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Message: 9500 - Contents - Hide Contents

Date: Thu, 29 Jan 2004 03:57:39

Subject: Re: 114 7-limit temperaments

From: Dave Keenan

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


> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
> wrote:


>> But you agree it's a perfectly valid interpretation, due to the
>> obvious generalisation of the normalising of weighted MAD, RMS etc. 
> to


>> the normalising of weighted minimax?
> 


> It's hard to say, because of course it's inconceivable for all of the 
> errors to be equal.



In our application perhaps, but not in general.

Here's another way to check the normalisation. Scaling all the weights
by the same factor (e.g. using a diffferent log base for the errors
and their weights) should not change the result.

Anyway, the main thing is that we're all happy with the scaling of the
error, irrespective of how we choose to interpret it.



>> Yes. Presumably any reasonable generating scheme should have some 
> such


>> interprtetation in terms of the TOP complexity,
> 
> Possibly.
> 


>> otherwise what use is it?
> 


> See above. This seems like a more direct definition of 'complexity' 
> to me. If I take Gene's confirmation to heart, it's the affine-
> geometrical size measure (length, area, volume, etc.) of the portion 
> of the lattice sufficient, under the relevant temperament, to 
> represent the entire lattice.



That sounds pretty good, in a hand-waving sort of way, but when I try
to pin down what it actually means in terms of how big a scale we
might need to make good use of some temperament, my brain just keeps
sliding off it.



>> I suspect one can still specify reasonable (maybe even optimal)
>> generating schemes with multiple generators. In fact since we're
>> tempering the octave, we apparently need to find such schemes 
> already


>> for so-called linear temperaments.
> 


> I don't think we need to do any of this, nor should we want to for 
> the purposes of a paper brief enough to be published.



So we're not goint to talk about how to obtain a finite scale from a
temperament. Now that the octave is just another generator (and the
pair of generators is no longer unique), don't we have some explaining
to do, about why we should iterate one particular generator modulo the
other?



>> In any case, I'd be happy to hear _any_ such interpretation for this
>> new complexity measure,
> 


> See Yahoo groups: /tuning-math/message/8781 * [with cont.]  and 
> Yahoo groups: /tuning-math/message/8806 * [with cont.]  . . .



Thanks. I may be some time ... :-)



>> even if it only works for linear >temperaments.
> 


> The whole point is that it works for equal, linear, planar . . . etc.
> 
> Oh, you said _such_ . . . Sorry. Well, I suspect that's still 
> possible, especially since Gene himself explained the lower 
> complexity of schismic vs. miracle in terms of generators-per-prime.



Yes. I meant "even if some particular _interpretation_ only works for
linear". I understand the _complexity_measure_ is intended to work for
all, and as such it sounds brilliant.
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Message: 9501 - Contents - Hide Contents

Date: Thu, 29 Jan 2004 21:57:51

Subject: Re: What the numbers mean

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
wrote:


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


>> Here is a no doubt long overdue discussion of what the numbers in 
the
>> wedgie for a linear temperament mean.
> ...
> 


> That was a good explanation. Thanks heaps Gene!
> 


>> If we take absolute values and normalize each by dividing by the 
log
>> base 2 of the two primes involved, we get
>> 
>> [6/p3 7/p5 2/p7 25/(p3p5) 20/(p3p7) 15/(p5p7)] =
>> 
>> [3.785578521, 3.014735907, .7124143742, 6.793166368, 4.494834256,
>> 2.301151281]
>> 
>> The complexity is defined by the weighted value for 5/3, which is 
the
>> worst case (as, in fact, it clearly is.)
> 


> You are dividing by the _product_ of the two logs, shouldn't you be
> dividing by their _sum_ (the log of the product of the primes)?



Product makes a lot more sense to me than sum, in my vague intuitive 
understanding of these things. You probably see that Gene's defining 
p2 as 1. Now the wedgie represents a bivector in 4-dimensional space, 
which means the relevant basis elements are *directed areas* in 4D, 
of which there are of course 6 "multilinearly-independent" ones. Now, 
the unit lengths in the lattice are scaled by (p2,) p3, p5, and p7, 
so the unit areas would have to be scaled by the 6 possible 
*products*, not *sums*, of these -- (p2)p3, (p2)p5, (p2)p7, p3p5, 
p3p7, and p5p7.
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Message: 9502 - Contents - Hide Contents

Date: Thu, 29 Jan 2004 04:08:22

Subject: Re: Attn: Gene 2

From: Dave Keenan

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


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


>> geometry, where angles are left undefined . . . Anyhow, since both of 
>> these methods could be used to address a 3-limit TOP temperament, in 
>> 5-limit could they be still both be expressible in a single form in a 
>> general enough framework, say exterior algebra?
> 


> That was my suggestion. You normalize by dividing each coordinate by
> log2(p) and take the wedge product up to the (normalized) multival,
> and then measure complexity by taking the max of the absolute values
> of the coefficients. If you start from the monzo side, you get the
> same normalized coefficients up to a constant factor, but now you
> might rather take the sum of the absolute values (L1 vs L infinity.)
> 
>  The goal, of course, is to produce 


>> complexity vs. TOP error graphs for 7-limit linear temperaments, 
>> something I currently don't know how to do. 
> 


> Why not use the formula I gave before:
> 
> || <<w1 w2 w3 w4 w5 w6|| || = max(|w1|/p3, |w2|/p5, |w3|/p7,
> |w4|/p3p5, |w5|/p3p7, |w6|/p5p7)



This formula looks like a good argument for using square brackets 
<<  ]] instead of vertical bars <<  || for multivectors. I though we
had agreed to use square brackets at one stage.
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Message: 9503 - Contents - Hide Contents

Date: Thu, 29 Jan 2004 22:01:11

Subject: Re: Maximum Error in TOP

From: Kalle Aho

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


> --- In tuning-math@xxxxxxxxxxx.xxxx "Kalle Aho" <kalleaho@m...> 
wrote:
>> Hi!
>> 


>> How do I get the maximum Tenney-weighted error in a TOP 
> temperament? 


>> Is it the same as the maximum weighted error of the primes
> 
> Yes.



Thanks, Paul. Can you explain why this is so? 

Kalle
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Message: 9504 - Contents - Hide Contents

Date: Thu, 29 Jan 2004 04:13:41

Subject: 60 for Dave (was: 41 "Hermanic" 7-limit linear temperaments)

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
wrote:



> One might try 
> 
> k*sqrt(comp) + sqrt(err) < x
> 
> for starters.



All right, just for you . . . This time I'll insist Waage just makes 
it in despite its high complexity. Then k=1.75 gives the most 
inclusive cutoff which is still more stringent than Gene's, so 
culling that list will work again.

Number 1 Meantone (Huygens)

[1, 4, 10, 4, 13, 12] [[1, 2, 4, 7], [0, -1, -4, -10]]
TOP tuning [1201.698521, 1899.262909, 2790.257556, 3370.548328]
TOP generators [1201.698520, 504.1341314]
bad: 4.6061 comp: 3.562072 err: 1.698521


Number 2 Magic

[5, 1, 12, -10, 5, 25] [[1, 0, 2, -1], [0, 5, 1, 12]]
TOP tuning [1201.276744, 1903.978592, 2783.349206, 3368.271877]
TOP generators [1201.276744, 380.7957184]
bad: 4.7480 comp: 4.274486 err: 1.276744

Number 3 Pajara

[2, -4, -4, -11, -12, 2] [[2, 3, 5, 6], [0, 1, -2, -2]]
TOP tuning [1196.893422, 1901.906680, 2779.100462, 3377.547174]
TOP generators [598.4467109, 106.5665459]
bad: 4.7881 comp: 2.988993 err: 3.106578

Number 4 Dominant Seventh

[1, 4, -2, 4, -6, -16] [[1, 2, 4, 2], [0, -1, -4, 2]]
TOP tuning [1195.228951, 1894.576888, 2797.391744, 3382.219933]
TOP generators [1195.228951, 495.8810151]
bad: 4.9260 comp: 2.454561 err: 4.771049

Number 5 Augmented

[3, 0, 6, -7, 1, 14] [[3, 5, 7, 9], [0, -1, 0, -2]]
TOP tuning [1199.976630, 1892.649878, 2799.945472, 3385.307546]
TOP generators [399.9922103, 107.3111730]
bad: 4.9876 comp: 2.147741 err: 5.870879

Number 6 Semisixths

[7, 9, 13, -2, 1, 5] [[1, -1, -1, -2], [0, 7, 9, 13]]
TOP tuning [1198.389531, 1903.732520, 2790.053107, 3364.304748]
TOP generators [1198.389531, 443.1602931]
bad: 5.0349 comp: 4.630693 err: 1.610469

Number 7 Schismic

[1, -8, -14, -15, -25, -10] [[1, 2, -1, -3], [0, -1, 8, 14]]
TOP tuning [1200.760625, 1903.401919, 2784.194017, 3371.388750]
TOP generators [1200.760624, 498.1193303]
bad: 5.1036 comp: 5.618543 err: .912904

Number 8 Injera

[2, 8, 8, 8, 7, -4] [[2, 3, 4, 5], [0, 1, 4, 4]]
TOP tuning [1201.777814, 1896.276546, 2777.994928, 3378.883835]
TOP generators [600.8889070, 93.60982493]
bad: 5.1411 comp: 3.445412 err: 3.582707

Number 9 Orwell

[7, -3, 8, -21, -7, 27] [[1, 0, 3, 1], [0, 7, -3, 8]]
TOP tuning [1199.532657, 1900.455530, 2784.117029, 3371.481834]
TOP generators [1199.532657, 271.4936472]
bad: 5.1530 comp: 5.706260 err: .946061

Number 10 Hemifourths

[2, 8, 1, 8, -4, -20] [[1, 2, 4, 3], [0, -2, -8, -1]]
TOP tuning [1203.668842, 1902.376967, 2794.832500, 3358.526166]
TOP generators [1203.668841, 252.4803582]
bad: 5.1637 comp: 3.445412 err: 3.668842

Number 11 Kleismic

[6, 5, 3, -6, -12, -7] [[1, 0, 1, 2], [0, 6, 5, 3]]
TOP tuning [1203.187308, 1907.006766, 2792.359613, 3359.878000]
TOP generators [1203.187309, 317.8344609]
bad: 5.1902 comp: 3.785579 err: 3.187309

Number 12 Negri

[4, -3, 2, -14, -8, 13] [[1, 2, 2, 3], [0, -4, 3, -2]]
TOP tuning [1203.187308, 1907.006766, 2780.900506, 3359.878000]
TOP generators [1203.187309, 124.8419629]
bad: 5.1986 comp: 3.804173 err: 3.187309

Number 13 Diminished

[4, 4, 4, -3, -5, -2] [[4, 6, 9, 11], [0, 1, 1, 1]]
TOP tuning [1194.128460, 1892.648830, 2788.245174, 3385.309404]
TOP generators [298.5321149, 101.4561401]
bad: 5.2032 comp: 2.523719 err: 5.871540

Number 14 Tripletone

[3, 0, -6, -7, -18, -14] [[3, 5, 7, 8], [0, -1, 0, 2]]
TOP tuning [1197.060039, 1902.640406, 2793.140092, 3377.079420]
TOP generators [399.0200131, 92.45965769]
bad: 5.2344 comp: 4.045351 err: 2.939961

Number 15 Blackwood

[0, 5, 0, 8, 0, -14] [[5, 8, 12, 14], [0, 0, -1, 0]]
TOP tuning [1195.893464, 1913.429542, 2786.313713, 3348.501698]
TOP generators [239.1786927, 83.83059859]
bad: 5.2708 comp: 2.173813 err: 7.239629

Number 16 Superpythagorean

[1, 9, -2, 12, -6, -30] [[1, 2, 6, 2], [0, -1, -9, 2]]
TOP tuning [1197.596121, 1905.765059, 2780.732078, 3374.046608]
TOP generators [1197.596121, 489.4271829]
bad: 5.3047 comp: 4.602303 err: 2.403879

Number 17 Miracle

[6, -7, -2, -25, -20, 15] [[1, 1, 3, 3], [0, 6, -7, -2]]
TOP tuning [1200.631014, 1900.954868, 2784.848544, 3368.451756]
TOP generators [1200.631014, 116.7206423]
bad: 5.3555 comp: 6.793166 err: .631014

Number 18 Porcupine

[3, 5, -6, 1, -18, -28] [[1, 2, 3, 2], [0, -3, -5, 6]]
TOP tuning [1196.905961, 1906.858938, 2779.129576, 3367.717888]
TOP generators [1196.905960, 162.3176609]
bad: 5.3860 comp: 4.295482 err: 3.094040

Number 19 Zeta Reticuli Trisex Porky

[6, 10, 10, 2, -1, -5] [[2, 4, 6, 7], [0, -3, -5, -5]]
TOP tuning [1196.893422, 1906.838962, 2779.100462, 3377.547174]
TOP generators [598.4467109, 162.3159606]
bad: 5.3943 comp: 4.306766 err: 3.106578

Number 20 Supermajor seconds

[3, 12, -1, 12, -10, -36] [[1, 1, 0, 3], [0, 3, 12, -1]]
TOP tuning [1201.698521, 1899.262909, 2790.257556, 3372.574099]
TOP generators [1201.698520, 232.5214630]
bad: 5.4159 comp: 5.522763 err: 1.698521

Number 21 Flattone

[1, 4, -9, 4, -17, -32] [[1, 2, 4, -1], [0, -1, -4, 9]]
TOP tuning [1202.536420, 1897.934872, 2781.593812, 3361.705278]
TOP generators [1202.536419, 507.1379663]
bad: 5.4700 comp: 4.909123 err: 2.536420

Number 22 Nonkleismic

[10, 9, 7, -9, -17, -9] [[1, -1, 0, 1], [0, 10, 9, 7]]
TOP tuning [1198.828458, 1900.098151, 2789.033948, 3368.077085]
TOP generators [1198.828458, 309.8926610]
bad: 5.4781 comp: 6.309298 err: 1.171542

Number 23

[6, 10, 3, 2, -12, -21] [[1, 2, 3, 3], [0, -6, -10, -3]]
TOP tuning [1202.659696, 1907.471368, 2778.232381, 3359.055076]
TOP generators [1202.659696, 82.97467050]
bad: 5.4973 comp: 4.306766 err: 3.480440

Number 24

[0, 0, 12, 0, 19, 28] [[12, 19, 28, 34], [0, 0, 0, -1]]
TOP tuning [1197.674070, 1896.317278, 2794.572829, 3368.825906]
TOP generators [99.80617249, 24.58395811]
bad: 5.5130 comp: 4.295482 err: 3.557008

Number 25 Decimal

[4, 2, 2, -6, -8, -1] [[2, 4, 5, 6], [0, -2, -1, -1]]
TOP tuning [1207.657798, 1914.092323, 2768.532858, 3372.361757]
TOP generators [603.8288989, 250.6116362]
bad: 5.5474 comp: 2.523719 err: 7.657798

Number 26 Quartaminorthirds

[9, 5, -3, -13, -30, -21] [[1, 1, 2, 3], [0, 9, 5, -3]]
TOP tuning [1199.792743, 1900.291122, 2788.751252, 3365.878770]
TOP generators [1199.792743, 77.83315314]
bad: 5.5686 comp: 6.742251 err: 1.049791

Number 27

[3, -7, -8, -18, -21, 1] [[1, 3, -1, -1], [0, -3, 7, 8]]
TOP tuning [1202.900537, 1897.357759, 2790.235118, 3360.683070]
TOP generators [1202.900537, 570.4479508]
bad: 5.5734 comp: 4.891080 err: 2.900537

Number 28

[3, 12, 11, 12, 9, -8] [[1, 3, 8, 8], [0, -3, -12, -11]]
TOP tuning [1202.624742, 1900.726787, 2792.408176, 3361.457323]
TOP generators [1202.624742, 569.0491468]
bad: 5.5985 comp: 5.168119 err: 2.624742

Number 29 Pelogic

[1, -3, -4, -7, -9, -1] [[1, 2, 1, 1], [0, -1, 3, 4]]
TOP tuning [1209.734056, 1886.526887, 2808.557731, 3341.498957]
TOP generators [1209.734056, 532.9412251]
bad: 5.6088 comp: 2.022675 err: 9.734056

Number 30 Dicot

[2, 1, 6, -3, 4, 11] [[1, 1, 2, 1], [0, 2, 1, 6]]
TOP tuning [1204.048158, 1916.847810, 2764.496143, 3342.447113]
TOP generators [1204.048159, 356.3998255]
bad: 5.6237 comp: 2.137243 err: 9.396316

Number 31

[5, 8, 2, 1, -11, -18] [[1, 2, 3, 3], [0, -5, -8, -2]]
TOP tuning [1194.335372, 1892.976778, 2789.895770, 3384.728528]
TOP generators [1194.335372, 99.13879319]
bad: 5.6284 comp: 3.445412 err: 5.664628

Number 32 Catakleismic

[6, 5, 22, -6, 18, 37] [[1, 0, 1, -3], [0, 6, 5, 22]]
TOP tuning [1200.536356, 1901.438376, 2785.068335, 3370.331646]
TOP generators [1200.536355, 316.9063960]
bad: 5.6313 comp: 7.836558 err: .536356

Number 33

[3, -5, -6, -15, -18, 0] [[1, 3, 0, 0], [0, -3, 5, 6]]
TOP tuning [1195.486066, 1908.381352, 2796.794743, 3356.153692]
TOP generators [1195.486066, 559.3589487]
bad: 5.6577 comp: 4.075900 err: 4.513934

Number 34

[8, 1, 18, -17, 6, 39] [[1, -1, 2, -3], [0, 8, 1, 18]]
TOP tuning [1201.135544, 1899.537544, 2789.855225, 3373.107814]
TOP generators [1201.135545, 387.5841360]
bad: 5.6663 comp: 6.411729 err: 1.525246

Number 35 Beatles

[2, -9, -4, -19, -12, 16] [[1, 1, 5, 4], [0, 2, -9, -4]]
TOP tuning [1197.104145, 1906.544822, 2793.037680, 3369.535226]
TOP generators [1197.104145, 354.7203384]
bad: 5.6780 comp: 5.162806 err: 2.895855

Number 36 Formerly Number 82

[6, -2, -2, -17, -20, 1] [[2, 2, 5, 6], [0, 3, -1, -1]]
TOP tuning [1203.400986, 1896.025764, 2777.627538, 3379.328030]
TOP generators [601.7004928, 230.8749260]
bad: 5.6954 comp: 4.619353 err: 3.740932

Number 37

[3, 5, 9, 1, 6, 7] [[1, 2, 3, 4], [0, -3, -5, -9]]
TOP tuning [1193.415676, 1912.390908, 2789.512955, 3350.341372]
TOP generators [1193.415676, 158.1468146]
bad: 5.6994 comp: 3.205865 err: 6.584324

Number 38

[3, 0, 9, -7, 6, 21] [[3, 5, 7, 9], [0, -1, 0, -3]]
TOP tuning [1193.415676, 1912.390908, 2784.636577, 3350.341372]
TOP generators [397.8052253, 76.63521863]
bad: 5.7070 comp: 3.221612 err: 6.584324

Number 39 {28/27, 50/49}

[2, 6, 6, 5, 4, -3] [[2, 3, 4, 5], [0, 1, 3, 3]]
TOP tuning [1191.599639, 1915.269258, 2766.808679, 3362.608498]
TOP generators [595.7998193, 127.8698005]
bad: 5.7115 comp: 2.584059 err: 8.400361

Number 40

[6, 0, 3, -14, -12, 7] [[3, 4, 7, 8], [0, 2, 0, 1]]
TOP tuning [1199.400031, 1910.341746, 2798.600074, 3353.970936]
TOP generators [399.8000105, 155.5708520]
bad: 5.7136 comp: 3.804173 err: 5.291448

Number 41 Superkleismic

[9, 10, -3, -5, -30, -35] [[1, 4, 5, 2], [0, -9, -10, 3]]
TOP tuning [1201.371917, 1904.129438, 2783.128219, 3369.863245]
TOP generators [1201.371918, 322.3731369]
bad: 5.7153 comp: 6.742251 err: 1.371918

Number 42

[8, 6, 6, -9, -13, -3] [[2, 5, 6, 7], [0, -4, -3, -3]]
TOP tuning [1198.553882, 1907.135354, 2778.724633, 3378.001574]
TOP generators [599.2769413, 272.3123381]
bad: 5.7395 comp: 5.047438 err: 3.268439

Number 43 Diaschismic

[2, -4, -16, -11, -31, -26] [[2, 3, 5, 7], [0, 1, -2, -8]]
TOP tuning [1198.732403, 1901.885616, 2789.256983, 3365.267311]
TOP generators [599.3662015, 103.7870123]
bad: 5.7450 comp: 6.966993 err: 1.267597

Number 44 Beep

[2, 3, 1, 0, -4, -6] [[1, 2, 3, 3], [0, -2, -3, -1]]
TOP tuning [1194.642673, 1879.486406, 2819.229610, 3329.028548]
TOP generators [1194.642673, 254.8994697]
bad: 5.7543 comp: 1.292030 err: 14.176105

Number 45

[1, -8, -2, -15, -6, 18] [[1, 2, -1, 2], [0, -1, 8, 2]]
TOP tuning [1195.155395, 1894.070902, 2774.763716, 3382.790568]
TOP generators [1195.155395, 496.2398890]
bad: 5.7634 comp: 4.075900 err: 4.974313

Number 46

[6, 0, 15, -14, 7, 35] [[3, 5, 7, 9], [0, -2, 0, -5]]
TOP tuning [1197.060039, 1902.856975, 2793.140092, 3360.572393]
TOP generators [399.0200131, 46.12154491]
bad: 5.7697 comp: 5.369353 err: 2.939961

Number 47

[6, 0, 0, -14, -17, 0] [[6, 10, 14, 17], [0, -1, 0, 0]]
TOP tuning [1194.473353, 1901.955001, 2787.104490, 3384.341166]
TOP generators [199.0788921, 88.83392059]
bad: 5.7715 comp: 3.820609 err: 5.526647

Number 48

[0, 12, 12, 19, 19, -6] [[12, 19, 28, 34], [0, 0, -1, -1]]
TOP tuning [1198.015473, 1896.857833, 2778.846497, 3377.854234]
TOP generators [99.83462277, 16.52294019]
bad: 5.7717 comp: 5.168119 err: 3.215955

Number 49 Supersupermajor

[3, 17, -1, 20, -10, -50] [[1, 1, -1, 3], [0, 3, 17, -1]]
TOP tuning [1200.231588, 1903.372996, 2784.236389, 3366.314293]
TOP generators [1200.231587, 234.3804692]
bad: 5.7926 comp: 7.670504 err: .894655

Number 50

[11, 13, 17, -5, -4, 3] [[1, 3, 4, 5], [0, -11, -13, -17]]
TOP tuning [1198.514750, 1899.600936, 2789.762356, 3371.570447]
TOP generators [1198.514750, 154.1766650]
bad: 5.8290 comp: 6.940227 err: 1.485250

Number 51 {1728/1715, 4000/3993}

[11, 18, 5, 3, -23, -39] [[1, 2, 3, 3], [0, -11, -18, -5]]
TOP tuning [1199.083445, 1901.293958, 2784.185538, 3371.399002]
TOP generators [1199.083445, 45.17026643]
bad: 5.8298 comp: 7.752178 err: .916555

Number 52 {25/24, 81/80} Jamesbond?

[0, 0, 7, 0, 11, 16] [[7, 11, 16, 20], [0, 0, 0, -1]]
TOP tuning [1209.431411, 1900.535075, 2764.414655, 3368.825906]
TOP generators [172.7759159, 86.69241190]
bad: 5.8344 comp: 2.493450 err: 9.431411

Number 53

[5, 3, 7, -7, -3, 8] [[1, 1, 2, 2], [0, 5, 3, 7]]
TOP tuning [1192.540126, 1890.131381, 2803.635005, 3361.708008]
TOP generators [1192.540126, 139.5182509]
bad: 5.8395 comp: 3.154649 err: 7.459874

Number 54

[9, -2, 14, -24, -3, 38] [[1, 3, 2, 5], [0, -9, 2, -14]]
TOP tuning [1201.918556, 1904.657347, 2781.858962, 3363.439837]
TOP generators [1201.918557, 189.0109248]
bad: 5.8541 comp: 6.521440 err: 1.918557


Number 55 Octacot

[8, 18, 11, 10, -5, -25] [[1, 1, 1, 2], [0, 8, 18, 11]]
TOP tuning [1199.031259, 1903.490418, 2784.064367, 3366.693863]
TOP generators [1199.031259, 88.05739491]
bad: 5.8567 comp: 7.752178 err: .968741

Number 56 Wizard

[12, -2, 20, -31, -2, 52] [[2, 1, 5, 2], [0, 6, -1, 10]]
TOP tuning [1200.639571, 1900.941305, 2784.828674, 3368.342104]
TOP generators [600.3197857, 216.7702531]
bad: 5.8788 comp: 8.423526 err: .639571

Number 57

[9, 0, 9, -21, -11, 21] [[9, 14, 21, 25], [0, 1, 0, 1]]
TOP tuning [1197.060039, 1897.499011, 2793.140092, 3360.572393]
TOP generators [133.0066710, 35.40561749]
bad: 5.8950 comp: 5.706260 err: 2.939961

Number 58 Squares

[4, 16, 9, 16, 3, -24] [[1, 3, 8, 6], [0, -4, -16, -9]]
TOP tuning [1201.698521, 1899.262909, 2790.257556, 3372.067656]
TOP generators [1201.698520, 426.4581630]
bad: 5.8971 comp: 6.890825 err: 1.698521

Number 59

[2, 1, -4, -3, -12, -12] [[1, 1, 2, 4], [0, 2, 1, -4]]
TOP tuning [1204.567524, 1916.451342, 2765.076958, 3394.502460]
TOP generators [1204.567524, 355.9419091]
bad: 5.8982 comp: 2.696901 err: 9.146173

Number 60 Waage

[0, 12, 24, 19, 38, 22] [[12, 19, 28, 34], [0, 0, -1, -2]]
TOP tuning [1200.617051, 1900.976998, 2785.844725, 3370.558188]
TOP generators [100.0514209, 16.55882096]
bad: 5.9023 comp: 8.548972 err: .617051




> I'd really like to see these on a chart.



I think Gene may be using the wrong norm to get his complexity 
values. I'll wait until I'm sure they're right or corrected.
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Message: 9505 - Contents - Hide Contents

Date: Thu, 29 Jan 2004 22:18:00

Subject: Re: Maximum Error in TOP

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Kalle Aho" <kalleaho@m...> wrote:


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


>> --- In tuning-math@xxxxxxxxxxx.xxxx "Kalle Aho" <kalleaho@m...> 
> wrote:
>>> Hi!
>>> 


>>> How do I get the maximum Tenney-weighted error in a TOP 
>> temperament? 


>>> Is it the same as the maximum weighted error of the primes
>> 
>> Yes.
> 


> Thanks, Paul. Can you explain why this is so? 
> 
> Kalle



It follows from the prime factorization theorem.

The maximum Tenney-weighted error being T implies that

error(p)/log(p) <= T

for all the primes p. If the factors in a chosen ratio are 2^a 3^b 
5^c . . . (each exponent may either be positive or negative), then 
the error of the chosen ratio cannot be greater than

T*(|a|/log(2) + |b|/log(3) + |c|/log(5) . . .)

since the errors in the primes that make up the chosen ratio, at 
worst, add up without cancellation to the error in the chosen ratio. 
The complexity of the ratio, meanwhile, is exactly

(|a|/log(2) + |b|/log(3) + |c|/log(5) . . .)

So the Tenney-weighted error in the ratio cannot be greater than the 
second-to-last expression divided by the last expression, i.e., T.
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Message: 9506 - Contents - Hide Contents

Date: Thu, 29 Jan 2004 00:28:39

Subject: 41 "Hermanic" 7-limit linear temperaments (was: Re: 114 7-limit temperaments)

From: Paul Erlich

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


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


>> Number 15 Porcupine
>> 
>> [3, 5, -6, 1, -18, -28] [[1, 2, 3, 2], [0, -3, -5, 6]]
>> TOP tuning [1196.905961, 1906.858938, 2779.129576, 3367.717888]
>> TOP generators [1196.905960, 162.3176609]
>> bad: 8.9144 comp: 4.295482 err: 3.094040
>> 
>> Number 16
>> 
>> [6, 10, 10, 2, -1, -5] [[2, 4, 6, 7], [0, -3, -5, -5]]
>> TOP tuning [1196.893422, 1906.838962, 2779.100462, 3377.547174]
>> TOP generators [598.4467109, 162.3159606]
>> bad: 8.9422 comp: 4.306766 err: 3.106578
> 


> These two are related, and very close in terms of error and 
> complexity. Bi porcupine makes me think of porcupine sex, and I'm 
not 
> sure I want to, but some sort of porky name seems apt.
> 


>> Number 25 Miracle
>> 
>> [6, -7, -2, -25, -20, 15] [[1, 1, 3, 3], [0, 6, -7, -2]]
>> TOP tuning [1200.631014, 1900.954868, 2784.848544, 3368.451756]
>> TOP generators [1200.631014, 116.7206423]
>> bad: 9.8358 comp: 6.793166 err: .631014
> 


> This seems absurdly far down the list. I think mine was better.



I don't think this is necessarily too far down the list for every 
conceivable user, including possibly Herman. As one would like all 
the harmonies to come from a single generalized-diatonic scale, some 
musicians may find the complexity of Miracle nearly prohibitive. 
However, I was by no means suggesting a ranking like this for our 
paper, and in fact I still believe we should avoid ranking 
altogether, simply giving a "survey" (ordered by complexity, perhaps, 
though almost certainly we would need to start with the most familiar 
examples before generalizing) for the systems that fall beneath 
whatever cutoff curve we choose to draw on the graph.
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Message: 9507 - Contents - Hide Contents

Date: Thu, 29 Jan 2004 04:19:32

Subject: Re: 114 7-limit temperaments

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
wrote:



> That sounds pretty good, in a hand-waving sort of way, but when I 
try
> to pin down what it actually means in terms of how big a scale we
> might need to make good use of some temperament, my brain just keeps
> sliding off it.



Might that be, in a way, a necessary result of dropping octave-
equivalence?



>>> I suspect one can still specify reasonable (maybe even optimal)
>>> generating schemes with multiple generators. In fact since we're
>>> tempering the octave, we apparently need to find such schemes 
>> already


>>> for so-called linear temperaments.
>> 


>> I don't think we need to do any of this, nor should we want to 
for 
>> the purposes of a paper brief enough to be published.
> 


> So we're not goint to talk about how to obtain a finite scale from a
> temperament. Now that the octave is just another generator (and the
> pair of generators is no longer unique), don't we have some 
explaining
> to do, about why we should iterate one particular generator modulo 
the
> other?



For linear temperaments, I agree that we may still want to give the 
period/generator specification, but there are many ways to justify 
the attention on this. For example, though we didn't assume octave-
equivalence, we may want to assume that the musician will generally 
be most interested in subsets of the temperament that repeat at the 
(tempered) octave.
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Message: 9508 - Contents - Hide Contents

Date: Thu, 29 Jan 2004 22:28:36

Subject: Re: Maximum Error in TOP

From: Paul Erlich

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


> --- In tuning-math@xxxxxxxxxxx.xxxx "Kalle Aho" <kalleaho@m...> 
wrote:


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


>>> --- In tuning-math@xxxxxxxxxxx.xxxx "Kalle Aho" <kalleaho@m...> 
>> wrote:
>>>> Hi!
>>>> 


>>>> How do I get the maximum Tenney-weighted error in a TOP 
>>> temperament? 


>>>> Is it the same as the maximum weighted error of the primes
>>> 
>>> Yes.
>> 


>> Thanks, Paul. Can you explain why this is so? 
>> 
>> Kalle



Whoops, I screwed up. Let me try again:

 It follows from the prime factorization theorem.
 
 The maximum Tenney-weighted error being T implies that
 
 error(p)/log(p) <= T

so

  error(p) <= T*log(p)

 for all the primes p. If the factors in a chosen ratio are 2^a 3^b 
 5^c . . . (each exponent may either be positive or negative), then 
 the error of the chosen ratio cannot be greater than
 
 T*(|a|*log(2) + |b|*log(3) + |c|*log(5) . . .)
 
 since the errors in the primes that make up the chosen ratio, at 
 worst, add up without cancellation to the error in the chosen ratio. 
 The complexity of the ratio, meanwhile, is exactly
 
 (|a|*log(2) + |b|*log(3) + |c|*log(5) . . .)
 
 So the Tenney-weighted error in the ratio cannot be greater than the 
 second-to-last expression divided by the last expression, i.e., T.
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Message: 9509 - Contents - Hide Contents

Date: Thu, 29 Jan 2004 07:10:08

Subject: Re: 60 for Dave (was: 41 "Hermanic" 7-limit linear temperaments)

From: Dave Keenan

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


> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
> wrote:
> 


>> One might try 
>> 
>> k*sqrt(comp) + sqrt(err) < x
>> 
>> for starters.
> 


> All right, just for you . . . This time I'll insist Waage just makes 
> it in despite its high complexity. Then k=1.75 gives the most 
> inclusive cutoff which is still more stringent than Gene's, so 
> culling that list will work again.
...



Well I wouldn't want as many as sixty. I think about half that would
be plenty.



> I think Gene may be using the wrong norm to get his complexity 
> values. I'll wait until I'm sure they're right or corrected.



Well something's wrong. Whether its the badness functions or only the
complexity I don't know. But Diaschismic shouldn't be so far down. I
don't think Miracle should be so far down either. Sure it gets a hit
for having 6 gens to the fifth, but not that much of a hit I would think.

And where's Shrutar?
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Message: 9510 - Contents - Hide Contents

Date: Thu, 29 Jan 2004 22:38:05

Subject: Re: 60 for Dave (was: 41 "Hermanic" 7-limit linear temperaments)

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
wrote:


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


>> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
>> wrote:
>> 


>>> One might try 
>>> 
>>> k*sqrt(comp) + sqrt(err) < x
>>> 
>>> for starters.
>> 


>> All right, just for you . . . This time I'll insist Waage just 
makes 
>> it in despite its high complexity. Then k=1.75 gives the most 
>> inclusive cutoff which is still more stringent than Gene's, so 
>> culling that list will work again.
> ...
> 


> Well I wouldn't want as many as sixty. I think about half that would
> be plenty.



Of course our paper should have even fewer. I'd like to see Injera 
but I won't insist on it. But I was just playing along for the 
purposes of this thread.



>> I think Gene may be using the wrong norm to get his complexity 
>> values. I'll wait until I'm sure they're right or corrected.
> 


> Well something's wrong. Whether its the badness functions or only 
the
> complexity I don't know.



Would you take a closer look, then? I think it's important to rethink 
the problem each time, to allow old prejudices a chance of dissolving.



> But Diaschismic shouldn't be so far down.



For 7-limit? This is only one of at least 3 possible 7-limit 
mappings, so it's weird to even use that name.



> And where's Shrutar?



It fell off the end, with a badness of 6.1221 in this formulation. 
Doesn't shine as a 7-limit linear temperament . . .
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Message: 9511 - Contents - Hide Contents

Date: Thu, 29 Jan 2004 07:30:08

Subject: Re: rank complexity explanation updated

From: Gene Ward Smith

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



> The interval matrix often, if not typically, has all 
unisons/octaves 
> along the diagonal. This one is merely a reshuffling so that the 
> diagonal becomes a right vertical.



That's what I used. Manuel's way is more useful for seeing how the 
harmony works out in the scale, however.
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Message: 9512 - Contents - Hide Contents

Date: Thu, 29 Jan 2004 01:17:16

Subject: Re: rank complexity explanation updated

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:


>>>>>>> When a listener hears a melody in a fixed scal... * [with cont.]  (Wayb.)
>>>>>> 
>>>>>>> Hi Paul,
>>>>> 


>>>>> I'm afraid I don't know what an "external" interval is.  
Here's
>>>>> the interval matrix of the diatonic scale in 12-equal, as 
given
>>>>> by Scala...
>>>>> 
>>>>> 100.0 : 2  4  5  7  9  11 12
>>>>> 200.0 : 2  3  5  7  9  10 12
>>>>> 400.0 : 1  3  5  7  8  10 12
>>>>> 500.0 : 2  4  6  7  9  11 12
>>>>> 700.0 : 2  4  5  7  9  10 12
>>>>> 900.0 : 2  3  5  7  8  10 12
>>>>> 1100.0: 1  3  5  6  8  10 12
>>>> 


>>>> Q: Shouldn't the first row be 000.0?
>>> 


>>> It is quite safe to ignore the values before the colon, as they
>>> are merely an artifact of scala's output.
>> 


>> What kind of artifact are they, if not an error?
> 


> They indicate the interval between 1/1 and the degree of the
> original 'native' mode, on which the mode shown on the particular
> line is based.



In that case, shouldn't the first entry be 000.0?



> In this case they are artifactal only because they are given in
> different units than is the interval matrix itself.



OK, but more serious would seem to be the actual error.
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Message: 9513 - Contents - Hide Contents

Date: Thu, 29 Jan 2004 07:28:20

Subject: Re: 114 7-limit temperaments

From: Dave Keenan

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


> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
> wrote:
> 


>> That sounds pretty good, in a hand-waving sort of way, but when I 
> try


>> to pin down what it actually means in terms of how big a scale we
>> might need to make good use of some temperament, my brain just keeps
>> sliding off it.
> 


> Might that be, in a way, a necessary result of dropping octave-
> equivalence?



It might be, but in any case I'd still like to get on top of it.

Without octave equivalence we can still make it finite by limiting it
to the range of human hearing, say 10 octaves.
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Message: 9514 - Contents - Hide Contents

Date: Thu, 29 Jan 2004 07:34:56

Subject: Re: 114 7-limit temperaments

From: Gene Ward Smith

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



> No particular generator basis is assumed in the TOP complexity 
> calculations. Instead, it's a direct measure of how much the 
> tempering simplifies the lattice, and reduces (Gene seems to 
> imply/agree) to the number of notes per acoustic whatever in the 
case 
> of equal (1-dimensional) temperaments.



I thought equal temperaments were "0-dimensional"?

The complexity measure for an equal temperament pretty much reduces 
to the division of the octave. You could take the maximum of 
|(mapping of p)/log2(p)| if you wanted to be precise about it.
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Message: 9515 - Contents - Hide Contents

Date: Thu, 29 Jan 2004 01:20:18

Subject: Re: 114 7-limit temperaments

From: Dave Keenan

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


> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
> It's the minimax over *all* intervals.



You mean the 1/log(product_complexity) weighted minimax over all
intervals in the prime limit.



>> Obviously you don't do an infinite
>> number of calculations. 
> 


> No, you only need to set the primes, and the rest falls out correctly.
> 



Yes, It's very clever in that way. But still it flies in the face of
Partch's "observation one" (or whatever it is called) that as the
complexity of a frequency ratio increases it must be tuned more and
more accurately to be perceived as just (or something), until it
becomes too complex to hear that way even when tuned precisely.

The very fact that TOP cannot distinguish between a good 7-limit
temperament and a good 9-limit temperament (nor 8 or 10 limit) should
make one suspicious.

I suspect that while the results of tenney weighting are quite good
for 5-limit, and may be acceptable for 7-limit, we might find it
agreeing less and less with our subjective experience when we go to 11
and 13 limits. But then again, maybe not. I'll wait and see (or hear).



>> But even so, the normalisation factor may
>> still converge.
> 


> Yes, it may.
> 


>> If it's a minimax type of thing then all we need is to divide the
>> current result by the maximum weight of any interval.
> 


> This doesn't seem right -- what's your reasoning?



I gave it in 
Yahoo groups: /tuning-math/message/8876 * [with cont.] 

It's what you get when you generalise the correctly normalised MAD
(p=1), RMS (p=2) etc. as p goes to infinity.

And in this application the interval with the maximum weight is the
one with the lowest product_complexity, namely the octave.



>> Let me guess:
>> this is lg2(2) = 1 so there would be no change?



I've just confirmed that for myself regarding errors (but not
complexity yet), by examining a few examples from Gene's list.
Although I should have given this maximum weight as 1/lg2(2).



> There is no need to use base 2 -- the result is the same regardless 
> of which base you use in the logarithms. The error is measured using 
> logarithms, but so is the complexity = log(n*d), so error divided by 
> complexity, which is what you're minimizing the maximum of, is 
> insensitive to choice of base.



Paul, that's clearly not what Gene has done in this list, otherwise
the error figures would be a factor of 1200 smaller than they are.
Gene has in fact already correctly normalised the errors and given
them in cents.

If you scan down the list 
Yahoo groups: /tuning-math/message/8809 * [with cont.] 
you will see that in most cases the error figure given for the
temperament happens to be the same as the cents error in the octave,
and is never less than it.

It's just that one now has to think like this:

If the minimax tenney-weighted error for the temperament is X cents,
then the errors in the following intervals could be as large as:

Interval  Error(cents)
----------------------
1:2      X
2:3  2.6 X
4:5  4.3 X
5:6  4.9 X
4:7  4.8 X
5:7  5.1 X
6:7  5.4 X
4:9  5.2 X
5:9  5.5 X
7:9  6.0 X

where the multiplication factor is the base-2 log of the product of
the two sides of the ratio in lowest terms.



>> Is the weighting the same for the complexity? Minimax where gens per
>> interval is divided by lg2(product_complexity(interval))?
> 


> No particular generator basis is assumed in the TOP complexity 
> calculations. Instead, it's a direct measure of how much the 
> tempering simplifies the lattice, and reduces (Gene seems to 
> imply/agree) to the number of notes per acoustic whatever in the case 
> of equal (1-dimensional) temperaments.



What's an "acoustic whatever"?  Anything that relates to sound??? I'm
being a pedant here in case you didn't guess. :-)

OK. That sounds alright, but how do I relate it to upper limits (or
whatever) on the numbers of notes before I get a particular interval?
i.e. in a similar way to what I just did above with the errors.
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Message: 9516 - Contents - Hide Contents

Date: Thu, 29 Jan 2004 07:38:32

Subject: Re: rank complexity explanation updated

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:



> They indicate the interval between 1/1 and the degree of the
> original 'native' mode, on which the mode shown on the particular
> line is based.



In which case the first line should have 0.000 cents, as people have 
been saying.
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Message: 9517 - Contents - Hide Contents

Date: Thu, 29 Jan 2004 07:40:30

Subject: Re: Graef article on rationalization of scales

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:



> That the diatonic scale is a 'good' PB seems like the best example.



Is PB a synonym for Fokker block, or is it more general, and if so, 
how precisely is it defined?
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Message: 9518 - Contents - Hide Contents

Date: Thu, 29 Jan 2004 01:29:27

Subject: Re: Graef article on rationalization of scales

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:


>>>>> This raises an interesting question.  What is our approved 
method
>>>>> for finding Fokker blocks for an arbitrary irrational scale?
>>>>> Such a method would surely make Graf's look silly.
>>>> 


>>>> All such methods are silly,
>>> 


>>> Perhaps you mean Graf's idea of people wanting "just" versions of
>>> arbitrary scales is silly.  That's for sure.
>> 
>> Yup.
>> 


>>>> but I prefer the hexagonal (rhombic dodecahedral, etc.) or Kees
>>>> blocks that result from the min-"odd-limit" criterion. But the
>>>> whole idea of rationalizing a tempered scale is completely
>>>> backwards and misses the point in a big way.
>>> 


>>> I think you missed my point.
>> 


>> Would you, then, clarify your point, perhaps with examples?
> 


> That the diatonic scale is a 'good' PB seems like the best example.



Sure, but when 81:80 is tempered out, it's a good "periodicity strip" 
(figure 5 in TFoT), which is even better, and doesn't require you to 
arbitrarily rationalize the pitches.



> Since you practically single-handedly launched the 'popular scales
> are good PBs' program, I find it highly unusual that you are now
> asking me what it is.



Now I understand you better. Yet, 'popular scales' will often have a 
large number of plausible derivations from a PB, in terms of its 
shape, its position in the lattice, and the unison vectors involved, 
so going from the scale to the PB still seems like a step backwards, 
a step from greater generality to lesser generality.
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Message: 9519 - Contents - Hide Contents

Date: Thu, 29 Jan 2004 07:42:41

Subject: Re: rank complexity explanation updated

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:



>>> I also see I had the wrong definition of interval matrix;
> 


> That's pretty incredible considering that you've got the
> Rothenberg papers on the matter.



Paul now tells us I had a correct definition. Did Rothenberg invent 
the idea of an interval matrix?
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Message: 9520 - Contents - Hide Contents

Date: Thu, 29 Jan 2004 01:32:27

Subject: Re: rank complexity explanation updated

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:



>> The interval matrix often, if not typically, has all 
unisons/octaves 
>> along the diagonal. This one is merely a reshuffling so that the 
>> diagonal becomes a right vertical.
> 


> No, the interval matrix is always written as above (the values
> after the colons, at least).



Where do you get *always*??

BTW, In matrix algebra, which is where you find characteristic 
polynomials and the like, the *diagonal*, not the *right vertical*, 
represents the relations between like elements.
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Message: 9521 - Contents - Hide Contents

Date: Thu, 29 Jan 2004 02:01:30

Subject: Re: Attn: Gene 2

From: Carl Lumma


>> >hy not use the formula I gave before:
>> 
>> || <<w1 w2 w3 w4 w5 w6|| || = max(|w1|/p3, |w2|/p5, |w3|/p7,
>> |w4|/p3p5, |w5|/p3p7, |w6|/p5p7)
>


>This formula looks like a good argument for using square brackets 
><<  ]] instead of vertical bars <<  || for multivectors. I though we
>had agreed to use square brackets at one stage.



I moved in favor of square brackets for this very reason, but nobody
paid any attention.

-Carl
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Message: 9522 - Contents - Hide Contents

Date: Thu, 29 Jan 2004 02:02:56

Subject: Re: 114 7-limit temperaments

From: Carl Lumma


>For linear temperaments, I agree that we may still want to give the 
>period/generator specification, but there are many ways to justify 
>the attention on this. For example, though we didn't assume octave-
>equivalence, we may want to assume that the musician will generally 
>be most interested in subsets of the temperament that repeat at the 
>(tempered) octave.



As long as the temperament maps the octave, will it not be periodic
at said octave?

-Carl
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Message: 9523 - Contents - Hide Contents

Date: Thu, 29 Jan 2004 21:51:46

Subject: Re: What the numbers mean

From: Herman Miller

On Thu, 29 Jan 2004 11:20:37 -0000, "Gene Ward Smith" <gwsmith@xxxxx.xxx>
wrote:



>Here is a no doubt long overdue discussion of what the numbers in the
>wedgie for a linear temperament mean.
>
>The wedgie for miracle is <<6 -7 -2 -25 -20 15||. Normally we would
>content ourselves with saying the mapping to primes is given by
>[<1 1 3 3|, <0 6 -7 -2|], but that is a 2-centric way of putting it;
>we have other maps for 3 and generator, 5 and generator, and 7 and
>generator:
>
>3&g: [<1 1 3 3|, <-6 0 -25 -20|] g ~ 7/72 ~ 11/114
>
>5&g: [<-2 -8 1 4|, <7 25 0 15|] g ~ 58/72 ~ 135/167
>
>7&g: [<1 7 -4 1|, <-2 -20 15 0|] g ~ 65/72 ~ 182/202



That's useful to know. I can see where the second part of the maps come
from, but how do you get the first part? It's clear that the element
corresponding to the period is always 1 in this example, which makes sense,
but is there any easy way to get the other three numbers other than trying
a few until you find one that works? In other words, it's easy to determine
that [<1 x y z|, <0 6 -7 -2|] is a possible mapping of miracle, as is [<x 1
y z|, <-6 0 -25 -20|], but I don't know how to get x, y, and z. I've been
trying to find something like this in the archives, but I don't know where
to look.

-- 
see my music page --->   ---<The Music Page * [with cont.]  (Wayb.)>--
hmiller (Herman Miller)   "If all Printers were determin'd not to print any
@io.com  email password: thing till they were sure it would offend no body,
\ "Subject: teamouse" /  there would be very little printed." -Ben Franklin
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Message: 9524 - Contents - Hide Contents

Date: Thu, 29 Jan 2004 02:07:00

Subject: Re: Graef article on rationalization of scales

From: Carl Lumma


>> >hat the diatonic scale is a 'good' PB seems like the best example.
>


>Is PB a synonym for Fokker block, or is it more general, and if so, 
>how precisely is it defined?



My impression was that PB is weaker than Fokker, the later requiring
epimorphism and monotonicity (neither of which I have a solid
understanding of) and that the former requires, well, nothing more
than the correct number of commas that, when all of them are tempered
out, gives an equal temperament.

-Carl
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