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Message: 7825 Date: Thu, 30 Oct 2003 19:05:14 Subject: Re: UVs for 46-ET 11-limit PB From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...>
wrote:
>
> > i grouped a few quartets of 11-limit commas together at random,
and
> > after a couple of 72s, found that 9801:9800, 3025:3024, 441:440
and
> > 176:175 together give 46. then i tried multiplying and dividing
pairs
> > of these to get simpler ratios (being sure to keep 4 linearly > > independent ones at each stage); one possibility is 896:891,
385:384,
> > 125:126, and 176:175.
> > If you TM reduce this, you get the 11-limit TM basis for h46, namely > [121/120, 126/125, 176/175, 245/243]. A Fokker block for this can be > obtained from > > q[i] = (81/80)^i (121/120)^round(-19i/46) (126/125)^round(-24i/46) > (176/175)^round(38i/46) (245/243)^round(31i/46) > > This gives the following scale: > > 1 [0, 0, 0, 0, 0] > 55/54 [-1, -3, 1, 0, 1] > 36/35 [2, 2, -1, -1, 0] > 25/24 [-3, -1, 2, 0, 0] > 35/33 [0, -1, 1, 1, -1] > 15/14 [-1, 1, 1, -1, 0] > 11/10 [-1, 0, -1, 0, 1] > 10/9 [1, -2, 1, 0, 0] > 198/175 [1, 2, -2, -1, 1] > 25/22 [-1, 0, 2, 0, -1] > 7/6 [-1, -1, 0, 1, 0] > 90/77 [1, 2, 1, -1, -1] > 6/5 [1, 1, -1, 0, 0] > 11/9 [0, -2, 0, 0, 1] > 216/175 [3, 3, -2, -1, 0] > 5/4 [-2, 0, 1, 0, 0] > 14/11 [1, 0, 0, 1, -1] > 9/7 [0, 2, 0, -1, 0] > 33/25 [0, 1, -2, 0, 1] > 4/3 [2, -1, 0, 0, 0] > 1188/875 [2, 3, -3, -1, 1] > 15/11 [0, 1, 1, 0, -1] > 7/5 [0, 0, -1, 1, 0] > 140/99 [2, -2, 1, 1, -1] > 10/7 [1, 0, 1, -1, 0] > 22/15 [1, -1, -1, 0, 1] > 875/594 [-1, -3, 3, 1, -1] > 3/2 [-1, 1, 0, 0, 0] > 50/33 [1, -1, 2, 0, -1] > 14/9 [1, -2, 0, 1, 0] > 11/7 [0, 0, 0, -1, 1] > 8/5 [3, 0, -1, 0, 0] > 175/108 [-2, -3, 2, 1, 0] > 18/11 [1, 2, 0, 0, -1] > 5/3 [0, -1, 1, 0, 0] > 77/45 [0, -2, -1, 1, 1] > 12/7 [2, 1, 0, -1, 0] > 44/25 [2, 0, -2, 0, 1] > 175/99 [0, -2, 2, 1, -1] > 9/5 [0, 2, -1, 0, 0] > 20/11 [2, 0, 1, 0, -1] > 28/15 [2, -1, -1, 1, 0] > 66/35 [1, 1, -1, -1, 1] > 48/25 [4, 1, -2, 0, 0] > 35/18 [-1, -2, 1, 1, 0] > 108/55 [2, 3, -1, 0, -1]
my fokker block for [121/120, 126/125, 176/175, 245/243] is
ratio 3^ 5^ 7^ 11^
891/875 4 -3 -1 1
45/44 2 1 0 -1
21/20 1 -1 1 0
77/72 -2 0 1 1
15/14 1 1 -1 0
11/10 0 -1 0 1
875/792 -2 3 1 -1
9/8 2 0 0 0
25/22 0 2 0 -1
7/6 -1 0 1 0
33/28 1 0 -1 1
6/5 1 -1 0 0
175/144 -2 2 1 0
27/22 3 0 0 -1
5/4 0 1 0 0
77/60 -1 -1 1 1
9/7 2 0 -1 0
33/25 1 -2 0 1
175/132 -1 2 1 -1
27/20 3 -1 0 0
15/11 1 1 0 -1
7/5 0 -1 1 0
99/70 2 -1 -1 1
36/25 2 -2 0 0
35/24 -1 1 1 0
81/55 4 -1 0 -1
3/2 1 0 0 0
55/36 -2 1 0 1
54/35 3 -1 -1 0
25/16 0 2 0 0
35/22 0 1 1 -1
45/28 2 1 -1 0
33/20 1 -1 0 1
5/3 -1 1 0 0
297/175 3 -2 -1 1
75/44 1 2 0 -1
7/4 0 0 1 0
135/77 3 1 -1 -1
9/5 2 -1 0 0
11/6 -1 0 0 1
324/175 4 -2 -1 0
15/8 1 1 0 0
21/11 1 0 1 -1
27/14 3 0 -1 0
99/50 2 -2 0 1
2 0 0 0 0
i transformed the lattice by the inverse of the fokker matrix and
threw out anything with a coordinate >.50000001 or <-.49999999. it
can't be that we differ only because you kept some or all of the -
0.5s, because that would only affect #23. hmm . . .
Message: 7826 Date: Thu, 30 Oct 2003 19:15:45 Subject: Re: UVs for 46-ET 11-limit PB From: Paul Erlich sorry folks -- there was a mysterious extra line in my program. when i removed it, i got the same block as gene, except 77/54 instead of 140/99 -- that's the point with a -.5 coordinate . . . --- 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:
> >
> > > i grouped a few quartets of 11-limit commas together at random,
> and
> > > after a couple of 72s, found that 9801:9800, 3025:3024, 441:440
> and
> > > 176:175 together give 46. then i tried multiplying and dividing
> pairs
> > > of these to get simpler ratios (being sure to keep 4 linearly > > > independent ones at each stage); one possibility is 896:891,
> 385:384,
> > > 125:126, and 176:175.
> > > > If you TM reduce this, you get the 11-limit TM basis for h46,
namely
> > [121/120, 126/125, 176/175, 245/243]. A Fokker block for this can
be
> > obtained from > > > > q[i] = (81/80)^i (121/120)^round(-19i/46) (126/125)^round(-24i/46) > > (176/175)^round(38i/46) (245/243)^round(31i/46) > > > > This gives the following scale: > > > > 1 [0, 0, 0, 0, 0] > > 55/54 [-1, -3, 1, 0, 1] > > 36/35 [2, 2, -1, -1, 0] > > 25/24 [-3, -1, 2, 0, 0] > > 35/33 [0, -1, 1, 1, -1] > > 15/14 [-1, 1, 1, -1, 0] > > 11/10 [-1, 0, -1, 0, 1] > > 10/9 [1, -2, 1, 0, 0] > > 198/175 [1, 2, -2, -1, 1] > > 25/22 [-1, 0, 2, 0, -1] > > 7/6 [-1, -1, 0, 1, 0] > > 90/77 [1, 2, 1, -1, -1] > > 6/5 [1, 1, -1, 0, 0] > > 11/9 [0, -2, 0, 0, 1] > > 216/175 [3, 3, -2, -1, 0] > > 5/4 [-2, 0, 1, 0, 0] > > 14/11 [1, 0, 0, 1, -1] > > 9/7 [0, 2, 0, -1, 0] > > 33/25 [0, 1, -2, 0, 1] > > 4/3 [2, -1, 0, 0, 0] > > 1188/875 [2, 3, -3, -1, 1] > > 15/11 [0, 1, 1, 0, -1] > > 7/5 [0, 0, -1, 1, 0] > > 140/99 [2, -2, 1, 1, -1] > > 10/7 [1, 0, 1, -1, 0] > > 22/15 [1, -1, -1, 0, 1] > > 875/594 [-1, -3, 3, 1, -1] > > 3/2 [-1, 1, 0, 0, 0] > > 50/33 [1, -1, 2, 0, -1] > > 14/9 [1, -2, 0, 1, 0] > > 11/7 [0, 0, 0, -1, 1] > > 8/5 [3, 0, -1, 0, 0] > > 175/108 [-2, -3, 2, 1, 0] > > 18/11 [1, 2, 0, 0, -1] > > 5/3 [0, -1, 1, 0, 0] > > 77/45 [0, -2, -1, 1, 1] > > 12/7 [2, 1, 0, -1, 0] > > 44/25 [2, 0, -2, 0, 1] > > 175/99 [0, -2, 2, 1, -1] > > 9/5 [0, 2, -1, 0, 0] > > 20/11 [2, 0, 1, 0, -1] > > 28/15 [2, -1, -1, 1, 0] > > 66/35 [1, 1, -1, -1, 1] > > 48/25 [4, 1, -2, 0, 0] > > 35/18 [-1, -2, 1, 1, 0] > > 108/55 [2, 3, -1, 0, -1]
> > my fokker block for [121/120, 126/125, 176/175, 245/243] is > > ratio 3^ 5^ 7^ 11^ > 891/875 4 -3 -1 1 > 45/44 2 1 0 -1 > 21/20 1 -1 1 0 > 77/72 -2 0 1 1 > 15/14 1 1 -1 0 > 11/10 0 -1 0 1 > 875/792 -2 3 1 -1 > 9/8 2 0 0 0 > 25/22 0 2 0 -1 > 7/6 -1 0 1 0 > 33/28 1 0 -1 1 > 6/5 1 -1 0 0 > 175/144 -2 2 1 0 > 27/22 3 0 0 -1 > 5/4 0 1 0 0 > 77/60 -1 -1 1 1 > 9/7 2 0 -1 0 > 33/25 1 -2 0 1 > 175/132 -1 2 1 -1 > 27/20 3 -1 0 0 > 15/11 1 1 0 -1 > 7/5 0 -1 1 0 > 99/70 2 -1 -1 1 > 36/25 2 -2 0 0 > 35/24 -1 1 1 0 > 81/55 4 -1 0 -1 > 3/2 1 0 0 0 > 55/36 -2 1 0 1 > 54/35 3 -1 -1 0 > 25/16 0 2 0 0 > 35/22 0 1 1 -1 > 45/28 2 1 -1 0 > 33/20 1 -1 0 1 > 5/3 -1 1 0 0 > 297/175 3 -2 -1 1 > 75/44 1 2 0 -1 > 7/4 0 0 1 0 > 135/77 3 1 -1 -1 > 9/5 2 -1 0 0 > 11/6 -1 0 0 1 > 324/175 4 -2 -1 0 > 15/8 1 1 0 0 > 21/11 1 0 1 -1 > 27/14 3 0 -1 0 > 99/50 2 -2 0 1 > 2 0 0 0 0 > > i transformed the lattice by the inverse of the fokker matrix and > threw out anything with a coordinate >.50000001 or <-.49999999. it > can't be that we differ only because you kept some or all of the - > 0.5s, because that would only affect #23. hmm . . .
Message: 7827
Date: Thu, 30 Oct 2003 22:31:34
Subject: Re: UVs for 46-ET 11-limit PB
From: Paul Erlich
i performed the same calculation but this time the transformed
coordinates were between zero (inclusive) and one (exclusive). this
was to check monz who did the same thing offlist. most of the pitches
of this block agree with his, but not all (i even used the same signs
on the unison vectors as him):
ratio 3^ 5^ 7^ 11^
1 0 0 0 0
782/779 -3 -3 1 2
77/75 -1 -2 1 1
847/810 -4 -1 1 2
2156/2025 -4 -2 2 1
242/225 -2 -2 0 2
1232/1125 -2 -3 1 1
539/486 -5 0 2 1
847/750 -1 -3 1 2
154/135 -3 -1 1 1
999/853 -4 -3 2 2
88/75 -1 -2 0 1
2137/1774 -2 -4 1 2
599/491 -5 -1 2 2
154/125 0 -3 1 1
847/675 -3 -2 1 2
2393/1873 -3 -3 2 1
484/375 -1 -3 0 2
2464/1875 -1 -4 1 1
539/405 -4 -1 2 1
847/625 0 -4 1 2
308/225 -2 -2 1 1
2971/2114 -3 -4 2 2
176/125 0 -3 0 1
77/54 -3 0 1 1
467/319 -4 -2 2 2
22/15 -1 -1 0 1
1694/1125 -2 -3 1 2
616/405 -4 -1 1 1
968/625 0 -4 0 2
847/540 -3 -1 1 2
1078/675 -3 -2 2 1
121/75 -1 -2 0 2
616/375 -1 -3 1 1
3388/2025 -4 -2 1 2
1056/625 1 -4 0 1
77/45 -2 -1 1 1
2932/1669 -3 -3 2 2
44/25 0 -2 0 1
3388/1875 -1 -4 1 2
1232/675 -3 -2 1 1
749/403 1 -5 0 2
847/450 -2 -2 1 2
2156/1125 -2 -3 2 1
242/125 0 -3 0 2
1232/625 0 -4 1 1
> > > If you TM reduce this, you get the 11-limit TM basis for h46,
> namely
> > > [121/120, 126/125, 176/175, 245/243]. A Fokker block for this
can
> be
> > > obtained from > > > > > > q[i] = (81/80)^i (121/120)^round(-19i/46) (126/125)^round(-
24i/46)
> > > (176/175)^round(38i/46) (245/243)^round(31i/46) > > > > > > This gives the following scale: > > > > > > 1 [0, 0, 0, 0, 0] > > > 55/54 [-1, -3, 1, 0, 1] > > > 36/35 [2, 2, -1, -1, 0] > > > 25/24 [-3, -1, 2, 0, 0] > > > 35/33 [0, -1, 1, 1, -1] > > > 15/14 [-1, 1, 1, -1, 0] > > > 11/10 [-1, 0, -1, 0, 1] > > > 10/9 [1, -2, 1, 0, 0] > > > 198/175 [1, 2, -2, -1, 1] > > > 25/22 [-1, 0, 2, 0, -1] > > > 7/6 [-1, -1, 0, 1, 0] > > > 90/77 [1, 2, 1, -1, -1] > > > 6/5 [1, 1, -1, 0, 0] > > > 11/9 [0, -2, 0, 0, 1] > > > 216/175 [3, 3, -2, -1, 0] > > > 5/4 [-2, 0, 1, 0, 0] > > > 14/11 [1, 0, 0, 1, -1] > > > 9/7 [0, 2, 0, -1, 0] > > > 33/25 [0, 1, -2, 0, 1] > > > 4/3 [2, -1, 0, 0, 0] > > > 1188/875 [2, 3, -3, -1, 1] > > > 15/11 [0, 1, 1, 0, -1] > > > 7/5 [0, 0, -1, 1, 0] > > > 140/99 [2, -2, 1, 1, -1] > > > 10/7 [1, 0, 1, -1, 0] > > > 22/15 [1, -1, -1, 0, 1] > > > 875/594 [-1, -3, 3, 1, -1] > > > 3/2 [-1, 1, 0, 0, 0] > > > 50/33 [1, -1, 2, 0, -1] > > > 14/9 [1, -2, 0, 1, 0] > > > 11/7 [0, 0, 0, -1, 1] > > > 8/5 [3, 0, -1, 0, 0] > > > 175/108 [-2, -3, 2, 1, 0] > > > 18/11 [1, 2, 0, 0, -1] > > > 5/3 [0, -1, 1, 0, 0] > > > 77/45 [0, -2, -1, 1, 1] > > > 12/7 [2, 1, 0, -1, 0] > > > 44/25 [2, 0, -2, 0, 1] > > > 175/99 [0, -2, 2, 1, -1] > > > 9/5 [0, 2, -1, 0, 0] > > > 20/11 [2, 0, 1, 0, -1] > > > 28/15 [2, -1, -1, 1, 0] > > > 66/35 [1, 1, -1, -1, 1] > > > 48/25 [4, 1, -2, 0, 0] > > > 35/18 [-1, -2, 1, 1, 0] > > > 108/55 [2, 3, -1, 0, -1]
Message: 7828 Date: Thu, 30 Oct 2003 00:04:52 Subject: Re: Linear temperament names? From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote: >
> > Yes. If my reader is to apply my method for themselves they will need > > to calculate monzos from ratios, and yes I use lattice diagrams to > > illustrate the process.
> > If they are going to turn monzos into octaves/generators, they will > need the information a mapping matrix gives, whether or not presented > as a matrix.
Yes indeed. I'm even using the MATLAB/Octave syntax, since it's the
most compact I know of ("Octave" is a free GNU clone of MATLAB). I'm
just not _calling_ them matrices, and I'm making them
octave-equivalent, which means there are no parameters for the prime
2, and periods are only counted modulo the octave.
> This idea you have rejected with contempt.
Well I wouldn't go that far. But I'm certainly leaving it up to someone else to teach the readers of Xenharmonicon, matrix arithmetic in its full generality.
> What to you > propose as a replacement?
I simply describe the individual scalar arithmetic operations involved, with examples, without ever ascending to the level of abstraction of vector or matrix operators. Of course such abstraction is a truly wonderful thing, and you and Graham have done more than anyone in helping us to appreciate that on tuning-math, but I don't have the space to introduce my readers to it. And I don't have the need since I'm only dealing with the simple cases in my examples - 5-limit, usually with an octave period.
> To be sure, the information can be extracted > using the generator size in cents, but that, as I pointed out, is > *harder*.
No, I'm not trying to do that.
Message: 7829 Date: Thu, 30 Oct 2003 22:35:48 Subject: Re: UVs for 46-ET 11-limit PB From: Paul Erlich d'oh! i was using format rat, which approximates ratios that are too complex with simpler ratios! the real ratios are . . . well who cares, they agree perfectly with monz's, and of course the monzos below agree . . . --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:
> i performed the same calculation but this time the transformed > coordinates were between zero (inclusive) and one (exclusive). this > was to check monz who did the same thing offlist. most of the
pitches
> of this block agree with his, but not all (i even used the same
signs
> on the unison vectors as him): > > ratio 3^ 5^ 7^ 11^ > 1 0 0 0 0 > 782/779 -3 -3 1 2 > 77/75 -1 -2 1 1 > 847/810 -4 -1 1 2 > 2156/2025 -4 -2 2 1 > 242/225 -2 -2 0 2 > 1232/1125 -2 -3 1 1 > 539/486 -5 0 2 1 > 847/750 -1 -3 1 2 > 154/135 -3 -1 1 1 > 999/853 -4 -3 2 2 > 88/75 -1 -2 0 1 > 2137/1774 -2 -4 1 2 > 599/491 -5 -1 2 2 > 154/125 0 -3 1 1 > 847/675 -3 -2 1 2 > 2393/1873 -3 -3 2 1 > 484/375 -1 -3 0 2 > 2464/1875 -1 -4 1 1 > 539/405 -4 -1 2 1 > 847/625 0 -4 1 2 > 308/225 -2 -2 1 1 > 2971/2114 -3 -4 2 2 > 176/125 0 -3 0 1 > 77/54 -3 0 1 1 > 467/319 -4 -2 2 2 > 22/15 -1 -1 0 1 > 1694/1125 -2 -3 1 2 > 616/405 -4 -1 1 1 > 968/625 0 -4 0 2 > 847/540 -3 -1 1 2 > 1078/675 -3 -2 2 1 > 121/75 -1 -2 0 2 > 616/375 -1 -3 1 1 > 3388/2025 -4 -2 1 2 > 1056/625 1 -4 0 1 > 77/45 -2 -1 1 1 > 2932/1669 -3 -3 2 2 > 44/25 0 -2 0 1 > 3388/1875 -1 -4 1 2 > 1232/675 -3 -2 1 1 > 749/403 1 -5 0 2 > 847/450 -2 -2 1 2 > 2156/1125 -2 -3 2 1 > 242/125 0 -3 0 2 > 1232/625 0 -4 1 1 >
> > > > If you TM reduce this, you get the 11-limit TM basis for h46,
> > namely
> > > > [121/120, 126/125, 176/175, 245/243]. A Fokker block for this
> can
> > be
> > > > obtained from > > > > > > > > q[i] = (81/80)^i (121/120)^round(-19i/46) (126/125)^round(-
> 24i/46)
> > > > (176/175)^round(38i/46) (245/243)^round(31i/46) > > > > > > > > This gives the following scale: > > > > > > > > 1 [0, 0, 0, 0, 0] > > > > 55/54 [-1, -3, 1, 0, 1] > > > > 36/35 [2, 2, -1, -1, 0] > > > > 25/24 [-3, -1, 2, 0, 0] > > > > 35/33 [0, -1, 1, 1, -1] > > > > 15/14 [-1, 1, 1, -1, 0] > > > > 11/10 [-1, 0, -1, 0, 1] > > > > 10/9 [1, -2, 1, 0, 0] > > > > 198/175 [1, 2, -2, -1, 1] > > > > 25/22 [-1, 0, 2, 0, -1] > > > > 7/6 [-1, -1, 0, 1, 0] > > > > 90/77 [1, 2, 1, -1, -1] > > > > 6/5 [1, 1, -1, 0, 0] > > > > 11/9 [0, -2, 0, 0, 1] > > > > 216/175 [3, 3, -2, -1, 0] > > > > 5/4 [-2, 0, 1, 0, 0] > > > > 14/11 [1, 0, 0, 1, -1] > > > > 9/7 [0, 2, 0, -1, 0] > > > > 33/25 [0, 1, -2, 0, 1] > > > > 4/3 [2, -1, 0, 0, 0] > > > > 1188/875 [2, 3, -3, -1, 1] > > > > 15/11 [0, 1, 1, 0, -1] > > > > 7/5 [0, 0, -1, 1, 0] > > > > 140/99 [2, -2, 1, 1, -1] > > > > 10/7 [1, 0, 1, -1, 0] > > > > 22/15 [1, -1, -1, 0, 1] > > > > 875/594 [-1, -3, 3, 1, -1] > > > > 3/2 [-1, 1, 0, 0, 0] > > > > 50/33 [1, -1, 2, 0, -1] > > > > 14/9 [1, -2, 0, 1, 0] > > > > 11/7 [0, 0, 0, -1, 1] > > > > 8/5 [3, 0, -1, 0, 0] > > > > 175/108 [-2, -3, 2, 1, 0] > > > > 18/11 [1, 2, 0, 0, -1] > > > > 5/3 [0, -1, 1, 0, 0] > > > > 77/45 [0, -2, -1, 1, 1] > > > > 12/7 [2, 1, 0, -1, 0] > > > > 44/25 [2, 0, -2, 0, 1] > > > > 175/99 [0, -2, 2, 1, -1] > > > > 9/5 [0, 2, -1, 0, 0] > > > > 20/11 [2, 0, 1, 0, -1] > > > > 28/15 [2, -1, -1, 1, 0] > > > > 66/35 [1, 1, -1, -1, 1] > > > > 48/25 [4, 1, -2, 0, 0] > > > > 35/18 [-1, -2, 1, 1, 0] > > > > 108/55 [2, 3, -1, 0, -1]
Message: 7830 Date: Thu, 30 Oct 2003 00:25:25 Subject: Re: Linear temperament names? From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...>
> I didn't call names,
That's true. And again I apologise for this. and you began the conversation by telling me to
> rub it in my hair.
Huh? You'd better go back and read it again. I wrote "Blow it out your ear". For a translation see Idioms: blow chunks -- blow my cover * [with cont.] (Wayb.)
> My impression still is that you are determined not > to listen to opinions other than your own.
A classic case of Jungian projection? I started this thread by asking for opinions other than my own on the naming of temperaments. I've accepted all of your suggestions but one (with some minor modifications). You might ask George Secor how I've managed to work with him on Sagittal for so long. But over this issue of whether I should use octave-equivalent vectors and matrices in my article, you're right, I am not interested in your opinion, and I didn't ask for it. But outside my article, I certainly agree with your standard format where 2's are explicit and the period comes before the generators.
Message: 7831
Date: Thu, 30 Oct 2003 22:47:40
Subject: Re: UVs for 46-ET 11-limit PB
From: Paul Erlich
when i get rid of that bad line in the program, this 896:891,
385:384, 125:126, and 176:175 block becomes:
ratio 3^ 5^ 7^ 11^
1 0 0 0 0
100/99 -2 2 0 -1
33/32 1 0 0 1
25/24 -1 2 0 0
16/15 -1 -1 0 0
15/14 1 1 -1 0
35/32 0 1 1 0
10/9 -2 1 0 0
9/8 2 0 0 0
8/7 0 0 -1 0
7/6 -1 0 1 0
33/28 1 0 -1 1
6/5 1 -1 0 0
40/33 -1 1 0 -1
99/80 2 -1 0 1
5/4 0 1 0 0
32/25 0 -2 0 0
9/7 2 0 -1 0
21/16 1 0 1 0
4/3 -1 0 0 0
27/20 3 -1 0 0
48/35 1 -1 -1 0
7/5 0 -1 1 0
64/45 -2 -1 0 0
10/7 0 1 -1 0
35/24 -1 1 1 0
40/27 -3 1 0 0
3/2 1 0 0 0
32/21 -1 0 -1 0
14/9 -2 0 1 0
25/16 0 2 0 0
8/5 0 -1 0 0
160/99 -2 1 0 -1
33/20 1 -1 0 1
5/3 -1 1 0 0
56/33 -1 0 1 -1
12/7 1 0 -1 0
7/4 0 0 1 0
16/9 -2 0 0 0
9/5 2 -1 0 0
64/35 0 -1 -1 0
28/15 -1 -1 1 0
15/8 1 1 0 0
48/25 1 -2 0 0
64/33 -1 0 0 -1
99/50 2 -2 0 1
i think this is the closest yet to fulfilling monz's original
requirements . . .
--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...>
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:
> > hi paul, > > > > --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > > wrote: > >
> > > --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich"
<perlich@a...>
> > > wrote:
> > > > --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote: > > > >
> > > > > i tried to derive a 46-note periodicity-block > > > > > which is a subset of this list,
> > > > > > > > of course, *any* 46-note subset which has each of > > > > the 46-equal degrees exactly once is already a > > > > periodicity block . . .
> > > > > > right, i know that ... but i deliberately left in > > the duplicates and triplicates to see what you, Gene, > > et al would come up with. > > > > > >
> > > > (by the way, i don't agree with your reckoning of > > > > "equally close" . . .)
> > > > > > i know ... you use the hexagonal reckoning rather than > > the rectangular one i used. i considered doing that > > from the start, but it was just easier for me to do > > the one i did since i'm still using Excel for this > > kind of stuff. > > > > > >
> > > >
> > > > > using our software, > > > > > with the following unison-vectors: > > > > > > > > > > [ 3 -6 1 0] > > > > > [-4 -2 0 0] (diaschisma) > > > > > [ 2 -3 1 0] (small septimal comma) > > > > > [ 3 -4 0 -1] > > > > > > > > > > > > > > > and the software gave me a nice 46-note > > > > > periodicity-block, but it was entirely > > > > > in the [3,5]-plane.
> > > > > > > > the determinant of this matrix is 14, so i'm not sure how
> you're
> > > > getting a 46-note periodicity block out of it!
> > > > > > > > a sign in the first comma was reversed. its complete > > [2,3,5,7,11]-monzo should be [12 3 -6 -1 0] . > > see my post to Gene. > > > > > > > >
> > > i grouped a few quartets of 11-limit commas together > > > at random, and after a couple of 72s, found that > > > 9801:9800, 3025:3024, 441:440 and 176:175 together > > > give 46. then i tried multiplying and dividing pairs > > > of these to get simpler ratios (being sure to keep > > > 4 linearly independent ones at each stage); one > > > possibility is 896:891, 385:384, 125:126, and 176:175. > > > the matrix of these: > > > > > > -4 0 1 -1 > > > -1 1 1 1 > > > -2 3 -1 0 > > > 0 -2 -1 1 > > > > > > now i looked at the periodicity block defined by the > > > unit hypercube lying between 0 and 1 (instead of the > > > usual -.5 and .5) along each of the four transformed > > > coordinate axes using the matrix above: > > > > > > numerator denominator > > > 55 54 > > > 33 32 > > > 25 24 > > > 16 15 > > > 275 256 > > > <etc., snip> > > > > > > this has plenty of ratios with factors of 7 and 11 -- > > > hopefully it's close to what you need!
> > > > > > > > thanks ... but it would be easier for me to tell if > > instead of ratios the notes had already been factored > > into monzos. if anyone else cares to do it for me, > > that would be nice! ;-) > > > > > > > > > > -monz
> > here are the factorizations: first column is 3, second column is 5, > third column is 7, fourth column is 11: > > > -3 1 0 1 > 1 0 0 1 > -1 2 0 0 > -1 -1 0 0 > 0 2 0 1 > 0 -1 0 1 > -2 1 0 0 > 2 0 0 0 > -1 1 0 1 > -1 0 1 0 > 1 0 -1 1 > 1 -1 0 0 > -2 2 1 0 > 2 -1 0 1 > 0 1 0 0 > 0 -2 0 0 > 1 1 0 1 > 1 -2 0 1 > -1 0 0 0 > 3 -1 0 0 > 0 0 0 1 > 0 -1 1 0 > 2 -1 -1 1 > -1 2 0 1 > -1 1 1 0 > 1 1 -1 1 > 1 0 0 0 > -2 1 0 1 > 2 0 0 1 > 0 2 0 0 > 0 -1 0 0 > 1 2 0 1 > 1 -1 0 1 > -1 1 0 0 > 3 -2 -1 1 > 0 1 0 1 > 0 0 1 0 > 2 0 -1 1 > 2 -1 0 0 > -1 0 0 1 > 3 -1 0 1 > 1 1 0 0 > 1 -2 0 0 > 2 1 0 1 > 2 -2 0 1 > 0 0 0 0
Message: 7832 Date: Thu, 30 Oct 2003 00:54:54 Subject: Re: Linear temperament names? From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >I might merely change a > >hemiwuerschmidt to semiwuerschmidt or trikleismic to triple kleismic. > >I can't see these causing any confusion.
> > Use a little imagination.
Would you care to explain what you mean by that cryptic remark? Do you mean that you think they _could_ cause confusion? If so, then I think it would be good if we agreed on some system for distinguishing a prefix implying a fractional octave period from one that implies a fractional interval generator. I've made one proposal. I don't think you can accuse me of insisting on rigour or taking the fun out of naming when I'm accepting names like miracle and magic and catakleismic, and some based on commas like schismic and kleismic and wuerschmidt, and others based on generators. But I think we can only afford the luxury of totally non-descriptive names for the most common or the best. If I come up with some obscure temperament and tell you I'm composing an algorithmic piece in the "Fart" temperament. The first thing you're going to ask me is "What's that?", and I'll say, "Oh it's generated by three chains of generators which are one third of a major third". But if we had a system and I said I'm composing a piece in the "triple-trithirds" temperament, you wouldn't even have to ask. And of course it's possible to have both common _and_ systematic names for things.
Message: 7833 Date: Thu, 30 Oct 2003 23:37:14 Subject: Re: Linear temperament names? From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:
> sometimes the difference is even smaller -- look at #4 and #5 here: > 4 5 6 9 10 12 15 16 18 19 22 26 27 29 31 35 36... * [with cont.] (Wayb.)
Sometimes it doesn't exist at all.
Message: 7834 Date: Thu, 30 Oct 2003 01:07:55 Subject: Re: UVs for 46-ET 11-limit PB From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >If you TM reduce this, you get the 11-limit TM basis for h46,
namely
> >[121/120, 126/125, 176/175, 245/243]. A Fokker block for this can
be
> >obtained from > > > >q[i] = (81/80)^i (121/120)^round(-19i/46) (126/125)^round(-24i/46) > >(176/175)^round(38i/46) (245/243)^round(31i/46)
> > Whoa, the makings of an intelligible method, safe to try at home. > > What's i?
Sorry. "i" is an integer which runs from 0 to 45.
> And where has 81/80 come from?
81/80 is one step in 46-et. The TM basis has 4 commas, which
> should be enough to enclose an 11-limit block.
Right. I'm giving the block. If you toss in the 81/80, giving you a unimodular matrix, and invert the matrix, you get the values I used in the above computation: 46, -19, -24, 38, 31. They are the top row of the inverted matrix.
Message: 7835 Date: Thu, 30 Oct 2003 23:39:27 Subject: Re: UVs for 46-ET 11-limit PB From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >> >> And where has 81/80 come from?
> >> > > >> >81/80 is one step in 46-et.
> >> > >> How are we supposed to know that?
> > > >Surely you can calculate that? What are you asking?
> > Calculate it from what? One step of 46-et could be lots > of things. Why did you pick 81:80?
The answer to "Why a duck?" is "Why not a duck?" I needed any 11- limit interval which counted as one step of 46-equal, and it didn't matter which one I picked.
Message: 7836 Date: Thu, 30 Oct 2003 01:29:01 Subject: Re: Linear temperament names? From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:
> Sorry about that. I guess I was just too busy with other things like > work, family and sagittal, around the time you were posting lists of > 11-limit LTs. But it isn't as if lots of people are already using
the
> name, so I don't see a problem with changing it.
I've already have up a posting devoted *solely* to unidec. Moreover, I rejected names like your proposal for a reason--it is likely to confuse unidec with the minortone/hemiminortone family. If you decide to go ahead and start unilaterally renaming things, be warned that I at least may not agree to it. That will likely sow confusion. If you propose a renaming, we should have some kind of consensus. To start with, we need an explanation of why we should want one.
> If I use names in my paper that are only my own construction, I will > indicate this. But that doesn't include where I might merely change
a
> hemiwuerschmidt to semiwuerschmidt or trikleismic to triple
kleismic.
> I can't see these causing any confusion.
Are you planning to aka these?
> By the way, I have credited you and Graham as follows. Please let me > know if this is not accurate:
Gene's software
> uses a different method to generate temperament candidates for > testing, and has served as a very important check that Graham's > algorithm is not missing anything important.
I used various methods to generate candidates, in order to be self- checking, but that has little to do with my software beyond the fact that is able to deal with a variety of methods. This also pretty well ignores the whole issue of theory, and seems to suggest Graham came up with a list, and I checked it or something like that. It was a little more complicated.
Message: 7837 Date: Thu, 30 Oct 2003 02:46:45 Subject: Re: Linear temperament names? From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> Sure. I can't count the number of times I've checked out a new > subject and observed a terminology difference that wasn't discussed > anywhere. It drives me crazy wondering whether they're actually > the same.
Good point.
> >I don't think you can accuse me of insisting on rigour or taking the > >fun out of naming when I'm accepting names like miracle and magic and > >catakleismic, and some based on commas like schismic and kleismic and > >wuerschmidt, and others based on generators. But I think we can only > >afford the luxury of totally non-descriptive names for the most common > >or the best.
> > That makes sense. But I don't think generators sizes are important, > especially re. diatonic names. We've been over this before, and it > boggles me how you can support such a program.
So when some minimally-mathematical musician wants to tune up their synthesizer or whatever, to my new Fart temperament. I should give them a list of six 19-limit commas and say, "Go to it bud!". Gimme a break.
> In my book commas are the way to name linear temperaments,
How do you make that work past 5-limit? i.e when there are several commas.
> and what > we actually need is a systematic way of naming commas. Which, IIRC, > you've also tried your hand at. Did it have something to do with > komma and/or quoma? If so, I think that's bad, 'cause it isn't > phonetic. The "a" vs. "ina" thing I thought went over better.
Nah. I only use the spelling "komma" sometimes when I want to distinguish the generic term from the term for a specific range of sizes. Go back and read from Yahoo groups: /tuning-math/message/6875 * [with cont.] if you want to understand my comma naming system.
> >If I come up with some obscure temperament and tell you I'm composing > >an algorithmic piece in the "Fart" temperament. The first thing you're > >going to ask me is "What's that?", and I'll say, "Oh it's generated by > >three chains of generators which are one third of a major third".
> > And once I knew that, I wouldn't soon forget it. It's why fantastic > absurdities are so common in advertising.
You'll remember the name, but why will you remember what it _is_. Did you find some association, some "reason" for the name? I didn't intend any. I suppose all the "th" sounds in the description? Advertising often uses off-the-wall associations, but they _are_ associations.
> >But > >if we had a system and I said I'm composing a piece in the > >"triple-trithirds" temperament, you wouldn't even have to ask.
> > Oh yes I would. I haven't the foggiest idea what this means,
I said "If we had a system". And even if you didn't know the system, at least you'd only have to learn it once.
> and I > hang out here regularly. Does "triple" mean three periods in an > octave?
Yes.
> And does "tri" mean 3 times the size of a third, or 1/3rd > the size of a third?
Which seems more likely given that we usually give the generator in lowest terms, or at least significantly smaller than the octave. And we're usually interested in how many generators make up common consonances, not the other way 'round. I admit it's not ideal, but I can't find any short prefixes that unambiguously indicate a fraction rather than a multiple or a diatonic interval, except for semi- (or hemi-) and quarter-.
> And is it a major or minor third?
That would be part of the "system" too, only being allowed to drop the "major" and "perfect" off interval names. This is something we already do in several existing temperament names.
> And even > if you answer all these questions I won't necc. be able to get close > to the optimal generator. Really a cent of resolution is needed > here, or it can mean the difference between maps.
That's true. But you can only pack so much information into a name. At least it does get you in the right ballpark.
Message: 7838 Date: Thu, 30 Oct 2003 17:25:14 Subject: Re: Eponyms From: Carl Lumma
>Now that I've found what I think is a good system for naming kommas, >I'd prefer it to be called "385-kleisma"
In your scheme, the term kleisma tells us that the denominator must be 384, and not 383?
>or "5.7.11-kleisma".
...tells us how to combine factors of 5, 7, and 11 to get the right ratio? Just asking. -Carl
Message: 7839 Date: Thu, 30 Oct 2003 03:53:26 Subject: Re: Linear temperament names? From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> > wrote:
> > But it isn't as if lots of people are already using the > > name, so I don't see a problem with changing it.
> > I've already have up a posting devoted *solely* to unidec.
Yes. I saw that one. The only posts containing the word "Unidec" come from you. In most cases, the name is buried in a great list of temperaments and their vectors and matrices. It seems no one ever responded regarding that particular temperament, and as far as I can tell, you never volunteered any reason for the name until now. You can't assume that a lack of response means that everyone assents to a name forever. I think you've gone overboard in naming so many things long before anyone actually _needed_ a name for them. Is it a territorial thing? You've gone and peed on all these temperaments and commas and now you think another doggie-come-lately is trying to overpower your scent? :-) All I'm asking is - if you have a system for the more descriptive names (in particular those based on the generator and period) what is it? And if you don't, can we make some improvements in that direction?
> Moreover, > I rejected names like your proposal for a reason--it is likely to > confuse unidec with the minortone/hemiminortone family.
OK! At last we're getting into the kind of discussion I had hoped for, when instead I got a rant about how "sloppy" I was being by assuming octave equivalence or something. What are the mappings for minortone and hemiminortone?
> If you decide > to go ahead and start unilaterally renaming things,
I never wanted to _unilaterally_ name anything. _That_ is what you seem to have done. I wanted to discuss your names. You apparently refused until now.
> be warned that I > at least may not agree to it. That will likely sow confusion. If you > propose a renaming, we should have some kind of consensus.
Gee I wish I'd thought of that? :-)
> To start > with, we need an explanation of why we should want one.
One what? A consensus? Oh I guess you mean the renaming of Unidec? Well in a way it's _worse_ than a meaningless name like "Fart", because it _looks_ like it is descriptive, but in fact it tells you nothing about the temperament that it doesn't have in common with zillions of others.
> > If I use names in my paper that are only my own construction, I will > > indicate this. But that doesn't include where I might merely change
> a
> > hemiwuerschmidt to semiwuerschmidt or trikleismic to triple
> kleismic.
> > I can't see these causing any confusion.
> > Are you planning to aka these?
Carl has convinced me to do so. But I'd prefer we just agreed on a few principles to make some of these names a little more systematic. Do you have some such principles that I am not understanding? I don't understand why you are unwilling to discuss this.
> I used various methods to generate candidates, in order to be self- > checking, but that has little to do with my software beyond the fact > that is able to deal with a variety of methods. This also pretty well > ignores the whole issue of theory, and seems to suggest Graham came > up with a list, and I checked it or something like that.
What is it about the theory that you think I should mention?
> It was a little more complicated.
It always is. But I'm not writing a history article. Please feel free to suggest alternate wording.
Message: 7840 Date: Thu, 30 Oct 2003 06:30:56 Subject: Re: Linear temperament names? From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >So when some minimally-mathematical musician wants to tune up their > >synthesizer or whatever, to my new Fart temperament. I should give > >them a list of six 19-limit commas and say, "Go to it bud!". Gimme a > >break.
> > No, you should give them the size of the generator in cents, and > probably the entire scale, and probably the entire fretboard, and > probably the name of a good luthier. :)
Well yeah. I guess that's what I _should_ do. :-)
> I thought you were using tertia for division, or some such.
Yeah, I was at one stage, but I thought it sounded dopey. If only there were more words like "half" (or semi- or hemi-) and "quarter" where the fractional differs from the ordinal (i.e. "second", "fourth"). Hence we have continuing confusion in ordinary language over what things like "triannual" mean.
> C'mon. You could just say "700 cents". But that's not much of a > name. So you've got to dress it up, and make it more ambiguous in > the process. Seems pretty silly to me.
I agree. But it seems to be what we humans do. For some reason it doesn't count as a name if it's a number.
Message: 7841 Date: Thu, 30 Oct 2003 06:34:45 Subject: Re: Linear temperament names? From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> Sorry; never put anything in here... >
> >>Go back and read from > >>Yahoo groups: /tuning-math/message/6875 * [with cont.] > >>if you want to understand my comma naming system.
> > Ouch. This hurts my brain. > > I love Paul's names, as shown on the xoomer charts. > > -Carl
It can be like chemistry. The common compounds have common names _and_ systematic names. But the uncommon ones have only systematic names. I think it is a mistake to start making up common names for things _before_ they become, well, common.
Message: 7842 Date: Thu, 30 Oct 2003 11:39:12 Subject: Re: UVs for 46-ET 11-limit PB From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >> And where has 81/80 come from?
> > > >81/80 is one step in 46-et.
> > How are we supposed to know that?
Surely you can calculate that? What are you asking?
Message: 7843 Date: Thu, 30 Oct 2003 12:08:59 Subject: Re: Linear temperament names? From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:
> Yes. I saw that one. The only posts containing the word "Unidec"
come
> from you.
Until now, I seem to have been the only person to take an interest in the temperament. Is that my fault?
> You can't assume that a lack of response means that everyone assents > to a name forever. I think you've gone overboard in naming so many > things long before anyone actually _needed_ a name for them.
If you want to blame someone for that, why not pick on Paul, who has named 5-limit nanotemperaments no one could possibly ever use in practice. I have NOT named things needlessly; I want to talk about temperaments, and while the wedgie or TM comma basis might supply a name, they really don't work with human beings. You can't tell someone "Oh yes, that's the [12, 22, -4, -6, 7, -40, -51, -71, -90, -3] temperament"; but saying "Oh yes, that's unidec" gives something you might be able to associate with.
> Is it a territorial thing? You've gone and peed on all these > temperaments and commas and now you think another doggie-come-lately > is trying to overpower your scent? :-)
Oh, please. I've accomodated you by adopting a number of your names for temperaments on my lists of temperaments, which in some cases meant I changed the name I had listed, despite the fact that I don't much like how you name things. Your names tend to be boring and difficult to remember. However, if you are going to adopt an Only Dave Keenan Names attitude, don't expect me to play along. I prefer calling things vulture, like Paul does, to calling them things which sound like an explosion in a chemical factory.
> All I'm asking is - if you have a system for the more descriptive > names (in particular those based on the generator and period) what
is
> it? And if you don't, can we make some improvements in that
direction? If you want to propose a completely systematic naming proceedure, then have at it. Name everything the Keenan way, and make every name tell you just exactly what the temperament is. Maybe people will like it, and adopt your scheme. However, I see little point in half- measures.
> > Moreover, > > I rejected names like your proposal for a reason--it is likely to > > confuse unidec with the minortone/hemiminortone family.
> > OK! At last we're getting into the kind of discussion I had hoped
for,
> when instead I got a rant about how "sloppy" I was being by assuming > octave equivalence or something. > > What are the mappings for minortone and hemiminortone?
Minortone is the 5-limit 50031545098999707/50000000000000000 comma system, extendible to 7-limit. 17 10/9's make up a 6, and 35 a 40.
> > If you decide > > to go ahead and start unilaterally renaming things,
> > I never wanted to _unilaterally_ name anything. _That_ is what you > seem to have done.
I said renaming. I wanted to discuss your names. You apparently
> refused until now.
What gives you that idea? Do
> you have some such principles that I am not understanding? I don't > understand why you are unwilling to discuss this.
Yes, pick memorable names.
Message: 7844 Date: Fri, 31 Oct 2003 10:59:52 Subject: Re: Eponyms From: Manuel Op de Coul Gene wrote:
>5.7.11-kleisma has no advantages over 385/384 that I can see.
I think so too. It looks like it's the simplest undecimal kleisma and there isn't another one called that so I'll change the name in "undecimal kleisma". Manuel
Message: 7845 Date: Fri, 31 Oct 2003 15:23:53 Subject: Re: Eponyms From: Manuel Op de Coul Dave wrote:
>Calling it a comma implies that it is bigger than it really is.
The list isn't systematic in that sense. It's more a list of historical names. So "comma" is used in the general sense and can be a comma of any size. I think "neutral third comma" was a name proposed by Brian.
>Is there a general rule as to how a semicomma >differs from a kleisma?
What you thought, no. "Semicomma majeur" comes from Rameau.
>And 2109375/2097152 (10.1 c) you call _the_ >semicomma or Fokker's comma.
Fokker called it the semicomma and I added "Fokker's comma" for he mentioned it in his books, noticed its vanishing in 31-tET, etc. Whether he borrowed the name from Rameau I don't know but it's quite possible.
>What will you call 2835/2816 (11.6 c). I call it the 11:35-kleisma. >There's also 2893401/2883584 (5.9 c). I call it the 11:49-kleisma. >Admittedly these last two are not very much in need of a name.
No, I haven't thought about it. Manuel
Message: 7846 Date: Fri, 31 Oct 2003 17:42:31 Subject: Re: Eponyms From: Carl Lumma
>> >>5.7.11-kleisma has no advantages over 385/384 that I can see.
>> >> The latter must be factored to see what it's good for, and >> log'ed to give an exact size. The former gives a size range, >> and with the addition of the 3 exponent tells you what it's >> good for
> >So these are two advantages of "5.7.11-kleisma" over "385/384".
Since factoring is as hard as extracting the 3 exponent given the size range, the only difference is whether you prefer to know a range by memorizing a few words, or guestimate a range by doing some quick division.
>> (otherwise how'reyou going to say what pythagorean >> commas are good for?).
> >There aren't too many of them that have come to my attention so far. >These are well known and have common names: Pythagorean-limma, >apotome, Pythagorean-comma. If the previously described naming system >was simply applied they would all be 1-<whatevers> where "<whatever>" >stands for the correct size category. But I would change this to >3-<whatevers> or even better, Pythagorean-<whatevers>. So the limma >and comma would be the same as their common names and the apotome >would probably have the systematic name: Pythagorean-semitone. The >3-exponent can be extracted from the size category if necessary.
The idea of leaving out the 3's is clever but not beneficial, in my opinion.
>> But with the addition of the 3 exponent, >> we loose the ability to draft size ranges. What say you to >> this, Dave?
> >I'm not sure what you mean by "to draft size ranges"? If you mean "to >decide the boundaries of size ranges", I don't understand why you >would lose that ability. You could keep the same ranges and the up >front information about the 3-exponent would simply be redundant.
With the wrong ranges, you wouldn't be able to extract the 3 exponent, I assumed. -Carl
Message: 7847 Date: Fri, 31 Oct 2003 11:35:20 Subject: Re: Eponyms From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Manuel Op de Coul" <manuel.op.de.coul@e...> wrote:
> > Gene wrote:
> >5.7.11-kleisma has no advantages over 385/384 that I can see.
So call it the 385-comma. I agree that in this case the systematic name is not much better than the ratio. But it does at least tell you its approximate size in cents. Manuel:
> I think so too. It looks like it's the simplest undecimal > kleisma and there isn't another one called that so I'll change > the name in "undecimal kleisma".
No it's not the simplest. What about 243/242 (7.1 c) which you call the neutral third comma. I call it the 121-kleisma. Calling it a comma implies that it is bigger than it really is. Although 896/891 (9.7 c) is more complex than either 243/242 (7.1 c) or 385/384 (4.5 c), it occurs more commonly as a notational comma relative to Pythagorean, since it serves to notate simpler ratios. I notice you call 896/891 (9.7 c) the undecimal semicomma. I call it the 7:11-kleisma. Is there a general rule as to how a semicomma differs from a kleisma? This seems unlikely since you list "semicomma majeur" as an alternate name for _the_ kleisma (the 5^6-kleisma, 15625/15552, 8.1 c). And 2109375/2097152 (10.1 c) you call _the_ semicomma or Fokker's comma. What will you call 2835/2816 (11.6 c). I call it the 11:35-kleisma. There's also 2893401/2883584 (5.9 c). I call it the 11:49-kleisma. Admittedly these last two are not very much in need of a name.
Message: 7849 Date: Fri, 31 Oct 2003 10:06:10 Subject: Re: Eponyms From: Carl Lumma At 06:30 PM 10/30/2003, you wrote:
>--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
>> >Now that I've found what I think is a good system for naming kommas, >> >I'd prefer it to be called "385-kleisma"
>> >> In your scheme, the term kleisma tells us that the denominator must >> be 384, and not 383? >>
>> >or "5.7.11-kleisma".
>> >> ...tells us how to combine factors of 5, 7, and 11 to get the >> right ratio?
> >I already did. Sorry I didn't give examples. > >The 5, 7 and 11 are all on the same side of the ratio, or there would >have been a colon ":" in there.
How do you pronounce that?
>They are all only to the power given, namely 1.
How do you do it with higher powers?
>It's a kleisma so it's in the range 4.5 c (a bit arbitrary at present) >to 11.7 c (actually, exactly half a pythagorean comma).
385:383 is in that range.
>That's how a dumb algorithm would have to do it, but you or I >(assuming we knew something about the system) would say: Its got 385 >as a factor along with some powers of 2 and 3. I know roughly how big >it is so I wonder if it's 386/385 or 385/384.
Oh, I thought you always gave the numerator.
>Oh 384 has prime factors of only 2's and 3's.
How do we know it's only got 2's and 3's if we're only given "385-kleisma"? -Carl
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