, 1000/729, 43046721/31250000, 390625000000/282429536481, 68630377364883/48828125000000, 250000/177147, 177147/125000, 97656250000000/68630377364883, 282429536481/195312500000, 62500000/43046721, 729/500, 40/27, 1162261467/781250000, 15625000000/10460353203, 3/2, 10000/6561, 4782969/3125000, 3906250000000/2541865828329, 2500000/1594323, 19683/12500, 617673396283947/390625000000000, 31381059609/19531250000, 625000000/387420489, 81/50, 2541865828329/1562500000000, 129140163/78125000, 156250000000/94143178827, 5/3, 100000/59049, 531441/312500, 39062500000000/22876792454961, 1250/729, 25000000/14348907, 2187/1250, 68630377364883/39062500000000, 3486784401/1953125000, 6250000000/3486784401, 9/5, 282429536481/156250000000, 14348907/7812500, 1562500000000/847288609443, 50/27, 59049/31250, 390625000000000/205891132094649, 12500/6561, 250000000/129140163, 243/125, 7625597484987/3906250000000, 3125000/1594323, 62500000000/31381059609] Generator order [3906250000/3486784401, 27/20, 2541865828329/1562500000000, 3125000/1594323, 59049/50000, 97656250000000/68630377364883, 1250/729, 129140163/125000000, 39062500000/31381059609, 3/2, 282429536481/156250000000, 15625000/14348907, 6561/5000, 617673396283947/390625000000000, 12500/6561, 14348907/12500000, 390625000000/282429536481, 5/3, 31381059609/31250000000, 156250000/129140163, 729/500, 68630377364883/39062500000000, 62500/59049, 1594323/1250000, 3906250000000/2541865828329, 50/27, 3486784401/3125000000, 1562500000/1162261467, 81/50, 7625597484987/3906250000000, 625000/531441, 177147/125000, 39062500000000/22876792454961, 250/243, 387420489/312500000, 15625000000/10460353203, 9/5, 847288609443/781250000000, 6250000/4782969, 19683/12500, 390625000000000/205891132094649, 2500/2187, 43046721/31250000, 156250000000/94143178827, 1, 94143178827/78125000000, 62500000/43046721, 2187/1250, 205891132094649/195312500000000, 25000/19683, 4782969/3125000, 1562500000000/847288609443, 10/9, 10460353203/7812500000, 625000000/387420489, 243/125, 22876792454961/19531250000000, 250000/177147, 531441/312500, 7812500000000/7625597484987, 100/81, 1162261467/781250000, 6250000000/3486784401, 27/25, 2541865828329/1953125000000, 2500000/1594323, 59049/31250, 78125000000000/68630377364883, 1000/729, 129140163/78125000, 62500000000/31381059609, 6/5, 282429536481/195312500000, 25000000/14348907, 6561/6250, 617673396283947/488281250000000, 10000/6561, 14348907/7812500, 312500000000/282429536481, 4/3, 31381059609/19531250000, 250000000/129140163, 729/625, 68630377364883/48828125000000, 100000/59049, 1594323/1562500, 3125000000000/2541865828329, 40/27, 3486784401/1953125000] Generator = 94143178827/78125000000 Generator circle of triads [1, 6/5, 36/25] [1, 6/5, 36/25] [1, 6/5, 36/25] [1, 6/5, 36/25] [1, 6/5, 36/25] [1, 6/5, 36/25] [1, 6/5, 36/25] [1, 6/5, 36/25] [1, 6/5, 36/25] [1, 6/5, 36/25] [1, 6/5, 36/25] [1, 6/5, 36/25] [1, 6/5, 36/25] [1, 6/5, 36/25] [1, 6/5, 36/25] [1, 6/5, 36/25] [1, 6/5, 36/25] [1, 6/5, 36/25] [1, 6/5, 36/25] [1, 6/5, 36/25] [1, 6/5, 36/25] [1, 6/5, 36/25] [1, 6/5, 36/25] [1, 6/5, 36/25] [1, 6/5, 36/25] [1, 6/5, 36/25] [1, 6/5, 36/25] [1, 6/5, 36/25] [1, 6/5, 36/25] [1, 6/5, 36/25] [1, 6/5, 36/25] [1, 6/5, 36/25] [1, 6/5, 36/25] [1, 6/5, 36/25] [1, 6/5, 36/25] [1, 6/5, 3/2] [1, 6/5, 3/2] [1, 6/5, 3/2] [1, 6/5, 3/2] [1, 6/5, 3/2] [1, 6/5, 3/2] [1, 6/5, 3/2] [1, 6/5, 3/2] [1, 6/5, 3/2] [1, 6/5, 3/2] [1, 6/5, 3/2] [1, 6/5, 3/2] [1, 6/5, 3/2] [1, 6/5, 3/2] [1, 6/5, 3/2] [1, 6/5, 3/2] [1, 6/5, 3/2] [1, 6/5, 3/2] [1, 6/5, 3/2] [1, 6/5, 3/2] [1, 6/5, 3/2] [1, 6/5, 3/2] [1, 6/5, 3/2] [1, 6/5, 3/2] [1, 6/5, 3/2] [1, 6/5, 3/2] [1, 6/5, 3/2] [1, 5/4, 3/2] [1, 5/4, 3/2] [1, 5/4, 3/2] [1, 5/4, 3/2] [1, 5/4, 3/2] [1, 5/4, 3/2] [1, 5/4, 3/2] [1, 5/4, 3/2] [1, 5/4, 3/2] [1, 5/4, 3/2] [1, 5/4, 3/2] [1, 5/4, 3/2] [1, 5/4, 3/2] [1, 5/4, 3/2] [1, 5/4, 3/2] [1, 5/4, 3/2] [1, 5/4, 3/2] [1, 5/4, 3/2] [1, 5/4, 3/2] [1, 5/4, 3/2] [1, 5/4, 3/2] [1, 5/4, 3/2] [1, 5/4, 3/2] [1, 5/4, 3/2] [1, 5/4, 3/2] [1, 5/4, 3/2] [1, 5/4, 3/2]
Message: 6661 - Contents - Hide Contents Date: Sun, 23 Mar 2003 04:25:49 Subject: Amity[21] and Hemisixths[11] From: Gene Ward Smith It should be noted when examining these that these are accurate temperaments. Hemisixths has an rms error of 1.575 cents, and amity, with an error of 0.383 cents, might be considered in the microtemperament range, though not to the extent of the nameless[89] system presented previously, which is 0.003 cents. Amity[21] 1600000/1594323 [1, 81/80, 800/729, 10/9, 9/8, 6/5, 243/200, 100/81, 4/3, 27/20, 1000/729, 729/500, 40/27, 3/2, 81/50, 400/243, 5/3, 16/9, 9/5, 729/400, 160/81] [9/8, 1000/729, 5/3, 81/80, 100/81, 3/2, 729/400, 10/9, 27/20, 400/243, 1, 243/200, 40/27, 9/5, 800/729, 4/3, 81/50, 160/81, 6/5, 729/500, 16/9] Generator = 243/200 [1, 6/5, 36/25] [1, 6/5, 36/25] [1, 6/5, 36/25] [1, 6/5, 36/25] [1, 6/5, 36/25] [1, 6/5, 3/2] [1, 6/5, 3/2] [1, 6/5, 3/2] [1, 6/5, 3/2] [1, 6/5, 3/2] [1, 6/5, 3/2] [1, 6/5, 3/2] [1, 6/5, 3/2] [1, 5/4, 3/2] [1, 5/4, 3/2] [1, 5/4, 3/2] [1, 5/4, 3/2] [1, 5/4, 3/2] [1, 5/4, 3/2] [1, 5/4, 3/2] [1, 5/4, 3/2] Hemisixths[11] 78732/78125 [1, 27/25, 10/9, 6/5, 162/125, 25/18, 36/25, 125/81, 5/3, 9/5, 50/27] [10/9, 36/25, 50/27, 6/5, 125/81, 1, 162/125, 5/3, 27/25, 25/18, 9/5] Generator = 162/125 [1, 5/4, 3/2] [1, 5/4, 3/2] [1, 6/5, 3/2] [1, 6/5, 3/2] [1, 6/5, 36/25] [1, 6/5, 36/25] [1, 6/5, 36/25] [1, 6/5, 36/25] [1, 6/5, 36/25] [1, 6/5, 36/25] [1, 6/5, 36/25]
Message: 6662 - Contents - Hide Contents Date: Sun, 23 Mar 2003 04:33:10 Subject: Re: this T[n] business From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>> 2. The last version of this thread thread (see msg. 6017) >>> left off with how to identify important commas. Gene's >>> mentioned square and triangular numbers as being better, >>> though I'm not sure why... >> >> don't know >> I suppose squareness and triangularity are types of > compositeness.It's nothing to worry about; n^2/(n^2-1) are commas between 9/8,8/7,7/6,6/5,5/4 etc. whereas Triangle(n)/(Triangle(n-1)) are commas between 3/2,5/3,7/4,9/5 etc. These are the ones which are going to turn up in this business, but it doesn't particularly help to know that. If we know that 15/14, 16/15, 21/20, 25/24, 28/27, 36/35, 45/44, 49/48 are worth checking, it doesn't add much to say that all of the numerators are eithers squares or triangles.
Message: 6663 - Contents - Hide Contents Date: Sun, 23 Mar 2003 04:56:53 Subject: Circle of fifths for amity[21] From: Gene Ward Smith Paul suggested looking at the circle of fifths for these things might be of interest, and since amity[21] has exactly three times the notes of meantone[7], this seems like a good place to start. You can see the funny effect a direct translation of diatonic music to amity[21] would apparently have; it might be worth a listen. Amity[21] circle of fifths [1, 5/4, 3/2] [1, 5/4, 3/2] [1, 6/5, 3/2] [1, 6/5, 3/2] [1, 6/5, 36/25] [1, 5/4, 3/2] [1, 6/5, 3/2] [1, 6/5, 3/2] [1, 6/5, 36/25] [1, 5/4, 3/2] [1, 6/5, 3/2] [1, 6/5, 3/2] [1, 6/5, 36/25] [1, 5/4, 3/2] [1, 5/4, 3/2] [1, 6/5, 3/2] [1, 6/5, 36/25] [1, 5/4, 3/2] [1, 5/4, 3/2] [1, 6/5, 3/2] [1, 6/5, 36/25]
Message: 6664 - Contents - Hide Contents Date: Mon, 24 Mar 2003 17:54:33 Subject: Re: Diaschismic, Negri and Blackwood 10 From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:> Gene Ward Smith wrote: > Sorry, I've lost the original message. 34-equal isn't consistent in the > 7-limit, so it won't define the system. Do you mean 46&58? > > 2 3 5 7 > 0 1 -2 -8Yes; I'm afraid all I meant is that h34^h46 (standard vals) will give the wedgie.
Message: 6665 - Contents - Hide Contents Date: Mon, 24 Mar 2003 17:57:20 Subject: Pajara[10] From: Gene Ward Smith This is a 25/24 chroma system, as well as being a 28/27 and 49/48 chroma system, so I don't give circles for major and minor tetrads, but rather another circle for subminorish stuff. Pajara[10] [2, -4, -4, -11, -12, 2] [1, 15/14, 8/7, 5/4, 4/3, 7/5, 3/2, 8/5, 7/4, 15/8] [[7/4, 4/3, 1, 3/2, 8/7], [5/4, 15/8, 7/5, 15/14, 8/5]] Circles of fifths [1, 6/5, 3/2, 12/7] [1, 8/7, 3/2, 8/5] [1, 6/5, 3/2, 12/7] [1, 8/7, 3/2, 8/5] [1, 5/4, 3/2, 7/4] [1, 8/7, 3/2, 8/5] [1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3] [1, 5/4, 14/9, 7/4] [1, 7/6, 14/9, 5/3]
Message: 6666 - Contents - Hide Contents Date: Mon, 24 Mar 2003 03:31:15 Subject: Tertiathirds[9] From: Gene Ward Smith When checking this out, bear in mind that 49/48 is a comma of tertiathirds, and so [1, 6/5, 3/2, 7/4] counts as a [1, 6/5, 3/2, 12/7] and etc. Tertiathirds[9] rms error 12.189 cents [4, -3, 2, -14, -8, 13] [1, 15/14, 7/6, 5/4, 4/3, 3/2, 8/5, 7/4, 15/8] [3/2, 8/5, 7/4, 15/8, 1, 15/14, 7/6, 5/4, 4/3] Generator = secor (15/14 ~ 16/15) [1, 5/4, 10/7, 5/3] [1, 7/6, 10/7, 5/3] [1, 5/4, 10/7, 5/3] [1, 7/6, 10/7, 5/3] [1, 5/4, 10/7, 7/4] [1, 7/6, 10/7, 7/4] [1, 5/4, 10/7, 7/4] [1, 7/6, 10/7, 7/4] [1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 7/4] [1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 7/4] [1, 9/7, 3/2, 7/4] [1, 7/6, 3/2, 7/4] [1, 9/7, 3/2, 7/4] [1, 6/5, 3/2, 7/4] [1, 9/7, 3/2, 7/4] [1, 6/5, 3/2, 7/4]
Message: 6667 - Contents - Hide Contents Date: Mon, 24 Mar 2003 21:51:38 Subject: Re: Pajara[10] From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> This is a 25/24 chroma system, as well as being a 28/27 and 49/48 > chroma system,as shown in "the forms of tonality"> so I don't give circles for major and minor tetrads, > but rather another circle for subminorish stuff.i don't get the logic of the "so". what implies what, exactly?> > Pajara[10] > > [2, -4, -4, -11, -12, 2] > [1, 15/14, 8/7, 5/4, 4/3, 7/5, 3/2, 8/5, 7/4, 15/8] > [[7/4, 4/3, 1, 3/2, 8/7], [5/4, 15/8, 7/5, 15/14, 8/5]] > > Circles of fifths > [1, 6/5, 3/2, 12/7] [1, 8/7, 3/2, 8/5] > [1, 6/5, 3/2, 12/7] [1, 8/7, 3/2, 8/5] > [1, 5/4, 3/2, 7/4] [1, 8/7, 3/2, 8/5] > [1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3] > [1, 5/4, 14/9, 7/4] [1, 7/6, 14/9, 5/3]
Message: 6668 - Contents - Hide Contents Date: Mon, 24 Mar 2003 07:00:17 Subject: Rule of 36 From: Gene Ward Smith Here is a curiosity--25/24 is very nearly three Pythagorean commas, the difference being the monzizma, a microcomma. Hence a temperament with pure or almost pure fifths as generator, which closes or nearly closes around a circle of N fifths, will have a "wolf" fifth about 25/24 sharp, making an augmented triad [1,5/4,25/16], for a temperament a multiple of N-36 notes, and one about 25/24 flat, making a diminised "wolf" triad [1,6/5,36/25], for a temperament a multiple of N+36 notes. The multiple aspect arises since the temperament may not have octave period. Taking the continued fraction for log2(3/2), we get 41-36=5, leading to diaschismic, 53-36=17, leading to schismic, 41+36=77 and 53+36=89 note scales; then 306+-36, 665+-36 and so on.
Message: 6669 - Contents - Hide Contents Date: Mon, 24 Mar 2003 07:01:45 Subject: Re: this T[n] business From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>>> So 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. > > -Carlsomething that *does* work like tenney in this way is the odd limit.
Message: 6670 - Contents - Hide Contents Date: Mon, 24 Mar 2003 07:04:48 Subject: Re: T[7] with 25/24 chroma From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> Here's something for Carl: > > Meantone[7] (Diatonic) > As they say about Kellogg's Corn Flakes, "The Original and Best" > > 81/80 > [1, 9/8, 6/5, 4/3, 3/2, 5/3, 9/5] > [6/5, 9/5, 4/3, 1, 3/2, 9/8, 5/3] > > [1, 5/4, 3/2] > [1, 5/4, 3/2] > [1, 5/4, 3/2] > [1, 6/5, 3/2] > [1, 6/5, 3/2] > [1, 6/5, 3/2] > [1, 6/5, 36/25] > > Pelogic[7] > > 135/128 > [1, 9/8, 5/4, 4/3, 3/2, 8/5, 15/8] > [5/4, 15/8, 4/3, 1, 3/2, 9/8, 8/5] > > Generator = 3/2 > [1, 6/5, 3/2] > [1, 6/5, 3/2] > [1, 6/5, 3/2] > [1, 5/4, 3/2] > [1, 5/4, 3/2] > [1, 5/4, 3/2] > [1, 5/4, 25/16] > > Porcupine[7] > > 250/243 > [1, 10/9, 6/5, 4/3, 3/2, 5/3, 9/5] > [3/2, 5/3, 9/5, 1, 10/9, 6/5, 4/3] > > Generator = 10/9 > [1, 6/5, 36/25] > [1, 6/5, 36/25] > [1, 6/5, 36/25] > [1, 6/5, 3/2] > [1, 6/5, 3/2] > [1, 5/4, 3/2] > [1, 5/4, 3/2]i'm proud to say i've created music in all three!
Message: 6671 - Contents - Hide Contents Date: Mon, 24 Mar 2003 07:08:16 Subject: Re: Diaschismic, Negri and Blackwood 10 From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> Diaschismic[10] looks like an attractive alternative to >meantone[7].especially if you look though 7-limit glasses :) :) there is evidence that the original 10 pitches implied by the ancient indian music treatises formed a 10-tone diaschsimic scale, though the diaschsima was probably only used to "average out" 16/15 and 135/128, the rest of the scale remaining just. see my paper.
Message: 6672 - Contents - Hide Contents Date: Mon, 24 Mar 2003 07:15:41 Subject: Re: Diaschismic, Negri and Blackwood 10 From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> > wrote: >>> Diaschismic[10] looks like an attractive alternative to >> meantone[7]. >> especially if you look though 7-limit glasses :) :)Very true--did I put something up about diaschismic[10] in the 7-limit yet?> there is evidence that the original 10 pitches implied by the ancient > indian music treatises formed a 10-tone diaschsimic scale, though the > diaschsima was probably only used to "average out" 16/15 and 135/128, > the rest of the scale remaining just. see my paper.Whether they did so or not, it's a very nice system with a stamp of approval from the Rule of 36, in case that helps.
Message: 6673 - Contents - Hide Contents Date: Mon, 24 Mar 2003 07:57:43 Subject: Re: Diaschismic, Negri and Blackwood 10 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: >>>>> Diaschismic[10] looks like an attractive alternative to >>> meantone[7]. >>>> especially if you look though 7-limit glasses :) :) >> Very true--did I put something up about diaschismic[10] in the 7-limit > yet?i wrote an entire paper about it :)
Message: 6674 - Contents - Hide Contents Date: Mon, 24 Mar 2003 08:00:21 Subject: Re: T[7] with 25/24 chroma From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>> i'm proud to say i've created music in all three! >> Do you have finished pieces in Porcupine or Pelogic? > > -Ca.part of "glassic" is in porcupine. you can hear the consecutive triads -- major, major, minor, minor -- as the root descends stepwise from "8ve" to "5th". thus part of you wants to believe it's mixolydian, but melodically it's singing quite a different tune.
6000 6050 6100 6150 6200 6250 6300 6350 6400 6450 6500 6550 6600 6650 6700 6750 6800 6850 6900 6950
6650 - 6675 -