Tuning-Math Digests messages 10300 - 10324

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Message: 10300

Date: Sat, 14 Feb 2004 21:23:56

Subject: Re: A modest proposal

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> What about using Herman's phenomenon as a high-error cutoff?

Works for linears -- but for other kinds of temperament?


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Message: 10302

Date: Sat, 14 Feb 2004 21:41:16

Subject: Re: A symmetric-based 7-limit temperament list

From: Paul Erlich

Injera involves two long chains of fifths, and fifths are just as 
long as any other consonance in the symmetric lattice Gene used here. 
In a 5-limit version of this list, 2187;2048 would surely score quite 
poorly because of the long chain of fifths it involves.

--- In tuning-math@xxxxxxxxxxx.xxxx Herman Miller <hmiller@I...> 
wrote:
> This ordering seems to be good at keeping similar/related 
temperaments 
> together. It's missing pelogic, injera, and dicot, though. I can 
> understand why pelogic and dicot might be missing, but injera [2, 
8, 8, 
> 8, 7, -4] is a good enough temperament that it should have made the 
list.
> 
> Gene Ward Smith wrote:
> 
> > 11: [1, -1, 3, -4, 2, 10] [[1, 2, 2, 4], [0, -1, 1, -3]]
> > rms: 43.659491 symcom: 35.000000 symbad: 1528.082200
> 
> Number 13 Father
> TOP tuning [1185.869125, 1924.351908, 2819.124589, 3401.317477]
> 
> > 12: [2, -2, -2, -8, -9, 1] [[2, 3, 5, 6], [0, 1, -1, -1]]
> > rms: 43.142169 symcom: 44.000000 symbad: 1898.255432
> 
> Number 62
> TOP tuning [1185.468457, 1924.986952, 2816.886876, 3409.621105]
> Audibly very similar to Number 13, and has a simpler mapping.
> 
> > 13: [2, -2, 1, -8, -4, 8] [[1, 2, 2, 3], [0, -2, 2, -1]]
> > rms: 41.524693 symcom: 35.000000 symbad: 1453.364254
> 
> Number 57
> TOP tuning [1185.869125, 1924.351909, 2819.124589, 3333.914203]
> Another member of the father temperament family, but the 7:1 
> approximation is worse than Number 13, and the 7:4 is 
unrecognizable.
> 
> > 14: [2, 3, 1, 0, -4, -6] [[1, 2, 3, 3], [0, -2, -3, -1]]
> > rms: 34.566097 symcom: 20.000000 symbad: 691.321943
> 
> Number 4 Beep
> TOP tuning [1194.642673, 1879.486406, 2819.229610, 3329.028548]
> 
> > 15: [4, 2, 2, -6, -8, -1] [[2, 4, 5, 6], [0, -2, -1, -1]]
> > rms: 23.945252 symcom: 32.000000 symbad: 766.248055
> 
> Number 32 Decimal
> TOP tuning [1207.657798, 1914.092323, 2768.532858, 3372.361757]
> 
> > 16: [1, 4, -2, 4, -6, -16] [[1, 2, 4, 2], [0, -1, -4, 2]]
> > rms: 20.163282 symcom: 75.000000 symbad: 1512.246136
> 
> Number 7 Dominant Seventh
> TOP tuning [1195.228951, 1894.576888, 2797.391744, 3382.219933]
> 
> > 17: [4, 4, 4, -3, -5, -2] [[4, 6, 9, 11], [0, 1, 1, 1]]
> > rms: 19.136993 symcom: 48.000000 symbad: 918.575644
> 
> Number 17 Diminished
> TOP tuning [1194.128460, 1892.648830, 2788.245174, 3385.309404]
> 
> > 18: [6, 0, 0, -14, -17, 0] [[6, 10, 14, 17], [0, -1, 0, 0]]
> > rms: 18.042924 symcom: 108.000000 symbad: 1948.635783
> 
> Number 85
> TOP tuning [1194.473353, 1901.955001, 2787.104490, 3384.341166]
> Would make a good 12-note keyboard mapping. There aren't many 
> temperaments based on 1/6-octave periods; this is the first one 
I've seen.
> 
> > 19: [6, 0, 3, -14, -12, 7] [[3, 4, 7, 8], [0, 2, 0, 1]]
> > rms: 16.786584 symcom: 99.000000 symbad: 1661.871769
> 
> Number 75
> TOP tuning [1199.400031, 1910.341746, 2798.600074, 3353.970936]
> No simpler than Augmented, but sounds a bit more warped.
> 
> > 20: [3, 0, 6, -7, 1, 14] [[3, 5, 7, 9], [0, -1, 0, -2]]
> > rms: 16.598678 symcom: 99.000000 symbad: 1643.269152
> 
> Number 5 Augmented
> TOP tuning [1199.976630, 1892.649878, 2799.945472, 3385.307546]
> 
> > 21: [0, 5, 0, 8, 0, -14] [[5, 8, 12, 14], [0, 0, -1, 0]]
> > rms: 15.815352 symcom: 75.000000 symbad: 1186.151431
> 
> Number 14 Blackwood
> TOP tuning [1195.893464, 1913.429542, 2786.313713, 3348.501698]
> 
> > 22: [2, 8, 1, 8, -4, -20] [[1, 2, 4, 3], [0, -2, -8, -1]]
> > rms: 12.690078 symcom: 155.000000 symbad: 1966.962143
> 
> Number 24 Hemifourths
> TOP tuning [1203.668842, 1902.376967, 2794.832500, 3358.526166]
> 
> > 23: [6, 5, 3, -6, -12, -7] [[1, 0, 1, 2], [0, 6, 5, 3]]
> > rms: 12.273810 symcom: 84.000000 symbad: 1031.000003
> 
> Number 27 Kleismic
> TOP tuning [1203.187308, 1907.006766, 2792.359613, 3359.878000]
> 
> > 24: [4, -3, 2, -14, -8, 13] [[1, 2, 2, 3], [0, -4, 3, -2]]
> > rms: 12.188571 symcom: 107.000000 symbad: 1304.177049
> 
> Number 28 Negri
> TOP tuning [1203.187308, 1907.006766, 2780.900506, 3359.878000]
> 
> > 25: [2, -4, -4, -11, -12, 2] [[2, 3, 5, 6], [0, 1, -2, -2]]
> > rms: 10.903177 symcom: 108.000000 symbad: 1177.543168
> 
> Number 6 Pajara
> TOP tuning [1196.893422, 1901.906680, 2779.100462, 3377.547174]
> 
> > 26: [8, 6, 6, -9, -13, -3] [[2, 5, 6, 7], [0, -4, -3, -3]]
> > rms: 10.132266 symcom: 144.000000 symbad: 1459.046340
> 
> Number 92
> TOP tuning [1198.553882, 1907.135354, 2778.724633, 3378.001574]
> Seems to be an alternate 22-ET-type temperament, not as good as 
Pajara.
> 
> > 27: [3, 0, -6, -7, -18, -14] [[3, 5, 7, 8], [0, -1, 0, 2]]
> > rms: 8.100679 symcom: 171.000000 symbad: 1385.216092
> 
> Number 31 Tripletone
> TOP tuning [1197.060039, 1902.640406, 2793.140092, 3377.079420]
> 
> > 28: [3, 5, -6, 1, -18, -28] [[1, 2, 3, 2], [0, -3, -5, 6]]
> > rms: 6.808962 symcom: 276.000000 symbad: 1879.273474
> 
> Number 42 Porcupine
> TOP tuning [1196.905961, 1906.858938, 2779.129576, 3367.717888]
> 
> > 29: [1, 9, -2, 12, -6, -30] [[1, 2, 6, 2], [0, -1, -9, 2]]
> > rms: 6.410458 symcom: 280.000000 symbad: 1794.928214
> 
> Number 34 Superpythagorean
> TOP tuning [1197.596121, 1905.765059, 2780.732078, 3374.046608]
> 
> > 30: [2, -9, -4, -19, -12, 16] [[1, 1, 5, 4], [0, 2, -9, -4]]
> > rms: 6.245316 symcom: 283.000000 symbad: 1767.424344
> 
> Number 79 Beatles
> TOP tuning [1197.104145, 1906.544822, 2793.037680, 3369.535226]
> 
> > 31: [7, 9, 13, -2, 1, 5] [[1, -1, -1, -2], [0, 7, 9, 13]]
> > rms: 5.052932 symcom: 355.000000 symbad: 1793.790776
> 
> Number 15 Semisixths
> TOP tuning [1198.389531, 1903.732520, 2790.053107, 3364.304748]
> 
> > 32: [5, 1, 12, -10, 5, 25] [[1, 0, 2, -1], [0, 5, 1, 12]]
> > rms: 4.139051 symcom: 356.000000 symbad: 1473.502082
> 
> Number 3 Magic
> TOP tuning [1201.276744, 1903.978592, 2783.349206, 3368.271877]
> 
> > 33: [14, 11, 9, -15, -25, -10] [[1, 5, 5, 5], [0, -14, -11, -9]]
> > rms: 4.006991 symcom: 436.000000 symbad: 1747.048215
> 
> Not in the 114 list. Seems overly complex to be of much use.
> 
> > 34: [1, 4, 10, 4, 13, 12] [[1, 2, 4, 7], [0, -1, -4, -10]]
> > rms: 3.665035 symcom: 243.000000 symbad: 890.603432
> 
> Almost goes without saying, but....
> Number 2 Meantone
> TOP tuning [1201.698521, 1899.262909, 2790.257556, 3370.548328]
> 
> > 35: [3, 12, -1, 12, -10, -36] [[1, 1, 0, 3], [0, 3, 12, -1]]
> > rms: 3.579262 symcom: 420.000000 symbad: 1503.290125
> 
> Number 35 Supermajor seconds
> TOP tuning [1201.698521, 1899.262909, 2790.257556, 3372.574099]
> 
> > 36: [4, 16, 9, 16, 3, -24] [[1, 3, 8, 6], [0, -4, -16, -9]]
> > rms: 3.443812 symcom: 571.000000 symbad: 1966.416662
> 
> Number 84 Squares
> TOP tuning [1201.698521, 1899.262909, 2790.257556, 3372.067656]
> Sounds practically identical to Number 35, but with a more complex 
mapping.
> 
> > 37: [10, 9, 7, -9, -17, -9] [[1, -1, 0, 1], [0, 10, 9, 7]]
> > rms: 3.320167 symcom: 244.000000 symbad: 810.120816
> 
> Number 29 Nonkleismic
> TOP tuning [1198.828458, 1900.098151, 2789.033948, 3368.077085]
> 
> > 38: [9, 5, -3, -13, -30, -21] [[1, 1, 2, 3], [0, 9, 5, -3]]
> > rms: 3.065962 symcom: 339.000000 symbad: 1039.361092
> 
> Number 30 Quartaminorthirds
> TOP tuning [1199.792743, 1900.291122, 2788.751252, 3365.878770]
> 
> > 39: [1, -8, -14, -15, -25, -10] [[1, 2, -1, -3], [0, -1, 8, 14]]
> > rms: 2.859338 symcom: 603.000000 symbad: 1724.180520
> 
> Number 8 Schismic
> TOP tuning [1200.760625, 1903.401919, 2784.194017, 3371.388750]
> 
> > 40: [7, -3, 8, -21, -7, 27] [[1, 0, 3, 1], [0, 7, -3, 8]]
> > rms: 2.589237 symcom: 344.000000 symbad: 890.697699
> 
> Number 10 Orwell
> TOP tuning [1199.532657, 1900.455530, 2784.117029, 3371.481834]
> 
> > 41: [17, 6, 15, -30, -24, 18] [[1, -5, 0, -3], [0, 17, 6, 15]]
> > rms: 2.469727 symcom: 756.000000 symbad: 1867.113518
> 
> Now we're starting to get into temperaments that are mostly too 
complex 
> to be of much interest. This is Number 66 from the big list, and 
doesn't 
> seem to be enough better than Orwell to justify its complexity. 
I'll 
> skip most of the rest.
> 
> > 45: [6, -7, -2, -25, -20, 15] [[1, 1, 3, 3], [0, 6, -7, -2]]
> > rms: 1.637405 symcom: 347.000000 symbad: 568.179603
> 
> Number 9 Miracle
> TOP tuning [1200.631014, 1900.954868, 2784.848544, 3368.451756]
> TOP generators [1200.631014, 116.7206423]
> bad: 29.119472 comp: 6.793166 err: .631014
> 
> > 49: [16, 2, 5, -34, -37, 6] [[1, -1, 2, 2], [0, 16, 2, 5]]
> > rms: .875363 symcom: 611.000000 symbad: 534.846775
> 
> Number 11 Hemiwuerschmidt
> TOP tuning [1199.692003, 1901.466838, 2787.028860, 3368.496143]
> 
> > 76: [18, 27, 18, 1, -22, -34] [[9, 15, 22, 26], [0, -2, -3, -2]]
> > rms: .130449 symcom: 1539.000000 symbad: 200.760896
> 
> Number 1 Ennealimmal
> TOP tuning [1200.036377, 1902.012656, 2786.350297, 3368.723784]


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Message: 10303

Date: Sat, 14 Feb 2004 00:00:25

Subject: Re: The same page

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >> >Erlich magic L1 norm; if
> >> >
> >> > <<a1 a2 a3 a4 a5 a6||
> >> >
> >> >is the wedgie, then complexity is
> >> >
> >> > |a1/p3|+|a2/p5|+|a3/p7|+|a4/p3p5|+|a5/p3p7|+|a6/p5p7|
> >> 
> >> Where wedgie is val-wedgie.  But apparently there's a monzo-
wedgie
> >> formualation...
> >
> >Simply reverse the order of the entries.
> 
> Not sure what you're saying.
> 
> monzo-wedgie = reverse(val-wedgie)

Up to some of the signs, yes. Since the above expression for 
complexity takes the absolute values anyway, you don't have to worry 
about the signs. The point is that the complexity you end up 
calculating is the same. We could write

If ||a6 a5 a4 a3 a2 a1>> is the monzo-wedgie, then the complexity is
|a1/p3|+|a2/p5|+|a3/p7|+|a4/p3p5|+|a5/p3p7|+|a6/p5p7|

and we'd be getting the same answer as above.


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Message: 10305

Date: Sat, 14 Feb 2004 00:15:22

Subject: Re: The same page

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> How am I ever going to find these posts of Dave's to get to
> 
> | a b c >  ~= | -b c a >
> 
> or whatever?

Try:
Yahoo groups: /tuning-math/message/7852 *


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Message: 10306

Date: Sat, 14 Feb 2004 00:18:08

Subject: Re: The same page

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> > How am I ever going to find these posts of Dave's to get to
> > 
> > | a b c >  ~= | -b c a >
> > 
> > or whatever?
> 
> Try:
> Yahoo groups: /tuning-math/message/7852 *

Thanks Dave, for looking in even after you've gone away!


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Message: 10307

Date: Sat, 14 Feb 2004 00:21:21

Subject: Re: Still another

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...>
wrote:
> I don't believe in Dave's 
> quantization of agony theory,

That's "quantification", not "quantization". These are very different
things.


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Message: 10322

Date: Tue, 17 Feb 2004 08:06:00

Subject: Re: rank complexity explanation updated

From: Carl Lumma

Welcome back, Manuel.

>Carl wrote on 29-1:
>>This happened because I wanted to give the interval matrix in
>>'steps of 12-tET' units.  Unfortunately (and one of my biggest
>>desired features) Scala does not offer 'degrees of n-ET' units.
>
>Fortunately it does, use
>set attribute et_step <steps/oct>
>
>To see the intervals in terms of these units, do
>show/attribute intervals

These don't look like units to me, but some secondary abstraction.
Can I author a scl file using them?  Does Scala display all its
output in them?  No, it still displays cents, with these in a
separate column.

-Carl


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Message: 10323

Date: Tue, 17 Feb 2004 12:00:04

Subject: Re: rank complexity explanation updated

From: Manuel Op de Coul

Carl wrote on 29-1:
>This happened because I wanted to give the interval matrix in
>'steps of 12-tET' units.  Unfortunately (and one of my biggest
>desired features) Scala does not offer 'degrees of n-ET' units.

Fortunately it does, use
set attribute et_step <steps/oct>

To see the intervals in terms of these units, do
show/attribute intervals

Manuel




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Message: 10324

Date: Tue, 17 Feb 2004 17:26:54

Subject: Re: rank complexity explanation updated

From: Manuel Op de Coul

Carl wrote:
>These don't look like units to me, but some secondary abstraction.

Ah, if you mean you expected non-integer numbers, that's also
possible. Then the command is:
set attribute <steps/oct>
For example: set attribute 12.0

>Does Scala display all its output in them?

No, that's not possible.
Note that you can use the input command to enter a scale in
any logarithmic unit if you convert it afterwards with
mult/abs 2/1, assuming 2/1 is the period you want.

>No, it still displays cents, with these in a
>separate column.

It gave me an idea for an enhancement though, showing the
attributes for the intervals of the interval matrix. I'll put that
in the next version under the command
show/attribute/line intervals.

Manuel


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