A top 20 11-limit superparticularly generated linear temperament list

I look all the 11-limit superparticulars >= 49/48 and found all the 11-limit linear temperaments they generated; there turned out to be 319 of them. The following is the top 20 in terms of low (logarithmically flat) badness, plus a special guest star "Monzo" which is what 45/44, 64/63 and 81/80 will give you. If Joe objects I will quit calling it that.

"Arabic", by the way, came in #32 but clearly would be much higher if we forgot about 7. Things to note are temperaments which don't seem to have much to do with "good" ets and temperaments which are close relatives of other temperaments. The first on the list, Hemiennealimmal, could certainly claim to be able to produce authentic Partch tunings of the 11-limit, and it would be interesting to check in how many keys 72 notes tempered in this way could play the Partch 43-tone scale to extreme accuracy. It is also interesting to note how many of the top temperaments are compatible with 72.

1. Hemiennealimmal

[36, 54, 36, 18, 2, -44, -96, -68, -145, -74]

[2401/2400, 3025/3024, 4375/4374, 9801/9800]

ets 72, 198, 270, 342, 612

[[0, 2, 3, 2, 1], [18, 12, 17, 34, 54]]

[.4591217954, 1/18]

a = 33.0568/72 = 280.9825/612 = 550.9491544 cents

badness 78.02778100
rms .1987978829
g 36.

2. Miracle

[6, -7, -2, 15, -25, -20, 3, 15, 59, 49]

[225/224, 243/242, 385/384, 441/440, 540/539, 2401/2400, 3025/3024]

ets 10, 31, 41, 72

[[0, 6, -7, -2, 15], [1, 1, 3, 3, 2]]

[.9722688696e-1, 1]

a = 7.0003/72 = 116.6722643 cents

badness 125.5016755
rms 1.901465778
g 12.35198075

3. Octoid

[24, 32, 40, 24, -5, -4, -45, 3, -55, -71]

[540/539, 3025/3024, 4375/4374, 9801/9800]

ets 72, 80, 152, 224, 296

[[0, -3, -4, -5, -3], [8, 16, 23, 28, 31]]

[.1383934690, 1/8]

a = 9.9643/72 = 31.0001/224 = 166.0721626

badness 147.3854996
rms .7687062948
g 23.42160176

4. Undecimal augmented fifth

[12, 22, -4, -6, 7, -40, -51, -71, -90, -3]

[385/384, 441/440, 3025/3024, 4375/4374, 9801/9800]

ets 26, 46, 72, 118, 190

[[0, 6, 11, -2, -3], [2, -1, -3, 7, 9]]

a = 25.0090/72 = 416.8172169 cents

[.3473476807, 1/2]

badness 169.9769111
rms 1.249416902
g 19.06380265

5.

[12, 34, 20, 30, 26, -2, 6, -49, -48, 15]

[243/242, 441/440, 540/539, 2401/2400, 9801/9800]

ets 58, 72, 130

[[0, -6, -17, -10, -15], [2, 4, 7, 7, 9]]

[.6933142420e-1, 1/2]

a = 4.9919/72 = 83.1977090 cents

badness 179.9856041
rms 1.462301383
g 17.95231779

6.

[12, -2, 20, -6, -31, -2, -51, 52, -7, -86]

[225/224, 385/384, 540/539, 9801/9800]

ets 22, 50, 72, 94

[[0, 6, -1, 10, -3], [2, 1, 5, 2, 8]]

[.1806533524, 1/2]

a = 13.0070/72 = 216.7840228 cents

badness 195.0280356
rms 1.584514315
g 17.95231779

7. Orwell

[7, -3, 8, 2, -21, -7, -21, 27, 15, -22]

[99/98, 121/120, 176/175, 225/224, 385/384, 540/539]

ets 9, 22, 31, 53

[[0, 7, -3, 8, 2], [1, 0, 3, 1, 3]]

[.2262038561, 1]

a = 11.9888/53 = 271.4446272 cents

badness 210.4954018
rms 5.548614670
g 8.860022575

8. Semihemimeantone

[4, 16, 9, 10, 16, 3, 2, -24, -32, -3]

[81/80, 99/98, 121/120, 243/242, 441/440, 540/539, 2401/2400]

ets 31

[[0, -4, -16, -9, -10], [1, 3, 8, 6, 7]]

[.3548751316, 1]

a = 11.0011/31 = 425.8501579 cents

badness 218.9540099
rms 6.965622568
g 7.914724072

9.

[9, 5, -3, 7, -13, -30, -20, -21, -1, 30]

ets 15, 31, 46

[121/120, 126/125, 176/175, 385/384, 441/440, 3025/3024]

[[0, 9, 5, -3, 7], [1, 1, 2, 3, 3]]

[.6494333856e-1, 1]

a = 2.0132/31 = 77.9320062 cents

badness 223.3668950
rms 4.418576095
g 10.52547929

10.

[1, -1, 3, 4, -4, 2, 3, 10, 13, 1]

[55/54, 56/55, 99/98, 3025/3024]

ets 5

[[0, -1, 1, -3, -4], [1, 2, 2, 4, 5]]

[.3798204598, 1]

11.0148/29 = 455.7845520 cents

badness 235.8100854
rms 44.34125247
g 2.725540575

11.

[6, 10, 10, 8, 2, -1, -8, -5, -16, -12]

[50/49, 55/54, 99/98, 100/99, 121/120, 540/539, 9801/9800]

ets 22

[[0, -3, -5, -5, -4], [2, 4, 6, 7, 8]]

[.1375489239, 1/2]

a = 3.0261/22 = 165.0587086 cents

badness 238.7261371
rms 11.89273384
g 6.047431569

12. Nonkleismic

[10, 9, 7, 25, -9, -17, 5, -9, 27, 46]

[126/125, 176/175, 243/242, 441/440, 540/539, 2401/2400]

ets 31, 58, 89

[[0, 10, 9, 7, 25], [1, -1, 0, 1, -3]]

[.2584558979, 1]

a = 23.0026/89 = 310.1470775 cents

badness 240.3019988
rms 3.316530191
g 13.06303399

13. Magic

[5, 1, 12, -8, -10, 5, -30, 25, -22, -64]

[100/99, 225/224, 385/384, 540/539]

ets 19, 22, 41

[[0, 5, 1, 12, -8], [1, 0, 2, -1, 6]]

[.3172615104, 1]

13.0077/41 = 380.7138126 cents

badness 242.7224832
rms 4.730404304
g 10.62006188

14. Septimal

[0, 0, 7, 0, 0, 11, 0, 16, 0, -24]

[55/54, 81/80, 100/99, 121/120, 243/242]

ets 7

[[0, 0, 0, -1, 0], [7, 11, 16, 21, 24]]

[.2141802354, 1/7]

a = 257.0162824 cents

badness 245.8506632
rms 22.63634705
g 4.183300133

15. Meanertone

[1, 4, 3, -1, 4, 2, -5, -4, -16, -13]

[55/54, 56/55, 81/80, 3025/3024]

ets 5

[[0, -1, -4, -3, 1], [1, 2, 4, 4, 3]]

[.4194849382, 1]

a = 13.0040/31 = 503.3819256 cents

badness 252.8666930
rms 47.54854253
g 2.725540575

16. Tweedledee

[3, 5, 9, 4, 1, 6, -4, 7, -8, -20]

ets 15

[55/54, 56/55, 100/99, 121/120, 126/125, 3025/3024]

[[0, -3, -5, -9, -4], [1, 2, 3, 4, 4]]

[.1329702752, 1]

a = 1.9946/15 = 159.5643303 cents

badness 255.7850727
rms 22.12985764
g 4.342481185

17. Tweedledum

[3, 5, -6, 4, 1, -18, -4, -28, -8, 32]

[55/54, 64/63, 100/99, 121/120, 176/175, 385/384]

ets 7, 15, 22

[[0, -3, -5, 6, -4], [1, 2, 3, 2, 4]]

[.1357721305, 1]

a = 2.9870/22 = 162.9265567 cents

badness 262.2914819
rms 11.79393546
g 6.430951940

18. Pentoid

[2, 3, 1, -2, 0, -4, -10, -6, -15, -9]

[49/48, 56/55, 99/98, 385/384]

ets 4, 5, 9

[[0, -2, -3, -1, 2], [1, 2, 3, 3, 3]]

[.2183480607, 1]

a = 5.0220/23 = 262.0176727 cents

badness 267.0829245
rms 40.16092708
g 3.116774888

19. Monzoid

[1, 4, -2, -1, 4, -6, -5, -16, -16, 4]

[55/54, 64/63, 81/80, 385/384]

ets 5, 7

[[0, -1, -4, 2, 1], [1, 2, 4, 2, 3]]

[.4181947520, 1]

a = 5.0183/12 = 501.8337024 cents

badness 269.9708171
rms 39.86372247
g 3.150963571

20. Catakleismic

[6, 5, 22, -21, -6, 18, -54, 37, -66, -135]

[225/224, 385/384, 540/539, 4375/4374]

ets 19, 72

[[0, 6, 5, 22, -21], [1, 0, 1, -3, 9]]

[.2639230436, 1]

a = 19.0025/72 = 316.7076522 cents

badness 271.0589693
rms 1.697136764
g 20.98979344

Number 46 Monzo

[64/63, 81/80, 100/99, 176/175]

ets 7, 12

[[0, -1, -4, 2, -6], [1, 2, 4, 2, 6]]

[.4190088422, 1]

a = 5.0281/12 = 502.8106107 cents

badness 312.5112733
rms 28.87226550
g 4.174754057

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:

> 16. Tweedledee
>
> [3, 5, 9, 4, 1, 6, -4, 7, -8, -20]
>
> ets 15
>
> [55/54, 56/55, 100/99, 121/120, 126/125, 3025/3024]
>
> [[0, -3, -5, -9, -4], [1, 2, 3, 4, 4]]
>
> [.1329702752, 1]
>
> a = 1.9946/15 = 159.5643303 cents
>
> badness 255.7850727
> rms 22.12985764
> g 4.342481185

The heptad in 15-et is 2222223 in its various flavors, and this is pretty much a 15-et system, though [7,7,7,7,7,7,11] in the 53-et and
[9,9,9,9,9,9,14] in the 68-et are interesting alternatives.

> 17. Tweedledum
>
> [3, 5, -6, 4, 1, -18, -4, -28, -8, 32]
>
> [55/54, 64/63, 100/99, 121/120, 176/175, 385/384]
>
> ets 7, 15, 22
>
> [[0, -3, -5, 6, -4], [1, 2, 3, 2, 4]]
>
> [.1357721305, 1]
>
> a = 2.9870/22 = 162.9265567 cents
>
> badness 262.2914819
> rms 11.79393546
> g 6.430951940

The 22-et version of this is 3333334, but 8/59 is closer to the rms optimal generator, and has the same 7s and 11s as the 118-et; this version of it goes [8,8,8,8,8,8,11].

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:

This one is related to Dum and Dee, so I guess it's just Tweedle.

> 11. Tweedle
>
> [6, 10, 10, 8, 2, -1, -8, -5, -16, -12]
>
> [50/49, 55/54, 99/98, 100/99, 121/120, 540/539, 9801/9800]
>
> ets 22
>
> [[0, -3, -5, -5, -4], [2, 4, 6, 7, 8]]
>
> [.1375489239, 1/2]
>
> a = 3.0261/22 = 165.0587086 cents
>
> badness 238.7261371
> rms 11.89273384
> g 6.047431569

Hi Gene,

> From: genewardsmith <genewardsmith@juno.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Friday, January 18, 2002 8:22 PM
> Subject: [tuning-math] A top 20 11-limit superparticularly generated
linear temperament list
>
>
> I look all the 11-limit superparticulars >= 49/48 and
> found all the 11-limit linear temperaments they generated;
> there turned out to be 319 of them. The following is
> the top 20 in terms of low (logarithmically flat) badness,
> plus a special guest star "Monzo" which is what 45/44,
> 64/63 and 81/80 will give you. If Joe objects I will
> quit calling it that.

Cool! I never object to having my name on a tuning thing
(something like the old show-biz dictum "even bad publicity
is good publicity").

But ... it would be really nice if you could explain, as
only tw examples, exactly what all this means. Since I've
already played around with these particular unison-vectors,
explaining what you did here would help me a lot to
understand the rest of your work.

>
> 19. Monzoid
>
> [1, 4, -2, -1, 4, -6, -5, -16, -16, 4]
>
> [55/54, 64/63, 81/80, 385/384]
>
> ets 5, 7
>
> [[0, -1, -4, 2, 1], [1, 2, 4, 2, 3]]
>
> [.4181947520, 1]
>
> a = 5.0183/12 = 501.8337024 cents
>
> badness 269.9708171
> rms 39.86372247
> g 3.150963571
>
> ...
>
> Number 46 Monzo
>
>
> [64/63, 81/80, 100/99, 176/175]
>
> ets 7, 12
>
> [[0, -1, -4, 2, -6], [1, 2, 4, 2, 6]]
>
> [.4190088422, 1]
>
> a = 5.0281/12 = 502.8106107 cents
>
> badness 312.5112733
> rms 28.87226550
> g 4.174754057

-monz

_________________________________________________________
Do You Yahoo!?
Get your free @yahoo.com address at http://mail.yahoo.com

> From: monz <joemonz@yahoo.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Saturday, January 19, 2002 7:52 AM
> Subject: Re: [tuning-math] A top 20 11-limit superparticularly generated
linear temperament list
>

>
> Hi Gene,
>
> > From: genewardsmith <genewardsmith@juno.com>
> > To: <tuning-math@yahoogroups.com>
> > Sent: Friday, January 18, 2002 8:22 PM
> > Subject: [tuning-math] A top 20 11-limit superparticularly generated
> linear temperament list
> >
> >
> But ... it would be really nice if you could explain, as
> only tw examples, exactly what all this means. Since I've
> already played around with these particular unison-vectors,
> explaining what you did here would help me a lot to
> understand the rest of your work.
> >
> > Number 46 Monzo
> >
> >
> > [64/63, 81/80, 100/99, 176/175]
> >
> > ets 7, 12
> >
> > [[0, -1, -4, 2, -6], [1, 2, 4, 2, 6]]
> >
> > [.4190088422, 1]
> >
> > a = 5.0281/12 = 502.8106107 cents
> >
> > badness 312.5112733
> > rms 28.87226550
> > g 4.174754057
>
>
>
>
> -monz

OK, I gave this a whirl thru my spreadsheet and this is
what I got:

kernel

2 3 5 7 11 unison vectors ~cents

[ 1 0 0 0 0 ] = 2:1 0
[ 4 0 -2 -1 1 ] = 176:175 9.864608166
[ 2 -2 2 0 -1 ] = 100:99 17.39948363
[ 6 -2 0 -1 0 ] = 64:63 27.2640918
[-4 4 -1 0 0 ] = 81:80 21.5062896

adjoint

[ 0 0 0 -0 0 ]
[ 0 1 1 -1 0 ]
[ 0 4 4 -4 0 ]
[ 0 -2 -2 2 0 ]
[ 0 6 6 -6 0 ]

determinant = | 0 |

mapping of Ets (top row above) to Uvs

[ 1 1/3 2/3 -2/3 0 ]
[ 4 2&2/3 2&2/3 -2&2/3 0 ]
[ 2 1&1/3 1&1/3 -1&1/3 0 ]
[ 6 4 4 -4 0 ]
[-4 -2&2/3 -2&2/3 2&2/3 0 ]

I don't really understand what this is saying either.

(Many of the "0"s were actually given by Excel as
tiny numbers such as "2.22045 * 10^-16", which is
what it actually gave as the determinant.)

-monz

_________________________________________________________
Do You Yahoo!?
Get your free @yahoo.com address at http://mail.yahoo.com

--- In tuning-math@y..., "monz" <joemonz@y...> wrote:

> But ... it would be really nice if you could explain, as
> only tw examples, exactly what all this means. Since I've
> already played around with these particular unison-vectors,
> explaining what you did here would help me a lot to
> understand the rest of your work.
>
>
> >
> > 19. Monzoid

This was number 19 on the list, in terms of a badness measure.

> > [1, 4, -2, -1, 4, -6, -5, -16, -16, 4]

This is the "wedgie". When standardized, there is a unique wedgie corresponding to each (non-torsion, non-equal) temperment. This is a linear temperament wedgie; in the 11-limit the planar temperament wedgies also have ten dimensions; however this wedgie is computed from three unisons or two ets, whereas a planar would be computed from two unisons or three ets.

> > [55/54, 64/63, 81/80, 385/384]

These are all the 11-limit superparticulars equal to or less than 49/48 which are commas of the temperament--meaning they are tempered out. Since there are four of them, there is a linear dependency, but we can generate Monzoid from three independent ones.

> > ets 5, 7

These are "standard" ets, which round off to the nearest integer when mapping primes; 12 is not on the list, but h5+h7 would be if I listed anything "nonstandard".

> > [[0, -1, -4, 2, 1], [1, 2, 4, 2, 3]]

This is the period matrix, in a way easier to print than as a
4x2 matrix. The first list is the first column, giving maps to primes of the generator, the second column is the octaves.

> > [.4181947520, 1]

> > a = 5.0183/12 = 501.8337024 cents

These are the two generators, the second being merely an octave, and the first being a slightly sharp fourth.

> > badness 269.9708171
> > rms 39.86372247
> > g 3.150963571

"Badness" is the flat badness measure, "rms" is an average value for how much, in cents, the 11-limit consonances are off (40 cents!), and
g is the average number of generator steps to get to a consonance (a mere 3.)

> > Number 46 Monzo
> >
> >
> > [64/63, 81/80, 100/99, 176/175]

45/44 does not appear only because 45/44 > 49/48, which I used as a cut-off.

> > ets 7, 12

This time, the "standard" h12 12-et map makes its appearance.

> > [[0, -1, -4, 2, -6], [1, 2, 4, 2, 6]]
> >
> > [.4190088422, 1]
> >
> > a = 5.0281/12 = 502.8106107 cents

A small difference in the size of the optimal generator, because Monzo maps 11 differently than Monzoid.

> > badness 312.5112733
> > rms 28.87226550
> > g 4.174754057

The different 11-map makes the 11-limit more accurate, but it takes more steps on average because 11 maps to -6 and not 1.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
Sorry about cutting your statement/question up as follows but I want
to address the accuracy and complexity points separately.

> The first on the list,
Hemiennealimmal, could certainly claim to be able to produce authentic
Partch tunings of the 11-limit, ... to extreme accuracy.
>

Yes. But even Partch didn't require such accuracy. I understand that
he couldn't tell the difference between his scale and either 41-tET or
72-tET versions of it. So the extreme accuracy doesn't mitigate the
badness of the extreme complexity, and Miracle leaves Hemiennealimmal
for dead with any reasonable badness measure that relates to human
beings.

> ... and it would be interesting to check
in how many keys 72 notes tempered in this way could play the Partch
43-tone scale ...
>

I think the answer is zero. But the question seems fairly irrelevant
of any temperament, since no-one I know wants to have as many as 72
notes per octave on a keyboard or fretboard and no composer I know
wants to have to deal with that many notes (choosing always some more
manageable subset). Also I don't find it likely that anyone would want
to play Partch's scale in more than one "key" per piece.

A more relevant question is how many notes of a given temperament does
it take to include _one_ version of Partch's scale (without
conflating any notes)? Since Partch's scale contains the 11-limit
diamond, if I'm reading your cryptic lists of numbers correctly, the
answer for hemiennealimmal cannot be less than (2*3+1)*18 = 126. The
answer for Miracle is (2*22+1)*1 = 45.

> 1. Hemiennealimmal
>
> [36, 54, 36, 18, 2, -44, -96, -68, -145, -74]
>
> [2401/2400, 3025/3024, 4375/4374, 9801/9800]
>
> ets 72, 198, 270, 342, 612
>
> [[0, 2, 3, 2, 1], [18, 12, 17, 34, 54]]
>
> [.4591217954, 1/18]
>
> a = 33.0568/72 = 280.9825/612 = 550.9491544 cents
>
> badness 78.02778100
> rms .1987978829
> g 36.
>
>
> 2. Miracle
>
> [6, -7, -2, 15, -25, -20, 3, 15, 59, 49]
>
> [225/224, 243/242, 385/384, 441/440, 540/539, 2401/2400, 3025/3024]
>
> ets 10, 31, 41, 72
>
> [[0, 6, -7, -2, 15], [1, 1, 3, 3, 2]]
>
> [.9722688696e-1, 1]
>
> a = 7.0003/72 = 116.6722643 cents
>
> badness 125.5016755
> rms 1.901465778
> g 12.35198075

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> Yes. But even Partch didn't require such accuracy.

If I remember correctly, he did require such accuracy. He considered the fifth of 53-et to be good enough, and the third to be close, but no cigar. The fifth is flat by .068 cents, and the major third is flat
1.408 cents. The rms for Hemiennealimmal is 1/5 cent, so it is in this range and well towards the small side. Hence it seems reasonable to conclude that using this tuning would be authentically Partch.

Of course, there's the question of how accurately his own instruments were tuned, which would give another handle on what authenticity would mean in this context.

I understand that
> he couldn't tell the difference between his scale and either 41-tET or
> 72-tET versions of it.

The question is one of his requirements, not his hearing.

> I think the answer is zero.

Here it is:

1--81/80--33/32--21/20--16/15--12/11--11/10--10/9--9/8--8/7--7/6
32/27--6/5--11/9--5/4--14/11--9/7--21/16--4/3--27/20--11/8--7/5
10/7--16/11--40/27--3/2--32/21--14/9--11/7--8/5--18/11--5/3--27/16
12/7--7/4--16/9--9/5--20/11--11/6--15/8--40/21--64/33--160/81--(2)

This is mapped to

[[0, 5, 3, 1, -5, 1, -2, -1, 4, -2, 0, -6, -1, -3, 3, 1, 2, 4, -2, 3, 1, -1, 1, -1, -3, 2, -4, -2, -1, -3, 3, 1, 6, 0, 2, -4, 1, 2, -1, 5,
-1, -3, -5]
, [0, -41, -24, -7, 43, -6, 19, 11, -30, 20, 4, 54, 13, 30, -19, -2,
-10, -26, 24, -17, 0, 17, 1, 18, 35, -6, 44, 28, 20, 37, -12, 5, -36, 14, -2, 48, 7, -1, 24, -25, 25, 42, 59]]

by Hemiennealimmal, so you are right, it won't fit--the first generator ranging from -6 to 6.

But the question seems fairly irrelevant
> of any temperament, since no-one I know wants to have as many as 72
> notes per octave on a keyboard or fretboard and no composer I know
> wants to have to deal with that many notes (choosing always some more
> manageable subset).

This is assuming that you must be using a keyboard or fretboard. Partch, after all, *did* have 43 actual tones per octave in play, so I don't see how this theory holds up.

Also I don't find it likely that anyone would want
> to play Partch's scale in more than one "key" per piece.

Even if they did not, tuning Partch's scale in this way would give you some equivalences for free (deriving from 2401/2400, 3025/3024,
4375/4374 and 9801/9800) which would make tempering Partch's 43 tones in this way a perfectly reasonable option.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> > But the question seems fairly irrelevant
> > of any temperament, since no-one I know wants to have as many as
72
> > notes per octave on a keyboard or fretboard and no composer I know
> > wants to have to deal with that many notes (choosing always some
more
> > manageable subset).
>
> This is assuming that you must be using a keyboard or fretboard.

Not in the case of the composers.

But I was wrong to say they always choose some more manageable subset.
Sometimes they just use continuously gliding tones etc.

> Partch, after all, *did* have 43 actual tones per octave in play, so
I
don't see how this theory holds up.
>

Huh? 43 is considerably less than 72, being only about 60% of it. So
it _supports_ this theory.

> Also I don't find it likely that anyone would want
> > to play Partch's scale in more than one "key" per piece.
>
> Even if they did not, tuning Partch's scale in this way would give
you some equivalences for free (deriving from 2401/2400, 3025/3024,
> 4375/4374 and 9801/9800) which would make tempering Partch's 43
tones in this way a perfectly reasonable option.

Yes. Certainly. But those equivalences would also be quite acceptable
_without_ tempering. Perhaps one day we'll have all of Partch's works
in some machine-readable form and can check to see if he ever used any
of them.

>But I was wrong to say they always choose some more manageable
>subset. Sometimes they just use continuously gliding tones etc.

What about Maneri and his students (72 tones)?

>>Partch, after all, *did* have 43 actual tones per octave in play,
>>so I don't see how this theory holds up.
>
>Huh? 43 is considerably less than 72, being only about 60% of it.
>So it _supports_ this theory.

Actually, Partch considered having far more than 43, stopping at
43 only for pragmatic reasons (according to him), and often using
far less. I can think of at least 4 separate diatribes given by
Partch at different times on the association of the 43-tone scale
with his music. He thought of his working area as the infinite
space of JI.

>>The numbers are edges/connectivity in the 5, 7, 9 and 11-limits.
>>I conclude that a great deal is gained by tempering in this way,
>>and nothing significant is conceded in terms of quality of
>>intonation. Of course, 72-et would do much better yet, but then
>>some concessions will have been made.
>
>I totally agree. With the discovery of microtemperaments like
>this, an insistence on strict RI starts to look more like a
>religion than an informed decision.

I agree. But there's something else that's starting to look like
a religion -- the insistence on re-casting everyone else's scale
choices in terms of temperament. If you've ever sat down to use
one of these scales, you know that they already have so many more
resources than a system which has been the life's-work of three
centuries of our best minds, that adding resources by tempering
may not be the first thing on the composer's mind. That's not to
say it isn't. But if it isn't, it is no sign of ignorance.

-Carl

--- In tuning-math@y..., "clumma" <carl@l...> wrote:

> >I totally agree. With the discovery of microtemperaments like
> >this, an insistence on strict RI starts to look more like a
> >religion than an informed decision.

> I agree. But there's something else that's starting to look like
> a religion -- the insistence on re-casting everyone else's scale
> choices in terms of temperament.

Hmmm? What's "religious" about looking at the mathematics of someone's scale?

This really isn't anything radical--the same point could have been made at any time with ets, by looking at the the versions of Partch's scale which results from using the likes of the 224, 270, 342, 494 or 612 ets.

> 1. Hemiennealimmal
[...]

They all had b = 1 octave?

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> > 1. Hemiennealimmal
> [...]
>
> They all had b = 1 octave?

This one has b = 1/18; it's in the [a, b] thing which gives the generators.

--- In tuning-math@y..., "monz" <joemonz@y...> wrote:
>
> > From: monz <joemonz@y...>
> > To: <tuning-math@y...>
> > Sent: Saturday, January 19, 2002 7:52 AM
> > Subject: Re: [tuning-math] A top 20 11-limit superparticularly
generated
> linear temperament list
> >
>
> >
> > Hi Gene,
> >
> > > From: genewardsmith <genewardsmith@j...>
> > > To: <tuning-math@y...>
> > > Sent: Friday, January 18, 2002 8:22 PM
> > > Subject: [tuning-math] A top 20 11-limit superparticularly
generated
> > linear temperament list
> > >
> > >
> > But ... it would be really nice if you could explain, as
> > only tw examples, exactly what all this means. Since I've
> > already played around with these particular unison-vectors,
> > explaining what you did here would help me a lot to
> > understand the rest of your work.
> > >
> > > Number 46 Monzo
> > >
> > >
> > > [64/63, 81/80, 100/99, 176/175]
> > >
> > > ets 7, 12
> > >
> > > [[0, -1, -4, 2, -6], [1, 2, 4, 2, 6]]
> > >
> > > [.4190088422, 1]
> > >
> > > a = 5.0281/12 = 502.8106107 cents
> > >
> > > badness 312.5112733
> > > rms 28.87226550
> > > g 4.174754057
> >
> >
> >
> >
> > -monz
>
>
> OK, I gave this a whirl thru my spreadsheet and this is
> what I got:
>
>
> kernel
>
> 2 3 5 7 11 unison vectors ~cents
>
> [ 1 0 0 0 0 ] = 2:1 0
> [ 4 0 -2 -1 1 ] = 176:175 9.864608166
> [ 2 -2 2 0 -1 ] = 100:99 17.39948363
> [ 6 -2 0 -1 0 ] = 64:63 27.2640918
> [-4 4 -1 0 0 ] = 81:80 21.5062896
>
> adjoint
>
> [ 0 0 0 -0 0 ]
> [ 0 1 1 -1 0 ]
> [ 0 4 4 -4 0 ]
> [ 0 -2 -2 2 0 ]
> [ 0 6 6 -6 0 ]
>
> determinant = | 0 |
>
>
> mapping of Ets (top row above) to Uvs

Which ETs?
>
> [ 1 1/3 2/3 -2/3 0 ]
> [ 4 2&2/3 2&2/3 -2&2/3 0 ]
> [ 2 1&1/3 1&1/3 -1&1/3 0 ]
> [ 6 4 4 -4 0 ]
> [-4 -2&2/3 -2&2/3 2&2/3 0 ]
>
>
> I don't really understand what this is saying either.
>
> (Many of the "0"s were actually given by Excel as
> tiny numbers such as "2.22045 * 10^-16", which is
> what it actually gave as the determinant.)

This is a linear temperament. If you want a periodicity block, you
have to add one "chromatic" unison vector to the list of "commatic"
unison vectors Gene gave.

>>I agree. But there's something else that's starting to look like
>>a religion -- the insistence on re-casting everyone else's scale
>>choices in terms of temperament.
>
>Hmmm? What's "religious" about looking at the mathematics of
>someone's scale?

I didn't bring up the term religion here, and bringing up religion
is a very religious thing to do. Yes, numerology has many traits
in common with some religions, and numerology has seeped in to RI.
But there are other traits of religion, including the re-casting of
history into one's own perspective. There's no doubt in my mind
that Partch's music can be said to ignore all sorts of commas -- who
cares? Will we then pronounce that Partch would have been better
off using temperament x? By what criteria will we say a composer's
work was not tempered? Temperament gets a lot of attention,
because RI is too simple to occupy theorists. But it is not too
simple to occupy composers, and theorists do a disservice when they
do not actively point that out.

-Carl

--- In tuning-math@y..., "clumma" <carl@l...> wrote:

> Actually, Partch considered having far more than 43, stopping at
> 43 only for pragmatic reasons (according to him), and often using
> far less. I can think of at least 4 separate diatribes given by
> Partch at different times on the association of the 43-tone scale
> with his music. He thought of his working area as the infinite
> space of JI.

Don't forget Ben Johnston, who often composed with 81 tones or more!

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning-math@y..., "clumma" <carl@l...> wrote:
>
> > Actually, Partch considered having far more than 43, stopping at
> > 43 only for pragmatic reasons (according to him), and often using
> > far less. I can think of at least 4 separate diatribes given by
> > Partch at different times on the association of the 43-tone scale
> > with his music. He thought of his working area as the infinite
> > space of JI.
>
> Don't forget Ben Johnston, who often composed with 81 tones or more!

OK. I stand corrected. And I apologise for the "religion" remark. One
might just as easily say "This subset of hemiennealimmal is so close
to RI we might as well try to tune the instruments to the RI scale
since that's easier to calculate and tunable by ear (assuming harmonic
timbres).

Hi Dave,

> From: dkeenanuqnetau <d.keenan@uq.net.au>
> To: <tuning-math@yahoogroups.com>
> Sent: Monday, January 21, 2002 2:35 PM
> Subject: Re: A top 20 11-limit superparticularly generated linear
temperament list
>
>
> ... One might just as easily say "This subset of
> hemiennealimmal is so close to RI we might as well try
> to tune the instruments to the RI scale since that's
> easier to calculate and tunable by ear (assuming
> harmonic timbres).

Hmmm ... given remarks on the tuning list in the past by Daniel
Wolf about the multiple senses (other than the two obvious
ones) or other ratios implied by Partch's use of his 43-tone
scale, now *that* sounds like something pretty close to the mark!

Partch apparently wove harmonic structures into his compositions
which sometimes require the listener to infer different rational
implications from his scale than the obvious ones. Without
examining the actual mathematics of it, your revised statement
here seems to me to be a good way to model that aspect of Partch's
compositional practice.

-monz

_________________________________________________________
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--- In tuning-math@y..., "monz" <joemonz@y...> wrote:
>
> Hi Dave,
>
>
> > From: dkeenanuqnetau <d.keenan@u...>
> > To: <tuning-math@y...>
> > Sent: Monday, January 21, 2002 2:35 PM
> > Subject: Re: A top 20 11-limit superparticularly generated linear
> temperament list
> >
> >
> > ... One might just as easily say "This subset of
> > hemiennealimmal is so close to RI we might as well try
> > to tune the instruments to the RI scale since that's
> > easier to calculate and tunable by ear (assuming
> > harmonic timbres).
>
>
> Hmmm ... given remarks on the tuning list in the past by Daniel
> Wolf about the multiple senses (other than the two obvious
> ones) or other ratios implied by Partch's use of his 43-tone
> scale, now *that* sounds like something pretty close to the mark!
>
> Partch apparently wove harmonic structures into his compositions
> which sometimes require the listener to infer different rational
> implications from his scale than the obvious ones. Without
> examining the actual mathematics of it, your revised statement
> here seems to me to be a good way to model that aspect of Partch's
> compositional practice.
>
>
>
> -monz

Well, Monz, it would be good to know if Partch exploited _only_ the
hemiennealimmal equivalencies, or _only_ the MIRACLE equivalencies,
or what. Wilson seems to have felt that he exploited enough
equivalencies that a closed 41-tone system (as in 41-tET) was
actually implied. But Wilson never seems to have thought much about
MIRACLE, let along hemiennealimmal.

> From: paulerlich <paul@stretch-music.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Tuesday, January 22, 2002 4:39 AM
> Subject: [tuning-math] Re: A top 20 11-limit superparticularly generated
linear temperament list
>
>
> > Partch apparently wove harmonic structures into his compositions
> > which sometimes require the listener to infer different rational
> > implications from his scale than the obvious ones. Without
> > examining the actual mathematics of it, your [Dave Keenan's]
> > revised statement here seems to me to be a good way to model
> > that aspect of Partch's compositional practice.
> >
> >
> >
> > -monz
>
> Well, Monz, it would be good to know if Partch exploited _only_ the
> hemiennealimmal equivalencies, or _only_ the MIRACLE equivalencies,
> or what. Wilson seems to have felt that he exploited enough
> equivalencies that a closed 41-tone system (as in 41-tET) was
> actually implied. But Wilson never seems to have thought much about
> MIRACLE, let along hemiennealimmal.

All good points, Paul. I was careful to add "without examining
the actual mathematics", because I don't even know what
"hemiennealimmal" is!!!!! (... but I'm still studying!)

Why don't you guys take a look at that? Sounds interesting.

-monz

_________________________________________________________
Do You Yahoo!?
Get your free @yahoo.com address at http://mail.yahoo.com

--- In tuning-math@y..., "monz" <joemonz@y...> wrote:

> > Well, Monz, it would be good to know if Partch exploited _only_
the
> > hemiennealimmal equivalencies, or _only_ the MIRACLE
equivalencies,
> > or what. Wilson seems to have felt that he exploited enough
> > equivalencies that a closed 41-tone system (as in 41-tET) was
> > actually implied. But Wilson never seems to have thought much
about
> > MIRACLE, let along hemiennealimmal.
>
>
> All good points, Paul. I was careful to add "without examining
> the actual mathematics", because I don't even know what
> "hemiennealimmal" is!!!!! (... but I'm still studying!)
>
> Why don't you guys take a look at that? Sounds interesting.

We would need Partch's scores in a readable form . . .