Tuning-Math Archive Section 11: 10650

Message: 10650 - Contents - Hide Contents Date: Fri, 19 Mar 2004 23:54:23 Subject: Re: 5-limit yantra commas From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote: If we take the consecutive commas in pairs as we did for the 7-limit, we get equal temperaments; they go 1, 1, 3, 3, 7, 12, 34, 53, 118, 441, 612, 612...

Message: 10652 - Contents - Hide Contents Date: Fri, 19 Mar 2004 01:10:01 Subject: Re: Minimal filled scale From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma" <ekin@l...> wrote:

>> I mean if we have for example a chord [0 1 4 10], we take >> [1 2 5 11] [1 3 6 12] etc. >

> You mean [2 3 6 12]? Right.

>> until we've filled all the holes, >

> I still don't get it. You're harmonizing every note of the > original chord?

No, I'm harmonizing everything with translates of the chord in a minimal contiguous-generator scale containing the chord.

>> and every >> note is harmonizable by at least one such chord. >

> The original chord has this property...

No, the numbers from 0 to 10 only find harmonies for 0, 1, 4 and 10. 2, 3, 5, 6, 7, 8 and 9 have no major tetrad. If, however, I take the numbers from 0 to 15, every one of them has a major tetrad to harmonize it. The union of the sets {i,i+1,i+4,i+10} as i ranges from 0 to 5 is {0..15}; no smaller value than 5 will work.

Message: 10653 - Contents - Hide Contents Date: Fri, 19 Mar 2004 01:20:09 Subject: Re: Interval Vectors - The Musical Set Theory Kind From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" <paul.hjelmstad@u...> wrote:

> For those who need a review on Z-relations: This is when 2 or more > sets (in the Dihedral Group) have the same interval vector.

Don't you mean two or more sets in the orbit of an action by the dihedral group? In other words, s1 Z s2 iff Intervalvector(s1) = Intervalvalvector(g(s2)) for some g in the dihedral group Dn?

> So far, I have analyzed Intvec(n,m) where n=2m. This gives the > series for Intvec(2,1), Intvec(4,2),...Intvec(24,12) as follows: > > 1,2,3,7,13,35,85,254,701,2376,7944,25220. These are the "master > subsets" for even sets. (hexachords are the "master subset" for the > 12 tone set because all other subsets (or supersets) can be derived > from them, where the superset is merely the complement of the subset > of this set). > > Anyone see a pattern to the above series?

No, but see Reply from On-Line Encyclopedia * [with cont.] (Wayb.)

Message: 10658 - Contents - Hide Contents Date: Sat, 20 Mar 2004 20:00:32 Subject: 7-limit yantra temperaments From: Gene Ward Smith I went through the 7-limit yantras up to yantra_7(1000), and took the two smallest interval steps, and found the corresponding temperament. This is what I got: dicot 9 father 10-11 decimal 12 pajara 13-14 dominant seventh 15-16 meantone 17-37 miracle 38-53 hemiwuerschmidt 54-58 ennealimmal 59-556 <<88 151 183 35 43 1|| 557-1000+ The last temperament has the following properties: Mapping: [<1 -19 -33 -40|, <0 88 151 183|] TM basis: {78125000/78121827, 645700815/645657712} generators: [1, 731/3125] As a 7-limit system it is pretty strongly wedded to the 3125-et, but 6079, a very strong 13-limit system which I used for the Mozart Victims piece, also supports it, which means it has a natural extension to the 13-limit. The generator then is 1422/6079, and since I know how eager people are going to be to rush out and tune their microtonal guitars to this thing, I give the mapping: [<1 -19 -33 -40 312 163|, <0 88 151 183 -1319 -681|]

Message: 10659 - Contents - Hide Contents Date: Sat, 20 Mar 2004 23:35:32 Subject: 11-limit yantra temperaments From: Gene Ward Smith Here are the 11-limit temperaments which arise from tempering out the three smallest independent steps of 11-limit yantras. [<1 1 2 1 1|, <0 2 1 6 8|] 10 [<1 1 2 1 4|, <0 2 1 6 -2|] dicot 11 [<1 1 2 4 3|, <0 -1 1 -3 1|] father 12 [<1 2 1 5 3|, <0 -1 3 -5 1|] hexidecimal 13 [<2 4 5 6 9|, <0 -2 -1 -1 -5|] decimal 14 [<2 3 4 5 7|, <0 1 3 3 0|] octokaidecal 15 [<5 8 12 14 17|, <0 0 -1 0 1|] blackwood 16-18 [<1 2 4 2 1|, <0 -1 -4 2 6|] dominant seventh 19 [<2 3 4 5 6|, <0 1 4 4 6|] injera 20-21 [<2 4 6 7 8|, <0 -3 -5 -5 -4|] biporcupine 22 [<1 2 3 4 4|, <0 -3 -5 -9 -4|] tweedledee 23-26 [<1 1 2 3 3|, <0 9 5 -3 7|] valentine (alpha) 27-28 [<1 2 4 7 11|, <0 -1 -4 -10 -18|] huygens 29-30 [<1 -1 2 -3 -3|, <0, 8, 1, 18, 20|] wuerschmidt 31-38 [<1 1 3 3 2|, <0 6 -7 -2 15|] miracle 39-94 [<1 -13 -14 -9 -8|, <0 42 47 34 33|] undecififth 95-97 [<18 28 41 50 62|, <0 2 3 2 1|] hemiennealimmal 98-509 [<6 11 17 20 24|, <0 -17 -35 -36 -37|] 510-1074 [<1 2 24 6 -2|, <0 -13 -679 -100 171|] 1075-1102 [<1 177 239 46 -127|, <0 -398 -537 -98 296|] 1103-1307

Message: 10662 - Contents - Hide Contents Date: Sun, 21 Mar 2004 01:25:44 Subject: The Mathematical Theory of Tone Systems From: Gene Ward Smith I got a brochure from Marcel Dekker, and this is the title of one of their new releases. I requested it by interlibrary loan, but if you want to buy a copy it is priced at the usual incredibly cheap Marcel Dekker price, in this case $165 for 302 pages, or 55 cents a page. The book is by a Slovak mathematician named Jan Haluska, and I'm wondering if anyone has heard of him. I haven't, but he lists his fields of interest on his home page as "Measure and Integration, Harmonic Analysis and Uncertainty-Based Information" so that's probably not surprising.

Message: 10663 - Contents - Hide Contents Date: Sun, 21 Mar 2004 05:41:36 Subject: 13-limit yantra temperaments From: Gene Ward Smith I took this up to the 2000th 13-limit yantra. Hanson/catakleismic seems pretty important, so does the 35th temperament on the list, which is a 494-et system with generators [1/2, 86/494] (the second being a 44/39 interval.) Less complex systems follow one after another, but supersupermajor does alright for itself, along with wuerschmidt and the two versions of miracle. [2, 1, -1, 1, 2, -3, -7, -5, -4, -5, -1, 1, 6, 9, 3] [[1, 1, 2, 3, 3, 3], [0, 2, 1, -1, 1, 2]] [10, 10] [0, 0, 0, 4, 0, 0, 0, 6, 0, 0, 9, 0, 10, 0, -14] [[4, 6, 9, 10, 13, 14], [0, 0, 0, 0, 1, 0]] [11, 11] [2, 3, 6, -2, 6, 0, 4, -10, 2, 6, -15, 3, -26, -6, 28] [[1, 2, 3, 4, 3, 5], [0, -2, -3, -6, 2, -6]] [12, 12] [1, -1, 3, -1, 3, -4, 2, -5, 1, 10, 1, 11, -13, -3, 14] [[1, 2, 2, 4, 3, 5], [0, -1, 1, -3, 1, -3]] [13, 13] [1, -1, 3, -1, -2, -4, 2, -5, -7, 10, 1, -1, -13, -17, -3] [[1, 2, 2, 4, 3, 3], [0, -1, 1, -3, 1, 2]] [14, 14] [1, -3, 5, -1, -4, -7, 5, -5, -10, 20, 8, 2, -20, -30, -10] [[1, 2, 1, 5, 3, 2], [0, -1, 3, -5, 1, 4]] [15, 15] [4, 2, 2, 10, -2, -6, -8, 2, -18, -1, 16, -12, 21, -13, -44] [[2, 4, 5, 6, 9, 7], [0, -2, -1, -1, -5, 1]] [16, 16] [0, 0, 0, 0, 10, 0, 0, 0, 16, 0, 0, 23, 0, 28, 35] [[10, 16, 23, 28, 35, 37], [0, 0, 0, 0, 0, 1]] [17, 18] [1, -1, -2, -1, 1, -4, -6, -5, -2, -2, 1, 6, 4, 10, 7] [[1, 2, 2, 2, 3, 4], [0, -1, 1, 2, 1, -1]] [19, 20] [1, 4, -2, -1, -4, 4, -6, -5, -10, -16, -16, -24, 4, -4, -10] [[1, 2, 4, 2, 3, 2], [0, -1, -4, 2, 1, 4]] [21, 22] [1, 4, -2, 11, 8, 4, -6, 14, 9, -16, 12, 4, 38, 30, -13] [[1, 2, 4, 2, 8, 7], [0, -1, -4, 2, -11, -8]] [23, 23] [2, 8, 8, 12, 4, 8, 7, 12, -1, -4, 0, -20, 6, -18, -30] [[2, 3, 4, 5, 6, 7], [0, 1, 4, 4, 6, 2]] [24, 24] [4, 2, 2, -4, 8, -6, -8, -20, -2, -1, -16, 11, -18, 15, 42] [[2, 4, 5, 6, 6, 9], [0, -2, -1, -1, 2, -4]] [25, 26] [6, 10, 10, 8, 26, 2, -1, -8, 19, -5, -16, 23, -12, 36, 60] [[2, 4, 6, 7, 8, 11], [0, -3, -5, -5, -4, -13]] [27, 27] [3, 5, 9, 4, 17, 1, 6, -4, 16, 7, -8, 21, -20, 14, 44] [[1, 2, 3, 4, 4, 6], [0, -3, -5, -9, -4, -17]] [28, 29] [9, 5, -3, 7, 11, -13, -30, -20, -16, -21, -1, 7, 30, 42, 12] [[1, 1, 2, 3, 3, 3], [0, 9, 5, -3, 7, 11]] [30, 32] [8, 8, 8, 8, 8, -6, -10, -15, -17, -4, -9, -11, -5, -7, -2] [[8, 13, 19, 23, 28, 30], [0, -1, -1, -1, -1, -1]] [33, 33] [13, 9, 1, 19, 7, -16, -35, -15, -37, -23, 13, -17, 50, 16, -46] [[1, 4, 4, 3, 7, 5], [0, -13, -9, -1, -19, -7]] [34, 35] [9, 0, 9, 9, 9, -21, -11, -17, -19, 21, 21, 21, -6, -8, -2] [[9, 14, 21, 25, 31, 33], [0, 1, 0, 1, 1, 1]] [36, 36] [4, -3, 2, -4, 3, -14, -8, -20, -10, 13, 1, 18, -18, 1, 25] [[1, 2, 2, 3, 3, 4], [0, -4, 3, -2, 4, -3]] [37, 38] [8, 1, 18, 20, 27, -17, 6, 4, 13, 39, 43, 59, -6, 9, 19] [[1, -1, 2, -3, -3, -5], [0, 8, 1, 18, 20, 27]] [39, 44] [10, 2, 24, 25, 36, -20, 10, 5, 20, 50, 51, 76, -13, 12, 32] [[1, 5, 3, 11, 12, 16], [0, -10, -2, -24, -25, -36]] [45, 46] [4, 9, -8, 10, -2, 5, -24, 2, -18, -44, -8, -38, 56, 24, -44] [[1, 1, 1, 4, 2, 4], [0, 4, 9, -8, 10, -2]] [47, 47] [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]] [48, 49] [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]] [50, 52] [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]] [53, 57] [6, -7, -2, 15, 38, -25, -20, 3, 38, 15, 59, 114, 49, 114, 76] [[1, 1, 3, 3, 2, 0], [0, 6, -7, -2, 15, 38]] [58, 61] [6, -7, -2, 15, -34, -25, -20, 3, -76, 15, 59, -53, 49, -88, -173] [[1, 1, 3, 3, 2, 7], [0, 6, -7, -2, 15, -34]] [62, 63] [24, 20, 16, 60, 56, -24, -42, 12, 0, -19, 70, 56, 113, 98, -28] [[4, 6, 9, 11, 13, 14], [0, 6, 5, 4, 15, 14]] [64, 66] [6, 5, 22, -21, 14, -6, 18, -54, 0, 37, -66, 14, -135, -42, 126] [[1, 0, 1, -3, 9, 0], [0, 6, 5, 22, -21, 14]] [67, 93] [12, 34, 20, 30, 52, 26, -2, 6, 38, -49, -48, -5, 15, 72, 69] [[2, 4, 7, 7, 9, 11], [0, -6, -17, -10, -15, -26]] [94, 111] [36, 73, 89, 119, 127, 32, 40, 64, 68, 2, 24, 25, 26, 27, -1] [[1, -4, -9, -11, -15, -16], [0, 36, 73, 89, 119, 127]] [112, 122] [36, 54, 36, 18, 108, 2, -44, -96, 38, -68, -145, 51, -74, 170, 307] [[18, 28, 41, 50, 62, 65], [0, 2, 3, 2, 1, 6]] [123, 138] [2, -57, -28, 46, 81, -95, -50, 66, 121, 95, 304, 399, 226, 331, 110] [[1, 1, 19, 11, -10, -20], [0, 2, -57, -28, 46, 81]] [139, 142] [22, 48, -38, -34, -54, 25, -122, -130, -167, -223, -245, -303, 36, -11, -61] [[2, 7, 13, -1, 1, -2], [0, -11, -24, 19, 17, 27]] [143, 435] [78, 72, -12, -96, -216, -67, -238, -422, -631, -230, -472, -768, -228, -562, -392] [[6, 15, 19, 16, 14, 7], [0, -13, -12, 2, 16, 36]] [436, 457] [208, 147, 58, -76, -81, -250, -492, -840, -898, -278, -685, -732, -414, -442, 1] [[1, -30, -20, -6, 15, 16], [0, 208, 147, 58, -76, -81]] [458, 475] [873, 790, 1894, 2792, 2986, -775, 551, 1405, 1502, 2180, 3750, 4010, 1286, 1374, -2] [[1, 104, 95, 225, 331, 354], [0, -873, -790, -1894, -2792, -2986]] [476, 526] [150, 180, 60, -60, -270, -63, -326, -614, -983, -366, -762, -1293, -376, -980, -712] [[30, 47, 69, 84, 104, 112], [0, 5, 6, 2, -2, -9]] [527, 949] [141, -225, -2, 586, -356, -684, -399, 441, -1086, 627, 2139, 6, 1652, -992, -3400] [[1, -12, 24, 3, -53, 38], [0, 141, -225, -2, 586, -356]] [950, 965] [79, 517, 374, -20, 457, 636, 371, -305, 432, -583, -1835, -852, -1350, -101, 1655] [[1, 35, 221, 161, -5, 197], [0, -79, -517, -374, 20, -457]] [966, 979] [184, 556, 242, 430, 340, 454, -133, 45, -142, -999, -925, -1268, 370, 59, -415] [[2, 43, 125, 58, 100, 81], [0, -92, -278, -121, -215, -170]] [980, 1012] [96, -60, 418, 166, -56, -318, 393, -69, -444, 1139, 593, 92, -980, -1704, -808] [[2, 9, 1, 31, 17, 4], [0, -48, 30, -209, -83, 28]] [1013, 1153] [441, 135, 848, 1196, -166, -810, 106, 370, -1895, 1590, 2310, -885, 424, -3604, -5000] [[1, -55, -15, -106, -150, 25], [0, 441, 135, 848, 1196, -166]] [1154, 1216] [79, -348, 786, 1510, 563, -735, 1024, 2120, 600, 2802, 4710, 2595, 1520, -1328, -3640] [[1, -19, 93, -202, -390, -143], [0, 79, -348, 786, 1510, 563]] [1217, 1328] [872, 391, 2366, 3509, 3752, -1405, 1302, 2545, 2720, 4396, 6795, 7265, 1666, 1778, -5] [[1, -211, -93, -574, -852, -911], [0, 872, 391, 2366, 3509, 3752]] [1329, 2000]

Message: 10664 - Contents - Hide Contents Date: Sun, 21 Mar 2004 08:32:42 Subject: Re: The Mathematical Theory of Tone Systems From: Graham Breed Gene Ward Smith wrote:

> The book is by a Slovak mathematician named Jan Haluska, and I'm > wondering if anyone has heard of him. I haven't, but he lists his > fields of interest on his home page as "Measure and Integration, > Harmonic Analysis and Uncertainty-Based Information" so that's > probably not surprising.

Yes, he contacted me a few years ago. He's a mathematician who was publishing about tuning in mathematical journals but wasn't in contact with musicians. He sent me a paper which I can't find now, but I think it's the one that's dead-linked to from Manuel's bibliography. As it's a sonic-arts URL, perhaps Monz knows something. I do have the envelope with his address on! I can send it privately if you like. I don't want to encourage hoards of Slovakian tuning fanatics to converge on his residence ... Graham

Message: 10665 - Contents - Hide Contents Date: Sun, 21 Mar 2004 08:37:42 Subject: Re: The Mathematical Theory of Tone Systems From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:

> I do have the envelope with his address on! I can send it privately if > you like. I don't want to encourage hoards of Slovakian tuning fanatics > to converge on his residence ...

I'll wait until I've read his book before considering that. He is easily contactable via email. ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ 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: 10666 - Contents - Hide Contents Date: Mon, 22 Mar 2004 18:58:27 Subject: Re: 5-limit yantra commas From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> > wrote: > > If we take the consecutive commas in pairs as we did for the 7- limit, > we get equal temperaments; they go > 1, 1, 3, 3, 7, 12, 34, 53, 118, 441, 612, 612...

This is the same set of equal temperaments found here, S235 * [with cont.] (Wayb.) which is referred to from here: Searching Small Intervals * [with cont.] (Wayb.) If you don't understand why this is so, it's time for you to reconsider Kees's work . . . ;)

Message: 10668 - Contents - Hide Contents Date: Mon, 22 Mar 2004 19:05:14 Subject: Re: The Mathematical Theory of Tone Systems From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:

> I got a brochure from Marcel Dekker, and this is the title of one of > their new releases. I requested it by interlibrary loan, but if you > want to buy a copy it is priced at the usual incredibly cheap Marcel > Dekker price, in this case $165 for 302 pages, or 55 cents a page. > > The book is by a Slovak mathematician named Jan Haluska, and I'm > wondering if anyone has heard of him. I haven't,

Monz had an article by him up, which could be gotten by clicking on the appropriate link here: Research on the work of other composers and th... * [with cont.] (Wayb.) . . . but the article appears to be offline right now. I don't remember anything terribly interesting or novel in that article.

Message: 10670 - Contents - Hide Contents Date: Mon, 22 Mar 2004 19:17:43 Subject: Re: 5-limit yantra commas From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> > wrote:

>> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> >> wrote: >> >> If we take the consecutive commas in pairs as we did for the 7- > limit,

>> we get equal temperaments; they go >> 1, 1, 3, 3, 7, 12, 34, 53, 118, 441, 612, 612... >

> This is the same set of equal temperaments found here, > > S235 * [with cont.] (Wayb.) > > which is referred to from here: > > Searching Small Intervals * [with cont.] (Wayb.) > > If you don't understand why this is so, it's time for you to > reconsider Kees's work . . . ;)

Yantras are naturally weighted by log(p), so it isn't surprising to find some connection. Are you saying the two will always lead to exactly identical results?

Message: 10671 - Contents - Hide Contents Date: Mon, 22 Mar 2004 19:39:47 Subject: Re: 5-limit yantra commas From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:

>> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> >> wrote:

>>> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> >>> wrote: >>> >>> If we take the consecutive commas in pairs as we did for the 7- >> limit,

>>> we get equal temperaments; they go >>> 1, 1, 3, 3, 7, 12, 34, 53, 118, 441, 612, 612... >>

>> This is the same set of equal temperaments found here, >> >> S235 * [with cont.] (Wayb.) >> >> which is referred to from here: >> >> Searching Small Intervals * [with cont.] (Wayb.) >> >> If you don't understand why this is so, it's time for you to >> reconsider Kees's work . . . ;) >

> Yantras are naturally weighted by log(p), so it isn't surprising to > find some connection. Are you saying the two will always lead to > exactly identical results? Yes.