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Message: 10400 - Contents - Hide Contents
Date: Sat, 28 Feb 2004 11:02:02
Subject: Harmonized melody in the 7-limit
From: Gene Ward Smith
Two 7-limit notes less than 2 apart in the symmetric lattice can be
harmonized by two tetrads sharing at least one common note, and notes
2 or more apart cannot be so harmonized. Hence, a scale consisting
only of such intervals has the property that stepwise progressions can
be harmonized by tetrads with common notes--though not, of course,
necessarily tetrads all of whose notes belong to the scale.
If list all notes reduced to an octave which are at a distance of less
than three from the unison and smaller than 200 cents in size, we
obtain this:
{28/25, 10/9, 35/32, 15/14, 16/15, 21/20, 25/24, 36/35, 49/48}
The possible types of JI scale with the above property, in terms of
the intervals and their multiplicities, with the above restriction on
step size are given below. We can obtain tempered versions of these by
temperaments which equate steps; we have (28/25)/(10/9) = 126/125,
(10/9)/(35/32) = 64/63, (35/32)/(15/14) = 49/48, (15/14)/(16/15) =
225/224, (16/15)/(21/20) = 64/63, (21/20)/(25/24) = 126/125,
(25/24)/(36/35) = 875/864, (36/35)/(49/48) = 1728/1715. Meantone,
magic, orwell, pajara, porcupine, blackwood, superpythagorean,
tripletone, kleismic or nonkleismic would all be reasonable linear
temperaments to try--or beep if you think that is reasonable.
[28/25, 10/9, 35/32, 36/35] [1, 3, 2, 3]
[28/25, 10/9, 25/24, 36/35] [3, 1, 4, 3]
[28/25, 15/14, 16/15, 49/48] [1, 5, 3, 2]
[28/25, 16/15, 25/24, 36/35] [3, 1, 5, 3]
[10/9, 35/32, 16/15, 36/35] [2, 3, 2, 3]
[10/9, 35/32, 36/35, 49/48] [4, 1, 5, 2]
[10/9, 35/32, 21/20, 36/35] [4, 1, 2, 3]
[10/9, 15/14, 16/15, 21/20] [2, 3, 2, 3]
[10/9, 15/14, 21/20, 36/35] [4, 1, 3, 2]
[10/9, 15/14, 36/35, 49/48] [4, 1, 5, 3]
[10/9, 21/20, 25/24, 36/35] [4, 3, 1, 3]
[10/9, 25/24, 36/35, 49/48] [4, 1, 6, 3]
[35/32, 15/14, 16/15, 36/35] [3, 2, 4, 1]
[35/32, 16/15, 25/24, 36/35] [3, 4, 2, 3]
[15/14, 16/15, 21/20, 25/24] [3, 4, 3, 2]
[15/14, 16/15, 21/20, 49/48] [5, 4, 1, 2]
[15/14, 16/15, 36/35, 49/48] [5, 4, 1, 3]
[16/15, 21/20, 25/24, 36/35] [4, 3, 5, 3]
[16/15, 25/24, 36/35, 49/48] [4, 5, 6, 3]
[28/25, 10/9, 15/14, 25/24] [3, 1, 3, 1]
[28/25, 15/14, 16/15, 25/24] [3, 3, 1, 2]
[28/25, 35/32, 15/14, 16/15] [1, 2, 3, 3]
[28/25, 10/9, 15/14, 21/20] [2, 2, 3, 1]
[35/32, 15/14, 16/15, 21/20] [2, 3, 4, 1]
Message: 10401 - Contents - Hide Contents Date: Sat, 28 Feb 2004 23:12:37 Subject: Re: Harmonized melody in the 7-limit From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>> Rad. No 6- or 7-toners, eh? I wonder about the "9-limit". >>> The 9-limit has the further advantage that you can hit more >>> fifths, and thus improve omnitetrachordality. >>>> The 9-limit would be different, for sure. The simple symmetrical >> lattice criterion wouldn't work, but it would be easy enough to >> find what does. >> Nobody ever answered me if symmetrical is synonymous with unweighted.Probably no one was sure what the question meant. It means 3, 5, 7, 5/3, 7/3 and 7/5 are all the same size, however.> The thing is to only store one permutation in memory at a time. > Alas, I haven't come up with an easy way to code these kinds of > evaluations in scheme. They're very natural in C, I think. The > present problem may still be hard on account of CPU cycles, tho.I'd certainly use C myself. On the other hand, I don't know scheme. :)
Message: 10402 - Contents - Hide Contents Date: Sat, 28 Feb 2004 11:08:07 Subject: Re: Harmonized melody in the 7-limit From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> If list all notes reduced to an octave which are at a distance of less > than threeLess than two. ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ 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: 10403 - Contents - Hide Contents Date: Sat, 28 Feb 2004 15:49:12 Subject: Re: DE scales with the stepwise harmonization property From: Carl Lumma>Augmented[9] >[28/25, 35/32, 15/14, 16/15] [1, 2, 3, 3] > >(28/25)/(35/32) = 128/125 >(15/14)/(16/15) = 225/224Augmented[9], eh? How far is the 7-limit TOP version from... ! TOP 5-limit Augmented[9]. 9 ! 93.15 306.77 399.92 493.07 706.69 799.84 892.99 1106.61 1199.76 ! ...? -Carl
Message: 10404 - Contents - Hide Contents Date: Sat, 28 Feb 2004 15:51:21 Subject: Re: Harmonized melody in the 7-limit From: Carl Lumma>>>> >ad. No 6- or 7-toners, eh? I wonder about the "9-limit". >>>> The 9-limit has the further advantage that you can hit more >>>> fifths, and thus improve omnitetrachordality. >>>>>> The 9-limit would be different, for sure. The simple symmetrical >>> lattice criterion wouldn't work, but it would be easy enough to >>> find what does. >>>> Nobody ever answered me if symmetrical is synonymous with unweighted. >>Probably no one was sure what the question meant. It means 3, 5, 7, >5/3, 7/3 and 7/5 are all the same size, however.As I thought then. -Carl ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ 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: 10405 - Contents - Hide Contents Date: Sun, 29 Feb 2004 14:36:25 Subject: Stepwise harmonizing property From: Gene Ward Smith I was implicitly assuming that one of the chords harmonized to the root, which doesn't make a lot of sense to assume. Dropping that makes the analysis far easier--steps have this property iff they are products (or ratios, but that adds nothing) of consonant intervals (including 1 as a consonant interval.) This is obvious enough if you think about it; the situation no longer depends on fine distinctions. You get one consonant interval from the unison to the common note in one chord, and from the common note to another interval in the next chord; that interval, by definition one reachable by a chord with a common note, is therefore a product of consonances of the system. In the 7-limit, this has the effect of adding 50/49 (= (10/7)(5/7)) to the list of harmonizable intervals. I did, and also extended the size up to 8/7, getting a considerably larger list this time, and including some six and seven note scales for Carl. 1 [50/49, 49/48, 36/35, 16/15] [5, 8, 6, 4] 23 2 [50/49, 49/48, 36/35, 10/9] [1, 4, 6, 4] 15 3 [50/49, 49/48, 36/35, 28/25] [5, 4, 2, 4] 15 4 [50/49, 49/48, 36/35, 8/7] [1, 4, 2, 4] 11 5 [50/49, 49/48, 21/20, 16/15] [5, 2, 6, 4] 17 6 [50/49, 49/48, 21/20, 28/25] [5, 2, 2, 4] 13 7 [50/49, 49/48, 21/20, 8/7] [1, 2, 2, 4] 9 8 [50/49, 49/48, 16/15, 9/8] [2, 2, 4, 3] 11 9 [50/49, 49/48, 15/14, 28/25] [3, 2, 2, 4] 11 10 [50/49, 49/48, 28/25, 9/8] [4, 2, 4, 1] 11 11 [50/49, 36/35, 25/24, 28/25] [1, 2, 4, 4] 11 12 [50/49, 36/35, 21/20, 10/9] [1, 2, 4, 4] 11 13 [50/49, 36/35, 16/15, 35/32] [1, 2, 4, 4] 11 14 [50/49, 25/24, 21/20, 16/15] [3, 2, 6, 4] 15 15 [50/49, 25/24, 21/20, 28/25] [3, 2, 2, 4] 11 16 [50/49, 25/24, 15/14, 28/25] [1, 2, 2, 4] 9 17 [50/49, 25/24, 28/25, 9/8] [2, 2, 4, 1] 9 18 [50/49, 21/20, 16/15, 35/32] [3, 4, 4, 2] 13 19 [50/49, 21/20, 16/15, 10/9] [3, 6, 2, 2] 13 20 [50/49, 21/20, 10/9, 28/25] [3, 4, 2, 2] 11 21 [50/49, 21/20, 10/9, 8/7] [1, 4, 2, 2] 9 22 [50/49, 16/15, 35/32, 9/8] [1, 4, 2, 2] 9 23 [50/49, 10/9, 28/25, 9/8] [1, 2, 2, 2] 7 24 [49/48, 36/35, 25/24, 16/15] [3, 6, 5, 4] 18 25 [49/48, 36/35, 25/24, 10/9] [3, 6, 1, 4] 14 26 [49/48, 36/35, 25/24, 8/7] [3, 2, 1, 4] 10 27 [49/48, 36/35, 16/15, 15/14] [3, 1, 4, 5] 13 28 [49/48, 36/35, 15/14, 10/9] [3, 5, 1, 4] 13 29 [49/48, 36/35, 15/14, 8/7] [3, 1, 1, 4] 9 30 [49/48, 36/35, 35/32, 10/9] [2, 5, 1, 4] 12 31 [49/48, 36/35, 35/32, 8/7] [2, 1, 1, 4] 8 32 [49/48, 36/35, 10/9, 9/8] [2, 4, 4, 1] 11 33 [49/48, 25/24, 21/20, 8/7] [1, 1, 2, 4] 8 34 [49/48, 21/20, 16/15, 15/14] [2, 1, 4, 5] 12 35 [49/48, 21/20, 15/14, 8/7] [2, 1, 1, 4] 8 36 [49/48, 21/20, 35/32, 8/7] [1, 1, 1, 4] 7 37 [49/48, 16/15, 15/14, 28/25] [2, 3, 5, 1] 11 38 [49/48, 16/15, 15/14, 9/8] [2, 4, 4, 1] 11 40 [49/48, 15/14, 28/25, 8/7] [2, 2, 1, 3] 8 41 [36/35, 25/24, 21/20, 16/15] [3, 5, 3, 4] 15 42 [36/35, 25/24, 21/20, 10/9] [3, 1, 3, 4] 11 43 [36/35, 25/24, 16/15, 35/32] [3, 2, 4, 3] 12 44 [36/35, 25/24, 16/15, 28/25] [3, 5, 1, 3] 12 45 [36/35, 25/24, 10/9, 28/25] [3, 4, 1, 3] 11 46 [36/35, 25/24, 28/25, 8/7] [2, 4, 3, 1] 10 47 [36/35, 21/20, 15/14, 10/9] [2, 3, 1, 4] 10 48 [36/35, 21/20, 35/32, 10/9] [3, 2, 1, 4] 10 49 [36/35, 21/20, 10/9, 9/8] [2, 2, 4, 1] 9 50 [36/35, 16/15, 15/14, 35/32] [1, 4, 2, 3] 10 51 [36/35, 16/15, 35/32, 10/9] [3, 2, 3, 2] 10 52 [36/35, 16/15, 35/32, 8/7] [1, 2, 3, 2] 8 53 [36/35, 35/32, 10/9, 28/25] [3, 2, 3, 1] 9 54 [36/35, 10/9, 28/25, 9/8] [1, 3, 1, 2] 7 55 [25/24, 21/20, 16/15, 15/14] [2, 3, 4, 3] 12 56 [25/24, 21/20, 16/15, 8/7] [2, 3, 1, 3] 9 57 [25/24, 21/20, 10/9, 8/7] [1, 3, 1, 3] 8 58 [25/24, 21/20, 28/25, 8/7] [2, 2, 1, 3] 8 59 [25/24, 16/15, 15/14, 28/25] [2, 1, 3, 3] 9 60 [25/24, 15/14, 10/9, 28/25] [1, 3, 1, 3] 8 61 [25/24, 15/14, 28/25, 8/7] [2, 2, 3, 1] 8 62 [25/24, 28/25, 9/8, 8/7] [2, 2, 1, 2] 7 63 [21/20, 16/15, 15/14, 35/32] [1, 4, 3, 2] 10 64 [21/20, 16/15, 15/14, 10/9] [3, 2, 3, 2] 10 65 [21/20, 16/15, 35/32, 8/7] [1, 1, 2, 3] 7 66 [21/20, 15/14, 10/9, 28/25] [1, 3, 2, 2] 8 67 [21/20, 15/14, 10/9, 8/7] [3, 1, 2, 2] 8 68 [21/20, 35/32, 10/9, 8/7] [2, 1, 1, 3] 7 69 [21/20, 10/9, 9/8, 8/7] [2, 2, 1, 2] 7 70 [16/15, 15/14, 35/32, 28/25] [3, 3, 2, 1] 9 71 [16/15, 15/14, 35/32, 9/8] [4, 2, 2, 1] 9 72 [16/15, 35/32, 9/8, 8/7] [2, 2, 1, 2] 7 74 [15/14, 10/9, 28/25, 9/8] [2, 2, 2, 1] 7 75 [10/9, 28/25, 9/8, 8/7] [2, 1, 2, 1] 6
Message: 10406 - Contents - Hide Contents Date: Sun, 29 Feb 2004 19:51:29 Subject: 9-limit stepwise From: Gene Ward Smith Here are some 9-limit stepwise harmonizable scales, with the same bound on size of steps--8/7 is the largest. In order to keep the numbers down, I also enforced that the size of the largest step (in cents) is less than four times that of the smallest step--the logarithmic ratio is the fourth number listed. In order I give scale type number on the list, scale steps, multiplicities, largest/smallest, and number of steps in the scale. As you can see, the largest scale listed has 41 steps, which is getting up there. (64/63)/(81/80)=5120/5103 and (49/48)/(50/49) = 2401/2400; putting these together gives us hemififths, and hence Hemififths[41] as a DE for this. Hemififths is into the microtemperament range by most standards; it has octave-generator with TOP values [1199.700, 351.365] and a mapping of [<1 1 -5 -1|, <0 2 25 13|]. I've never tried to use it, and so far as I know neither has anyone else, but this certainly gives a motivation. Ets for hemififths are 41, 58, 99 and 140. We also get Schismic[29] out of scale 4, Diaschismic[22] out of scale 5, Orwell[31] out of scale 8 and Orwell[22] out of scale 24, Meantone[19] out of scales 10, 11, 13, or 19, Superkleismic[26] out of scale 12, Octacot[27] out of scale 14, Semisixths[19] out of scale 29. 1 [81/80, 64/63, 50/49, 49/48] [10, 14, 5, 12] 1.659831 41 2 [81/80, 64/63, 50/49, 28/27] [10, 2, 5, 12] 2.927558 29 3 [81/80, 64/63, 49/48, 25/24] [10, 14, 7, 5] 3.286128 36 4 [81/80, 64/63, 28/27, 25/24] [10, 7, 7, 5] 3.286128 29 5 [81/80, 64/63, 25/24, 21/20] [3, 7, 5, 7] 3.927558 22 6 [81/80, 50/49, 36/35, 28/27] [8, 5, 2, 12] 2.927558 27 7 [81/80, 36/35, 28/27, 25/24] [3, 7, 7, 5] 3.286128 22 8 [64/63, 50/49, 49/48, 36/35] [4, 5, 12, 10] 1.788814 31 9 [64/63, 50/49, 49/48, 21/20] [4, 5, 2, 10] 3.098111 21 10 [64/63, 50/49, 28/27, 21/20] [2, 5, 2, 10] 3.098111 19 11 [64/63, 50/49, 25/24, 21/20] [4, 3, 2, 10] 3.098111 19 12 [64/63, 49/48, 36/35, 25/24] [4, 7, 10, 5] 2.592143 26 13 [64/63, 36/35, 25/24, 21/20] [4, 3, 5, 7] 3.098111 19 14 [50/49, 49/48, 36/35, 28/27] [5, 8, 10, 4] 1.800137 27 15 [50/49, 49/48, 36/35, 16/15] [5, 8, 6, 4] 3.194548 23 16 [50/49, 49/48, 28/27, 27/25] [5, 3, 4, 5] 3.809442 17 17 [50/49, 49/48, 21/20, 16/15] [5, 2, 6, 4] 3.194548 17 18 [50/49, 49/48, 16/15, 27/25] [5, 5, 4, 3] 3.809442 17 19 [50/49, 36/35, 28/27, 21/20] [5, 2, 4, 8] 2.415031 19 20 [50/49, 28/27, 25/24, 27/25] [2, 4, 3, 5] 3.809442 14 21 [50/49, 28/27, 21/20, 16/15] [5, 2, 8, 2] 3.194548 17 22 [50/49, 28/27, 21/20, 27/25] [5, 4, 6, 2] 3.809442 17 23 [50/49, 25/24, 21/20, 16/15] [3, 2, 6, 4] 3.194548 15 24 [49/48, 36/35, 28/27, 25/24] [3, 10, 4, 5] 1.979797 22 25 [49/48, 36/35, 28/27, 15/14] [3, 5, 4, 5] 3.346036 17 26 [49/48, 36/35, 25/24, 16/15] [3, 6, 5, 4] 3.130007 18 27 [49/48, 36/35, 16/15, 15/14] [3, 1, 4, 5] 3.346036 13 28 [49/48, 21/20, 16/15, 15/14] [2, 1, 4, 5] 3.346036 12 29 [36/35, 28/27, 25/24, 21/20] [7, 4, 5, 3] 1.731936 19 30 [36/35, 28/27, 25/24, 27/25] [4, 4, 5, 3] 2.731936 16 31 [36/35, 28/27, 25/24, 49/45] [7, 1, 5, 3] 3.022902 16 32 [36/35, 28/27, 25/24, 35/32] [7, 4, 2, 3] 3.181021 16 33 [36/35, 28/27, 21/20, 15/14] [2, 4, 3, 5] 2.449085 14 34 [36/35, 28/27, 15/14, 49/45] [2, 1, 5, 3] 3.022902 11 35 [36/35, 28/27, 15/14, 35/32] [5, 4, 2, 3] 3.181021 14 36 [36/35, 28/27, 35/32, 54/49] [3, 4, 3, 2] 3.449085 12 37 [36/35, 28/27, 35/32, 10/9] [5, 2, 3, 2] 3.740051 12 38 [36/35, 25/24, 21/20, 16/15] [3, 5, 3, 4] 2.290966 15 39 [36/35, 25/24, 21/20, 10/9] [3, 1, 3, 4] 3.740051 11 40 [36/35, 25/24, 16/15, 49/45] [6, 5, 1, 3] 3.022902 15 41 [36/35, 25/24, 16/15, 35/32] [3, 2, 4, 3] 3.181021 12 42 [36/35, 25/24, 49/45, 10/9] [6, 4, 3, 1] 3.740051 14 43 [36/35, 21/20, 15/14, 10/9] [2, 3, 1, 4] 3.740051 10 44 [36/35, 21/20, 35/32, 10/9] [3, 2, 1, 4] 3.740051 10 45 [36/35, 21/20, 54/49, 10/9] [1, 3, 1, 4] 3.740051 9 46 [36/35, 16/15, 15/14, 49/45] [1, 1, 5, 3] 3.022902 10 47 [36/35, 16/15, 15/14, 35/32] [1, 4, 2, 3] 3.181021 10 48 [36/35, 16/15, 35/32, 10/9] [3, 2, 3, 2] 3.740051 10 49 [36/35, 15/14, 49/45, 10/9] [2, 4, 3, 1] 3.740051 10 50 [36/35, 27/25, 35/32, 10/9] [1, 2, 1, 4] 3.740051 8 51 [36/35, 49/45, 35/32, 10/9] [4, 1, 2, 3] 3.740051 10 52 [28/27, 25/24, 15/14, 27/25] [4, 1, 4, 3] 2.116195 12 53 [28/27, 25/24, 15/14, 28/25] [1, 1, 4, 3] 3.116195 9 54 [28/27, 25/24, 27/25, 54/49] [4, 3, 3, 2] 2.671709 12 55 [28/27, 25/24, 27/25, 8/7] [2, 3, 3, 2] 3.671709 10 56 [28/27, 25/24, 54/49, 28/25] [1, 3, 2, 3] 3.116195 9 57 [28/27, 21/20, 16/15, 15/14] [2, 3, 2, 5] 1.897095 12 58 [28/27, 21/20, 15/14, 27/25] [4, 1, 5, 2] 2.116195 12 59 [28/27, 21/20, 15/14, 54/49] [4, 3, 3, 2] 2.671709 12 60 [28/27, 21/20, 15/14, 28/25] [2, 1, 5, 2] 3.116195 10 61 [28/27, 21/20, 15/14, 8/7] [2, 3, 3, 2] 3.671709 10 62 [28/27, 21/20, 54/49, 10/9] [1, 3, 2, 3] 2.897095 9 63 [28/27, 16/15, 15/14, 9/8] [2, 2, 2, 3] 3.238677 9 64 [28/27, 16/15, 35/32, 54/49] [1, 3, 3, 2] 2.671709 9 65 [28/27, 15/14, 27/25, 49/45] [3, 5, 2, 1] 2.341582 11 66 [28/27, 15/14, 27/25, 9/8] [4, 4, 2, 1] 3.238677 11 67 [28/27, 15/14, 49/45, 54/49] [1, 3, 3, 2] 2.671709 9 68 [28/27, 15/14, 49/45, 28/25] [1, 5, 1, 2] 3.116195 9 69 [28/27, 15/14, 28/25, 9/8] [2, 4, 2, 1] 3.238677 9 70 [28/27, 27/25, 35/32, 54/49] [4, 1, 2, 3] 2.671709 10 71 [28/27, 27/25, 35/32, 8/7] [1, 1, 2, 3] 3.671709 7 72 [28/27, 35/32, 54/49, 28/25] [3, 2, 3, 1] 3.116195 9 73 [25/24, 21/20, 16/15, 15/14] [2, 3, 4, 3] 1.690091 12 74 [25/24, 21/20, 16/15, 8/7] [2, 3, 1, 3] 3.271065 9 75 [25/24, 21/20, 10/9, 8/7] [1, 3, 1, 3] 3.271065 8 76 [25/24, 21/20, 28/25, 8/7] [2, 2, 1, 3] 3.271065 8 77 [25/24, 16/15, 15/14, 28/25] [2, 1, 3, 3] 2.776167 9 78 [25/24, 16/15, 49/45, 54/49] [2, 1, 3, 3] 2.380181 9 79 [25/24, 15/14, 10/9, 28/25] [1, 3, 1, 3] 2.776167 8 80 [25/24, 15/14, 28/25, 8/7] [2, 2, 3, 1] 3.271065 8 81 [25/24, 27/25, 28/25, 8/7] [3, 1, 2, 2] 3.271065 8 82 [25/24, 49/45, 54/49, 10/9] [1, 3, 3, 1] 2.580974 8 83 [25/24, 54/49, 28/25, 8/7] [3, 1, 3, 1] 3.271065 8 84 [25/24, 28/25, 9/8, 8/7] [2, 2, 1, 2] 3.271065 7 85 [21/20, 16/15, 15/14, 49/45] [1, 2, 5, 2] 1.745389 10 86 [21/20, 16/15, 15/14, 35/32] [1, 4, 3, 2] 1.836685 10 87 [21/20, 16/15, 15/14, 10/9] [3, 2, 3, 2] 2.159462 10 88 [21/20, 16/15, 35/32, 8/7] [1, 1, 2, 3] 2.736851 7 89 [21/20, 15/14, 27/25, 10/9] [1, 1, 2, 4] 2.159462 8 90 [21/20, 15/14, 49/45, 8/7] [1, 3, 2, 2] 2.736851 8 91 [21/20, 15/14, 10/9, 28/25] [1, 3, 2, 2] 2.322777 8 92 [21/20, 15/14, 10/9, 8/7] [3, 1, 2, 2] 2.736851 8 93 [21/20, 27/25, 54/49, 10/9] [2, 1, 1, 4] 2.159462 8 94 [21/20, 49/45, 54/49, 10/9] [2, 1, 2, 3] 2.159462 8 95 [21/20, 35/32, 10/9, 8/7] [2, 1, 1, 3] 2.736851 7 96 [21/20, 54/49, 10/9, 8/7] [3, 1, 3, 1] 2.736851 8 97 [21/20, 10/9, 9/8, 8/7] [2, 2, 1, 2] 2.736851 7 98 [16/15, 15/14, 49/45, 54/49] [1, 4, 3, 1] 1.505516 9 99 [16/15, 15/14, 49/45, 28/25] [1, 5, 2, 1] 1.755985 9 100 [16/15, 15/14, 49/45, 9/8] [2, 4, 2, 1] 1.825004 9 101 [16/15, 15/14, 35/32, 54/49] [4, 1, 3, 1] 1.505516 9 102 [16/15, 15/14, 35/32, 28/25] [3, 3, 2, 1] 1.755985 9 103 [16/15, 15/14, 35/32, 9/8] [4, 2, 2, 1] 1.825004 9 104 [16/15, 35/32, 54/49, 8/7] [3, 3, 1, 1] 2.069018 8 105 [16/15, 35/32, 9/8, 8/7] [2, 2, 1, 2] 2.069018 7 106 [15/14, 27/25, 49/45, 10/9] [2, 2, 1, 3] 1.527122 8 107 [15/14, 49/45, 54/49, 10/9] [2, 3, 2, 1] 1.527122 8 108 [15/14, 49/45, 54/49, 8/7] [3, 3, 1, 1] 1.935438 8 109 [15/14, 49/45, 10/9, 28/25] [4, 1, 1, 2] 1.642614 8 110 [15/14, 49/45, 28/25, 8/7] [4, 2, 1, 1] 1.935438 8 111 [15/14, 49/45, 9/8, 8/7] [2, 2, 1, 2] 1.935438 7 112 [15/14, 10/9, 28/25, 9/8] [2, 2, 2, 1] 1.707177 7 113 [27/25, 35/32, 10/9, 8/7] [2, 1, 3, 1] 1.735052 7 114 [49/45, 35/32, 9/8, 8/7] [1, 1, 1, 3] 1.568046 6 115 [49/45, 54/49, 10/9, 9/8] [2, 2, 2, 1] 1.383115 7 116 [49/45, 10/9, 9/8, 8/7] [1, 1, 2, 2] 1.568046 6 117 [10/9, 28/25, 9/8, 8/7] [2, 1, 2, 1] 1.267376 6
Message: 10407 - Contents - Hide Contents Date: Sun, 29 Feb 2004 12:01:00 Subject: Re: Stepwise harmonizing property From: Carl Lumma>I was implicitly assuming that one of the chords harmonized to the >root, which doesn't make a lot of sense to assume. Dropping that makes >the analysis far easier--steps have this property iff they are >products (or ratios, but that adds nothing) of consonant intervals >(including 1 as a consonant interval.)How is this any different than a symmetric lattice distance of 2, which is what I thought you used in the first place.>This is obvious enough if you >think about it; the situation no longer depends on fine distinctions. >You get one consonant interval from the unison to the common note in >one chord, and from the common note to another interval in the next >chord; that interval, by definition one reachable by a chord with a >common note, is therefore a product of consonances of the system.I can't parse this.>In the 7-limit, this has the effect of adding 50/49 (= (10/7)(5/7)) to >the list of harmonizable intervals. I did, and also extended the size >up to 8/7, getting a considerably larger list this time, and including >some six and seven note scales for Carl.Well this is cool, but since many classic scales contain steps up to a minor third apart, perhaps 6/5 should be the cutoff. -Carl
Message: 10408 - Contents - Hide Contents Date: Sun, 29 Feb 2004 12:07:59 Subject: Re: Harmonized melody in the 7-limit From: Carl Lumma>>>>> >ad. No 6- or 7-toners, eh? I wonder about the "9-limit". >>>>> The 9-limit has the further advantage that you can hit more >>>>> fifths, and thus improve omnitetrachordality. >>>>>>>> The 9-limit would be different, for sure. The simple symmetrical >>>> lattice criterion wouldn't work, but it would be easy enough to >>>> find what does.And why, pray tell, does symmetrical lattice distance not work in the 9-limit? -Carl
Message: 10409 - Contents - Hide Contents Date: Sun, 29 Feb 2004 20:31:26 Subject: Re: 9-limit stepwise From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote: On the other end of the size scale we have these. Paul, have you ever considered Pajara[6] as a possible melody scale?> 88 [21/20, 16/15, 35/32, 8/7] [1, 1, 2, 3] 2.736851 7 Dom7[7] > 111 [15/14, 49/45, 9/8, 8/7] [2, 2, 1, 2] 1.935438 7 Beatles[7] > 112 [15/14, 10/9, 28/25, 9/8] [2, 2, 2, 1] 1.707177 7 Dicot[7] > 114 [49/45, 35/32, 9/8, 8/7] [1, 1, 1, 3] 1.568046 6 Pajara[6] > 115 [49/45, 54/49, 10/9, 9/8] [2, 2, 2, 1] 1.383115 7 Squares[7] > 116 [49/45, 10/9, 9/8, 8/7] [1, 1, 2, 2] 1.568046 6 Pajara[6] > 117 [10/9, 28/25, 9/8, 8/7] [2, 1, 2, 1] 1.267376 6 Tripletone[6]
Message: 10410 - Contents - Hide Contents Date: Sun, 29 Feb 2004 20:39:27 Subject: Re: Stepwise harmonizing property From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>> I was implicitly assuming that one of the chords harmonized to the >> root, which doesn't make a lot of sense to assume. Dropping that makes >> the analysis far easier--steps have this property iff they are >> products (or ratios, but that adds nothing) of consonant intervals >> (including 1 as a consonant interval.) >> How is this any different than a symmetric lattice distance of 2, > which is what I thought you used in the first place.50/49 has a distance of exactly 2; I said less than 2.> Well this is cool, but since many classic scales contain steps up to a > minor third apart, perhaps 6/5 should be the cutoff.Hmmm. If you want to temper the results, you should bound the steps pairwise if you are going to go this big--meaning the ratio of the biggest to second-biggest is not allowed to be too large.
Message: 10411 - Contents - Hide Contents Date: Sun, 29 Feb 2004 20:40:55 Subject: Re: Harmonized melody in the 7-limit From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>>>>> Rad. No 6- or 7-toners, eh? I wonder about the "9-limit". >>>>>> The 9-limit has the further advantage that you can hit more >>>>>> fifths, and thus improve omnitetrachordality. >>>>>>>>>> The 9-limit would be different, for sure. The simple symmetrical >>>>> lattice criterion wouldn't work, but it would be easy enough to >>>>> find what does. >> And why, pray tell, does symmetrical lattice distance not work in > the 9-limit?If you call something which makes 3 half as large as 5 or 7 "symmetrical", it does.
Message: 10412 - Contents - Hide Contents Date: Sun, 29 Feb 2004 13:19:30 Subject: Re: Harmonized melody in the 7-limit From: Carl Lumma>>>>>>> >ad. No 6- or 7-toners, eh? I wonder about the "9-limit". >>>>>>> The 9-limit has the further advantage that you can hit more >>>>>>> fifths, and thus improve omnitetrachordality. >>>>>>>>>>>> The 9-limit would be different, for sure. The simple symmetrical >>>>>> lattice criterion wouldn't work, but it would be easy enough to >>>>>> find what does. >>>> And why, pray tell, does symmetrical lattice distance not work in >> the 9-limit? >>If you call something which makes 3 half as large as 5 or 7 >"symmetrical", it does.One of us is still misunderstanding Paul Hahn's 9-limit approach. In the unweighted version 3, 5, 7 and 9 are all the same length. If you prefer I think you can just use your product-of-two-consonances rule where the ratios of 9 have been included. -Carl
Message: 10413 - Contents - Hide Contents Date: Sun, 29 Feb 2004 21:28:32 Subject: Re: Harmonized melody in the 7-limit From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:> One of us is still misunderstanding Paul Hahn's 9-limit approach.What in the world makes you think this has anything to do with me or anything I've said?
Message: 10414 - Contents - Hide Contents Date: Sun, 29 Feb 2004 22:24:54 Subject: Hanzos From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:> One of us is still misunderstanding Paul Hahn's 9-limit approach. > In the unweighted version 3, 5, 7 and 9 are all the same length.In this system you don't exactly, have 7-limit notes and intervals. You do have "hanzos", with basis 2,3,5,7,9. The hahnzo |0 -2 0 0 1> is a comma, 9/3^2, which obviously would play a special role. Hahnzos map onto 7-limit intervals, but not 1-1. Are you happy with the idea that two scales could be different, since they have steps and notes which are distinct as hahnzos, even though they have exactly the same steps and notes in the 7-limit? We've got three hahnzos corresponding to 81/80; if we take any two of them and wedge, we get the planar wedgie <<<0 1 0 4 0 -2 4 0 0 8|||. For 126/125 we get both |1 2 -3 1 0> and |1 0 -3 1 1> as a hahnzo. Going through all six combinations, I get <<1 4 10 2 4 13 0 12 -8 -26|| as the wedgie, leading to [<1 2 4 7 4|, <0 -1 -4 -10 -2|] as the mapping. The mechanism seems to work for hahnzos; here it is telling us the generator is a fourth, and 16*(4/3)^(-2) = 9. I can also get this by sticking in the dummy hahnzo <0 -2 0 0 1| as a comma. If I try three hahnzo commas which don't have a dummy relationship, I get a disguised et as a "linear" temperament.
Message: 10415 - Contents - Hide Contents Date: Mon, 01 Mar 2004 04:58:54 Subject: Re: Harmonized melody in the 7-limit From: Carl Lumma>> >hat you said was that symmetrical lattice distance won't work. > >It doesn't. >>> I asked why, and said Paul Hahn's version works. >>That's a symmetrical lattice, but it isn't a lattice of note-classes.If I didn't know better I'd say you were trying to BS me. What is a lattice of note classes? -Carl
Message: 10416 - Contents - Hide Contents Date: Mon, 01 Mar 2004 23:42:32 Subject: Re: DE scales with the stepwise harmonization property From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>> Augmented[9], eh? How far is the 7-limit TOP version >>> from... >>> >>> ! >>> TOP 5-limit Augmented[9]. >>> 9 >>> ! >>> 93.15 >>> 306.77 >>> 399.92 >>> 493.07 >>> 706.69 >>> 799.84 >>> 892.99 >>> 1106.61 >>> 1199.76 >>> ! >>> >>> ...? >>> >>> -Carl >>>> Just look at the horagram, Carl! >> >> 107.31 >> 292.68 >> 399.99 >> 507.3 >> 692.67 >> 799.98 >> 907.29 >> 1092.66 >> 1199.97 >> Oh! Where are the 7-limit horagrams? > > -C.Some of them are in Yahoo groups: /tuning/files/perlich/ * [with cont.] some of them are in Yahoo groups: /tuning_files/files/Erlich/seven... * [with cont.] and this one, aug7.gif, is in both.
Message: 10417 - Contents - Hide Contents Date: Mon, 01 Mar 2004 04:02:32 Subject: Re: Hanzos From: Carl Lumma>> >y recollection is that Paul H.'s algorithm assigns a unique >> lattice route (and therefore hanzo) to each 9-limit interval. >>So what? You still get an infinite number representing each interval, >since you can multiply by arbitary powers of the dummy comma 9/3^2.An infinite number from where? If you look at the algorithm, that dummy comma has zero length.>> Certainly it can be used to find the set of lattice points >> within distance <= 2 of a given point. >>Hahn's alogorithm can, or hahnzos can, or what?The algorithm can. -Carl
Message: 10418 - Contents - Hide Contents Date: Mon, 01 Mar 2004 23:44:21 Subject: Re: 9-limit stepwise 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: >>> >>> On the other end of the size scale we have these. Paul, have you >> ever>>> considered Pajara[6] as a possible melody scale? >>>> Seems awfully improper, but descending it resembles a famous >> Stravisky theme. >> What I should have asked was if you've tried 443443 as a melody scale > in 22-et.Right; I thought you were talking about 2 2 7 2 2 7, but then you clarified. See Yahoo groups: /tuning-math/message/9886 * [with cont.] .
Message: 10420 - Contents - Hide Contents Date: Mon, 01 Mar 2004 23:47:10 Subject: Re: 9-limit stepwise From: Paul Erlich --- 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 "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: >>>> >>>> On the other end of the size scale we have these. Paul, have > you >>> ever>>>> considered Pajara[6] as a possible melody scale? >>>>>> Seems awfully improper, but descending it resembles a famous >>> Stravisky theme. >>>> What I should have asked was if you've tried 443443 as a melody > scale >> in 22-et. >> Right; I thought you were talking about 2 2 7 2 2 7, but then you > clarified. See Yahoo groups: /tuning- * [with cont.] math/message/9886.Oh yeah, the answer is yes.
Message: 10421 - Contents - Hide Contents Date: Mon, 01 Mar 2004 19:04:54 Subject: Re: Harmonized melody in the 7-limit From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:> If I didn't know better I'd say you were trying to BS me. What > is a lattice of note classes?It's the kind of lattice I was talking about--for each octave ewquivalence class, we have a lattice point. Hence there is a lattice point representing 9,9/4,9/8... etc but only one.
Message: 10422 - Contents - Hide Contents Date: Mon, 01 Mar 2004 19:14:52 Subject: Re: Hanzos From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>> My recollection is that Paul H.'s algorithm assigns a unique >>> lattice route (and therefore hanzo) to each 9-limit interval. >>>> So what? You still get an infinite number representing each interval, >> since you can multiply by arbitary powers of the dummy comma 9/3^2. >> An infinite number from where? If you look at the algorithm, that > dummy comma has zero length.If it has length zero then we are not talking about a lattice at all, though a quotient of it (modding out the dummy comma) might be. In a symmetrical lattice it necessarily has the same length as, for example, 11/3^2, which is of length sqrt(1^2+2^2-1*2)=sqrt(3). It does *not* have the same length as 11/9, which is of length one, of course.>>> Certainly it can be used to find the set of lattice points >>> within distance <= 2 of a given point. >>>> Hahn's alogorithm can, or hahnzos can, or what? >> The algorithm can.Why do you think Hahn's definition of "distance" would work for this problem? If you tell me what it is, we could check and see. However, you've just told me it does not have the basic properties of a metric, so I'm inclined to object to calling it a "distance".
Message: 10423 - Contents - Hide Contents Date: Mon, 01 Mar 2004 19:17:53 Subject: Re: Canonical generators for 7-limit planar temperaments From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" <paul.hjelmstad@u...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> > wrote: >>> If we have a comma which does not (as 50/49, for instance, will) > lead>> to a part-octave period, we can find generators which are analogous >> to the period-and-generator generator of a linear temperament, by >> projecting orthogonally onto a plane in a way which makes the comma >> dissapper (people may recall my lattice diagrams of this.) We can > >> Gene, could you point me to this lattice diagram/message?You can find the plots in the files section of this newsgroup, in the planarplots directory.> So projecting onto the plane, in such a way as to make the comma > disappear, gives us the generators, based on the projected values, > which are minimized, as close to the origin as possible? Does the > lattice diagram mentioned above explain all?The diagrams are just jpg files, but you can see the result of projecting, and what lies near the origin.
Message: 10424 - Contents - Hide Contents Date: Mon, 01 Mar 2004 18:45:24 Subject: Re: Hanzos From: Carl Lumma>>>> >y recollection is that Paul H.'s algorithm assigns a unique >>>> lattice route (and therefore hanzo) to each 9-limit interval. >>>>>> So what? You still get an infinite number representing each >>> interval, since you can multiply by arbitary powers of the dummy >>> comma 9/3^2. >>>> An infinite number from where? If you look at the algorithm, that >> dummy comma has zero length. >>If it has length zero then we are not talking about a lattice at all, >though a quotient of it (modding out the dummy comma) might be. In a >symmetrical lattice it necessarily has the same length as, for >example, 11/3^2, which is of length sqrt(1^2+2^2-1*2)=sqrt(3). It >does *not* have the same length as 11/9, which is of length one, of >course.Yes, you're onto something here. In the unweighted lattice there is a point for 9/3^2, which lies on the diameter-1 hull.>>>> Certainly it can be used to find the set of lattice points >>>> within distance <= 2 of a given point. >>>>>> Hahn's alogorithm can, or hahnzos can, or what? >>>> The algorithm can. >>Why do you think Hahn's definition of "distance" would work for this >problem? If you tell me what it is, we could check and see. However, >you've just told me it does not have the basic properties of a >metric, so I'm inclined to object to calling it a "distance".I thought you acknowledged the receipt of the algorithm...>Given a Fokker-style interval vector (I1, I2, . . . In): > >1. Go to the rightmost nonzero exponent; add its absolute value >to the total. > >2. Use that exponent to cancel out as many exponents of the opposite >sign as possible, starting to its immediate left and working right; >discard anything remaining of that exponent. > >3. If any nonzero exponents remain, go back to step one, otherwise >stop.As for the problem, let's start over. Call the position occupied by 1/1 in 1/1,8/7,4/3,8/5 the root of utonal tetrads. Now 7-limit tetrads sharing a common dyad (pair of pitches) with an otonal tetrad rooted on 1/1 will have roots... 5/4, 3/2, 15/8, 7/4, 35/32, 21/16 ...and those sharing a common tone (single pitch) will have roots... ack, this is visually exhausting... am I correct that they are the 1- and 2-combinations of: 1, 3, 5, 7, 1/3, 1/5, 1/7 ? If so, can you say why adding 9 and 1/9 to this list will not produce an equivalent 9-limit result? -Carl
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