86 17-limit linear temperaments

Another long list of meaningless numbers for your enjoyment! I left
off the wedgies, though; figuring the mapping would do fine by itself. The
numbers on the bottom row are, in order, TOP L1 complexity, TOP error,
TOP L1 badness, and TOP L-inf badness. The list is ordered in terms of
increasing L1 badness, and is bounded by a maximum error of 10, a
maximum compexity of 250 and the fact that I used standard vals and
only up to 111 to generate candidates.

It's interesting to see the lowest L1 badness figure is picked up by a
sort of mutant cross of 22 and 46 equal; the period being 1/22 octave,
and the generator pretty close to a step of 46. Of course one could
simply use 22-equal as a 17-limit no-13's system, which clearly it
would shine at. The second temperament, 68&72, is close to 72-et, and
the third, 72&87, with the lowest L-inf badness score, associated also.

[[22, 35, 51, 62, 76, 81, 90], [0, 0, 0, 0, 0, 1, 0]]
35.703694 3.278354 489.163754 39.979127

[[4, 6, 9, 11, 14, 14, 16], [0, 6, 5, 4, -3, 14, 6]]
177.955931 .367138 519.118970 21.990568

[[3, 6, 8, 8, 11, 14, 16], [0, -6, -5, 2, -3, -14, -18]]
154.939685 .477860 556.580799 21.845733

[[2, 4, 7, 7, 9, 11, 9], [0, -6, -17, -10, -15, -26, -6]]
135.370886 .604709 583.017919 25.907693

[[3, 2, 1, 2, 9, 1, 4], [0, 6, 13, 14, 3, 22, 18]]
170.996698 .439897 588.211869 24.840356

[[2, 5, 8, 5, 6, 8, 10], [0, -6, -11, 2, 3, -2, -6]]
109.033694 .899546 640.631307 26.495232

[[1, -8, -10, 6, 3, 11, 5], [0, 21, 27, -7, 1, -16, -2]]
163.423313 .531637 667.197921 29.376897

[[1, 1, 3, 3, 2, 7, 7], [0, 6, -7, -2, 15, -34, -30]]
131.591478 .720856 667.986215 27.602718

[[1, 4, 14, 2, -5, 19, 21], [0, -6, -29, 2, 21, -38, -42]]
175.740816 .481058 668.372771 28.698745

[[1, 1, 3, 3, 2, 0, 0], [0, 6, -7, -2, 15, 38, 42]]
132.257325 .717180 669.291986 26.786942

[[2, 1, 5, 2, 8, 11, 6], [0, 6, -1, 10, -3, -10, 6]]
120.797165 .824643 677.868575 29.773085

[[2, 6, 7, 16, 14, 14, 11], [0, -6, -5, -22, -15, -14, -6]]
123.400956 .806425 682.983358 37.999833

[[2, 1, 5, 2, 8, -2, 6], [0, 6, -1, 10, -3, 26, 6]]
142.625716 .661231 685.851560 31.454571

[[12, 19, 28, 34, 42, 45, 49], [0, 0, -1, -2, -3, -4, 0]]
136.469761 .711644 693.926551 25.730342

[[1, 1, 3, 3, 2, 4, 4], [0, 6, -7, -2, 15, -3, 1]]
64.432370 2.060988 702.782857 33.610370

[[24, 38, 56, 67, 83, 89, 98], [0, 0, -1, 1, 0, -1, 0]]
139.150771 .719074 720.532396 43.937595

[[1, 2, 1, 1, 2, 3, 2], [0, -6, 19, 26, 21, 10, 30]]
120.192493 .887948 724.796028 32.325398

[[8, 13, 19, 23, 28, 30, 33], [0, -3, -4, -5, -3, -4, -3]]
137.980889 .739142 731.938592 33.189238

[[1, 0, 1, -3, 9, 0, -7], [0, 6, 5, 22, -21, 14, 42]]
132.588713 .786945 736.976175 39.296968

[[2, 3, 5, 3, 5, 6, 8], [0, 1, -2, 15, 11, 8, 1]]
107.611216 1.076337 752.573201 36.645673

[[2, 2, 5, 6, 5, 7, 7], [0, 3, -1, -1, 5, 1, 3]]
45.159589 3.740932 775.585731 31.870837

[[2, 3, 4, 6, 7, 8, 8], [0, 4, 15, -9, -2, -14, 4]]
174.955968 .563584 778.142105 38.630777

[[1, 2, 2, 3, 4, 4, 4], [0, -4, 3, -2, -5, -3, 1]]
31.246124 6.360941 787.473592 41.293889

[[2, 4, 3, 11, 9, 7, 9], [0, -2, 4, -13, -5, 1, -2]]
95.103636 1.361277 800.612517 47.956912

[[2, 1, 2, 2, 5, 5, 6], [0, 9, 11, 15, 8, 10, 9]]
103.031825 1.218960 801.953999 36.588485

[[2, 3, 5, 7, 9, 10, 8], [0, 1, -2, -8, -12, -15, 1]]
97.383202 1.350896 821.295726 31.013330

[[1, -1, -1, 15, 9, 17, 10], [0, 7, 9, -33, -15, -36, -16]]
160.181500 .676755 825.826100 33.346784

[[2, 4, 3, 7, 5, 3, 9], [0, -3, 6, -5, 7, 16, -3]]
120.645210 1.020023 836.997127 34.808123

[[1, 0, 1, -3, 9, 0, 12], [0, 6, 5, 22, -21, 14, -30]]
137.941373 .848226 839.622411 38.161814

[[9, 14, 21, 25, 31, 33, 37], [0, 2, -1, 2, 1, 2, -2]]
145.339391 .788787 840.032641 37.405701

[[31, 49, 72, 87, 107, 115, 127], [0, 0, 0, 0, 0, -1, -1]]
80.520912 1.805191 846.209131 35.403452

[[1, 4, 4, 3, 7, 5, 8], [0, -13, -9, -1, -19, -7, -21]]
76.358795 1.946389 841.845424 37.044502

[[1, -1, 0, 1, -3, 5, 9], [0, 10, 9, 7, 25, -5, -19]]
109.741255 1.171542 841.929200 37.458353

[[12, 19, 28, 34, 42, 44, 49], [0, 0, -1, -2, -3, 2, 0]]
145.750522 .788787 843.361281 41.212924

[[1, 3, 12, 8, 7, 7, 14], [0, -6, -41, -22, -15, -14, -42]]
139.922888 .835162 843.364272 46.502903

[[2, 3, 5, 5, 7, 8, 8], [0, 2, -4, 7, -1, -7, 2]]
79.621702 1.845166 846.212537 38.505472

[[2, 2, 7, 6, 3, 7, 7], [0, 3, -6, -1, 10, 1, 3]]
86.614132 1.646850 849.724600 47.003992

[[2, 3, 4, 6, 7, 7, 8], [0, 2, 8, -5, -1, 5, 2]]
80.788009 1.823424 853.440744 48.595456

[[1, 1, -1, 3, 6, 8, 8], [0, 3, 17, -1, -13, -22, -20]]
114.267037 1.122802 853.871257 41.901242

[[2, 2, 6, 6, 4, 7, 7], [0, 6, -7, -2, 15, 2, 6]]
129.949925 .949097 864.165921 40.846098

[[1, 0, 3, 1, 3, 1, 5], [0, 7, -3, 8, 2, 12, -4]]
56.855653 3.046034 871.802085 34.884053

[[3, 4, 10, 16, 10, 13, 10], [0, 2, -8, -20, 1, -5, 6]]
221.067026 .455256 872.145973 43.093565

[[3, 6, 8, 8, 11, 14, 11], [0, -6, -5, 2, -3, -14, 6]]
161.338934 .708877 873.787882 33.616266

[[1, 3, 6, -2, 6, 2, -1], [0, -5, -13, 17, -9, 6, 18]]
108.582248 1.235451 874.758078 38.473359

[[1, 0, -6, 4, -12, -7, 14], [0, 4, 21, -3, 39, 27, -25]]
159.484019 .725551 879.977995 39.567164

[[1, 1, 2, 3, 3, 3, 3], [0, 9, 5, -3, 7, 11, 17]]
59.021911 2.924896 882.122665 42.309444

[[1, -4, 5, -3, 1, -3, 9], [0, 25, -12, 26, 11, 30, -22]]
200.792766 .527708 883.570786 37.821689

[[1, 1, 0, 6, 2, 4, 7], [0, 2, 8, -11, 5, -1, -10]]
62.815909 2.705591 890.347596 44.278744

[[2, 3, 5, 5, 7, 6, 8], [0, 2, -4, 7, -1, 16, 2]]
99.892546 1.425313 897.958926 46.775010

[[1, 6, 8, 10, 8, 9, 8], [0, -35, -45, -57, -36, -42, -31]]
201.356562 .537870 904.126791 40.958676

[[29, 46, 67, 81, 100, 107, 119], [0, 0, 1, 1, 1, 1, -1]]
175.678772 .651558 904.815340 41.980486

[[1, 1, 0, 3, 5, 1, 1], [0, 3, 12, -1, -8, 14, 16]]
69.480184 2.388465 905.157771 39.878221

[[3, 5, 7, 9, 11, 11, 12], [0, -1, 0, -2, -2, 0, 1]]
28.261816 8.441030 907.978436 37.039351

[[2, 6, 7, -1, 14, 14, 11], [0, -6, -5, 14, -15, -14, -6]]
151.298701 .806425 908.517849 48.427415

[[1, 2, 4, 7, -2, 10, 15], [0, -1, -4, -10, 13, -15, -26]]
82.696553 1.884632 911.399698 48.090021

[[1, 1, 2, 3, 3, 5, 5], [0, 9, 5, -3, 7, -20, -14]]
94.984749 1.552975 911.758743 45.046593

[[1, -13, -4, -4, 2, -7, 7], [0, 30, 13, 14, 3, 22, -6]]
155.530567 .778880 912.037830 52.992251

[[1, 0, 3, 1, 3, 8, 5], [0, 7, -3, 8, 2, -19, -4]]
76.988597 2.088840 913.907267 49.179844

[[2, 4, 3, 5, 3, 7, 9], [0, -4, 8, 3, 19, 2, -4]]
135.968025 .945066 916.797784 46.036739

[[9, 14, 21, 25, 31, 33, 36], [0, 2, -1, 2, 1, 2, 6]]
155.469387 .788787 923.128869 41.884843

[[5, 8, 12, 14, 17, 19, 20], [0, 0, -1, 0, 1, -1, 1]]
31.961332 7.239629 925.105237 43.679594

[[1, 1, -5, -1, 2, 4, 12], [0, 2, 25, 13, 5, -1, -27]]
126.837248 1.056164 929.559068 57.542054

[[36, 57, 83, 101, 124, 133, 147], [0, 0, 1, 0, 1, 0, 0]]
134.546283 .972879 929.992127 45.153780

[[1, 5, -3, -1, -3, -2, -2], [0, -9, 14, 10, 17, 15, 16]]
90.728296 1.692526 931.912699 52.865988

[[1, 19, 8, 10, 8, 9, -5], [0, -46, -15, -19, -12, -14, 24]]
242.344021 .431130 939.318852 62.272699

[[1, 3, 2, 4, 4, 5, 3], [0, -13, 3, -11, -5, -12, 10]]
91.202462 1.697662 941.586814 46.577693

[[2, 3, 5, 6, 6, 7, 8], [0, 1, -2, -2, 5, 2, 1]]
40.599676 5.277847 942.732373 44.786211

[[6, 10, 14, 17, 21, 23, 25], [0, -6, -1, -2, -3, -10, -6]]
199.769151 .571898 950.732600 45.301805

[[8, 14, 19, 22, 29, 30, 34], [0, -3, -1, 1, -3, -1, -3]]
143.008253 .913426 950.996544 52.162277

[[1, 4, 1, 5, 5, 7, 1], [0, -11, 6, -10, -7, -15, 14]]
95.799140 1.607178 954.927417 42.316223

[[1, -5, 0, -3, -7, 13, 13], [0, 17, 6, 15, 27, -24, -23]]
167.786486 .734851 956.884090 39.506471

[[1, -1, 3, -1, 4, 9, 3], [0, 19, -5, 28, -4, -39, 8]]
189.817715 .619626 958.960867 48.456461

[[1, 2, 16, 14, -4, -5, -10], [0, -1, -33, -27, 18, 21, 34]]
208.113646 .549094 966.645465 43.052870

[[1, 2, 3, 2, 4, 4, 3], [0, -3, -5, 6, -4, -2, 8]]
43.550433 4.913207 968.176518 37.807834

[[1, 4, 5, 2, 4, 8, 10], [0, -9, -10, 3, -2, -16, -22]]
69.479312 2.562574 971.122633 37.068352

[[2, 7, 10, 6, 5, 12, 12], [0, -10, -14, -1, 5, -12, -10]]
152.817550 .852432 973.872559 39.556045

[[1, 4, 1, 5, 5, 7, 12], [0, -11, 6, -10, -7, -15, -36]]
100.118407 1.546473 977.376691 46.549188

[[2, 4, 11, 7, 13, 11, 9], [0, -3, -23, -5, -22, -13, -3]]
170.834824 .732328 977.941293 47.904743

[[1, 4, 5, 2, 4, 8, 3], [0, -9, -10, 3, -2, -16, 4]]
70.235360 2.543188 978.490291 36.787926

[[24, 38, 56, 68, 83, 89, 98], [0, 0, -1, -2, 0, -1, 0]]
156.779130 .826783 979.027978 44.006115

[[3, 5, 7, 9, 11, 12, 12], [0, -1, 0, -2, -2, -3, 1]]
31.375786 7.871799 980.181379 39.648038

[[4, 6, 9, 11, 13, 14, 16], [0, 1, 1, 1, 3, 3, 1]]
32.019593 7.673091 982.997639 43.774122

[[1, -4, 0, 7, 3, -7, 12], [0, 12, 5, -9, 1, 23, -17]]
112.503764 1.326638 987.157494 41.670592

[[1, 5, 15, 15, 2, -8, -12], [0, -7, -26, -25, 3, 24, 33]]
187.888751 .647154 987.344983 40.758270

[[1, 1, 0, 2, 3, 3, 2], [0, 5, 20, 7, 4, 6, 18]]
67.751799 2.709169 991.117895 55.219170

[[2, 4, 3, 5, 9, 7, 9], [0, -4, 8, 3, -10, 2, -4]]
110.938917 1.361277 993.262173 47.774013

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

> [[12, 19, 28, 34, 42, 45, 49], [0, 0, -1, -2, -3, -4, 0]]
> 136.469761 .711644 693.926551 25.730342

This, a 17-limit version of Waage, which seems interesting from the
notation angle as well as as a temperament; it has a TM basis
{221/220, 225/224, 289/288, 351/350, 441/440}, and can be decribed,
not very well, as 72&84. Like a lot of these, it seems we might as
well simply use an equal temperament (in this case 72) and regard the
linear temperament as an organizing concept.

hi Gene,

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

> Another long list of meaningless numbers for your enjoyment!
> I left off the wedgies, though; figuring the mapping would
> do fine by itself.

i thought the wedgies were important and useful as
identifiers. why not just include them as well?

> The numbers on the bottom row are, in order,
> TOP L1 complexity, TOP error, TOP L1 badness,
> and TOP L-inf badness.

i'm still waiting for definitions of "badness" and
"logflat badness". now what's this "L1" business?

-monz

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:

> i thought the wedgies were important and useful as
> identifiers. why not just include them as well?

The size of the wedgies grows quadratically, and the size of the
mapping linearlly. I used the wedgies to compute this, but figured at
21 integers long, it would began to seem excessive to give them; they
can, of course, be obtained as the wedge product of the two vals in
the mapping.

> i'm still waiting for definitions of "badness" and
> "logflat badness". now what's this "L1" business?

"Badness" is a function of the complexity and error of a temperament,
which increases if you increase either complexity or error. A common
example of a badness function is Complexity^e * Error, where e is a
fixed exponent. The idea of a badness function is to compare and rank
temperaments, with a lower badness number being better.

A logflat badness measure chooses the exponent "e" of complexity so
that there is a badness such that below it an infinite number of
temperaments can be found, but if "e" were to be raised, only a finite
number could be found below any badness cutoff. If L is a regular
temperament of rank r (or dimension r-1), so that for an equal
temperament r is 1, for a linear temperament it is 2, and so forth;
and if d is the number of primes in the prime limit p, so that d =
pi(p), then the exponent e should be chosen as d/(d-r) to achieve
logflatness.

hi Gene,

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

> --- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
>
> > i thought the wedgies were important and useful as
> > identifiers. why not just include them as well?
>
> The size of the wedgies grows quadratically, and the size
> of the mapping linearlly. I used the wedgies to compute
> this, but figured at 21 integers long, it would began to
> seem excessive to give them; they can, of course, be obtained
> as the wedge product of the two vals in the mapping.

ok, gotcha.

> > i'm still waiting for definitions of "badness" and
> > "logflat badness". now what's this "L1" business?
>
> "Badness" is a function of the complexity and error
> of a temperament, which increases if you increase either
> complexity or error. A common example of a badness function
> is Complexity^e * Error, where e is a fixed exponent.
> The idea of a badness function is to compare and rank
> temperaments, with a lower badness number being better.

ok, thanks for that.

can (should?) you give any other examples of badness function,
or do the other ones not really matter? please describe
the criteria for deciding which type of badness function
to choose.

also, i'd like you to go into more depth about how you
choose a certain type of complexity, because there are many.

> A logflat badness measure chooses the exponent "e" of
> complexity so that there is a badness such that below it
> an infinite number of temperaments can be found, but if
> "e" were to be raised, only a finite number could be
> found below any badness cutoff. If L is a regular
> temperament of rank r (or dimension r-1), so that for
> an equal temperament r is 1, for a linear temperament it
> is 2, and so forth; and if d is the number of primes in
> the prime limit p, so that d = pi(p), then the exponent e
> should be chosen as d/(d-r) to achieve logflatness.

this paragraph is a separate definition for "logflat badness",
and my thinking is that it should be a second entry on the
"badness" page. does anyone think it should be a separate
entry in the Encyclopaedia?

http://tonalsoft.com/enc/badness.htm

Gene, the link i have for "regular temperament" goes
to my "icon" page which temporarily copies your page
on regular temperaments. please help me sort out the
definition of "icon" so that i can set up a correct
"regular temperament" page.

feel free to add to my definition of "complexity"
after you've checked the links to all the specific types.

-monz

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:

> can (should?) you give any other examples of badness function,
> or do the other ones not really matter?

Dave or Paul might like to supply their current favorite, or you could
check Paul's draft for the Xenharmonikon paper and see if it is in
there. However, one line of thought about badness is that you don't
want it to allow an infinite list of temperaments, since you require a
finite list for your purposes, and would like to do the filtering with
some badness function. Logflat badness has the advantage of not being
so ad hoc.

please describe
> the criteria for deciding which type of badness function
> to choose.

Depends on what you want to use it for, and what type of error and
complexity measure you choose.

> http://tonalsoft.com/enc/badness.htm
>
> Gene, the link i have for "regular temperament" goes
> to my "icon" page which temporarily copies your page
> on regular temperaments. please help me sort out the
> definition of "icon" so that i can set up a correct
> "regular temperament" page.

Why not link to this instead:

http://en.wikipedia.org/wiki/Regular_temperament

hi Gene,

> > http://tonalsoft.com/enc/badness.htm
> >
> > Gene, the link i have for "regular temperament" goes
> > to my "icon" page which temporarily copies your page
> > on regular temperaments. please help me sort out the
> > definition of "icon" so that i can set up a correct
> > "regular temperament" page.
>
> Why not link to this instead:
>
> http://en.wikipedia.org/wiki/Regular_temperament

i could do that ... but the Encyclopaedia of Tuning
*should* have its own definition of "regular temperament"
anyway.

in any case, i don't want to keep your entire essay
on my "icon" page, since it's about much more than only
icons ... please help me with the editing of that definition.

-monz

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:

> in any case, i don't want to keep your entire essay
> on my "icon" page, since it's about much more than only
> icons ... please help me with the editing of that definition.

I proposed the name "icon", since it defines an image, but no one ever
uses it. It would make more sense to have a definition for the "prime
mapping" or "mapping to primes" of a temperament, which is what people
actually say.

Prime mapping

For a regular temperament of rank n, or in other words of "dimension"
n-1, the prime mapping is a list of n vals which defines how the prime
intervals are to be expressed in terms of the particular set of
generators chosen. For instance, [<1 2 4 7}, <0 -1 -4 -10|] is a prime
mapping for 7-limit meantone; in terms of the two generators P and G,
it says (expressed additively) that a 2 is given by P, a 3 by 2P-G, a
5 by 4P-4G, and a 7 by 7P-10G; so the generators are an octave and a
fourth. We could equally well express meantone in terms of 12P+7G and
19P+8G, in which case the prime mapping would be given by
[<12 19 28 34|, <19 30 44 53|]. Either system could be useful; using
the second one any meantone tuning between 12 and 19 is expressed as a
linear combination using non-negative integer coefficients of the 12
and 19 vals.

Gene Ward Smith wrote:
> Another long list of meaningless numbers for your enjoyment!

And here's a list of mostly meaningless names to relate these to more familiar 7- and 11-limit temperaments.

> [[1, 1, 3, 3, 2, 7, 7], [0, 6, -7, -2, 15, -34, -30]]
> 131.591478 .720856 667.986215 27.602718

This is a version of our old friend miracle. Three of these are extensions of miracle, plus a fourth which looks like a double miracle.

> [[1, 1, 3, 3, 2, 0, 0], [0, 6, -7, -2, 15, 38, 42]]
> 132.257325 .717180 669.291986 26.786942

The second of the miracle temperaments.

> [[2, 1, 5, 2, 8, 11, 6], [0, 6, -1, 10, -3, -10, 6]]
> 120.797165 .824643 677.868575 29.773085

This is one of two versions of wizard (50&72).

> [[2, 1, 5, 2, 8, -2, 6], [0, 6, -1, 10, -3, 26, 6]]
> 142.625716 .661231 685.851560 31.454571

The second version of wizard (72&94).

> [[12, 19, 28, 34, 42, 45, 49], [0, 0, -1, -2, -3, -4, 0]]
> 136.469761 .711644 693.926551 25.730342

As noted, this is a version of compton/waage/duodecimal/whatever it's being called.

> [[1, 1, 3, 3, 2, 4, 4], [0, 6, -7, -2, 15, -3, 1]]
> 64.432370 2.060988 702.782857 33.610370

The third miracle.

> [[1, 0, 1, -3, 9, 0, -7], [0, 6, 5, 22, -21, 14, 42]]
> 132.588713 .786945 736.976175 39.296968

This is one of two 17-limit versions of hanson (aka catakleismic).

> [[2, 2, 5, 6, 5, 7, 7], [0, 3, -1, -1, 5, 1, 3]]
> 45.159589 3.740932 775.585731 31.870837

Lemba (16&26).

> [[1, 2, 2, 3, 4, 4, 4], [0, -4, 3, -2, -5, -3, 1]]
> 31.246124 6.360941 787.473592 41.293889

Negri.

> [[2, 3, 5, 7, 9, 10, 8], [0, 1, -2, -8, -12, -15, 1]]
> 97.383202 1.350896 821.295726 31.013330

A 17-limit version of diaschismic (46&58).

> [[1, 0, 1, -3, 9, 0, 12], [0, 6, 5, 22, -21, 14, -30]]
> 137.941373 .848226 839.622411 38.161814

Another catakleismic/hanson variant.

> [[1, -1, 0, 1, -3, 5, 9], [0, 10, 9, 7, 25, -5, -19]]
> 109.741255 1.171542 841.929200 37.458353

Myna ("nonkleismic").

> [[12, 19, 28, 34, 42, 44, 49], [0, 0, -1, -2, -3, 2, 0]]
> 145.750522 .788787 843.361281 41.212924

Another duodecimal temperament.

> [[2, 3, 5, 5, 7, 8, 8], [0, 2, -4, 7, -1, -7, 2]]
> 79.621702 1.845166 846.212537 38.505472

One of two versions of shrutar.

> [[1, 1, -1, 3, 6, 8, 8], [0, 3, 17, -1, -13, -22, -20]]
> 114.267037 1.122802 853.871257 41.901242

Supersupermajor.

> [[2, 2, 6, 6, 4, 7, 7], [0, 6, -7, -2, 15, 2, 6]]
> 129.949925 .949097 864.165921 40.846098

This is the one that looks like a double miracle; it's got two periods per octave and the usual miracle generator mapping.

> [[1, 0, 3, 1, 3, 1, 5], [0, 7, -3, 8, 2, 12, -4]]
> 56.855653 3.046034 871.802085 34.884053

One of two versions of orwell.

> [[1, 3, 6, -2, 6, 2, -1], [0, -5, -13, 17, -9, 6, 18]]
> 108.582248 1.235451 874.758078 38.473359

Amity.

> [[1, 1, 2, 3, 3, 3, 3], [0, 9, 5, -3, 7, 11, 17]]
> 59.021911 2.924896 882.122665 42.309444

One of two versions of valentine (aka quartaminorthirds).

> [[2, 3, 5, 5, 7, 6, 8], [0, 2, -4, 7, -1, 16, 2]]
> 99.892546 1.425313 897.958926 46.775010

The second version of shrutar.

> [[1, 1, 0, 3, 5, 1, 1], [0, 3, 12, -1, -8, 14, 16]]
> 69.480184 2.388465 905.157771 39.878221

Supermajor (26&57).

> [[3, 5, 7, 9, 11, 11, 12], [0, -1, 0, -2, -2, 0, 1]]
> 28.261816 8.441030 907.978436 37.039351

One of two versions of augmented.

> [[1, 2, 4, 7, -2, 10, 15], [0, -1, -4, -10, 13, -15, -26]]
> 82.696553 1.884632 911.399698 48.090021

Meantone.

> [[1, 1, 2, 3, 3, 5, 5], [0, 9, 5, -3, 7, -20, -14]]
> 94.984749 1.552975 911.758743 45.046593

Another 17-limit extension of valentine.

> [[1, 0, 3, 1, 3, 8, 5], [0, 7, -3, 8, 2, -19, -4]]
> 76.988597 2.088840 913.907267 49.179844

Anoteher version of orwell.

> [[5, 8, 12, 14, 17, 19, 20], [0, 0, -1, 0, 1, -1, 1]]
> 31.961332 7.239629 925.105237 43.679594

Blackwood.

> [[1, 1, -5, -1, 2, 4, 12], [0, 2, 25, 13, 5, -1, -27]]
> 126.837248 1.056164 929.559068 57.542054

Hemififths.

> [[2, 3, 5, 6, 6, 7, 8], [0, 1, -2, -2, 5, 2, 1]]
> 40.599676 5.277847 942.732373 44.786211

Pajara.

> [[1, 2, 16, 14, -4, -5, -10], [0, -1, -33, -27, 18, 21, 34]]
> 208.113646 .549094 966.645465 43.052870

Kwai.

> [[1, 2, 3, 2, 4, 4, 3], [0, -3, -5, 6, -4, -2, 8]]
> 43.550433 4.913207 968.176518 37.807834

Porcupine.

> [[1, 4, 5, 2, 4, 8, 10], [0, -9, -10, 3, -2, -16, -22]]
> 69.479312 2.562574 971.122633 37.068352

One of two versions of superkleismic.

> [[1, 4, 5, 2, 4, 8, 3], [0, -9, -10, 3, -2, -16, 4]]
> 70.235360 2.543188 978.490291 36.787926

The other version of superkleismic.

> [[24, 38, 56, 68, 83, 89, 98], [0, 0, -1, -2, 0, -1, 0]]
> 156.779130 .826783 979.027978 44.006115

Appears to be a double version of compton/waage/duodecimal.

> [[3, 5, 7, 9, 11, 12, 12], [0, -1, 0, -2, -2, -3, 1]]
> 31.375786 7.871799 980.181379 39.648038

The second version of augmented.

> [[4, 6, 9, 11, 13, 14, 16], [0, 1, 1, 1, 3, 3, 1]]
> 32.019593 7.673091 982.997639 43.774122

Diminished.

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...> wrote:

> And here's a list of mostly meaningless names to relate these to more
> familiar 7- and 11-limit temperaments.

Looking at the unfamiliar temperaments has its virtues also. Consider
this one, the seventh on the list:

[[1, -8, -10, 6, 3, 11, 5], [0, 21, 27, -7, 1, -16, -2]]
163.423313 .531637 667.197921 29.376897

As a 7-limit temperament with commas 1029/1024 and 78732/78125 it
isn't very interesting. As an 11-limit temperament, things start to
improve, as we have 11/8 as a generator. When we get up to the
17-limit, we also have an uncomplex 17. If we take the {2,11,17}
subtemperament, we get 2057/2048 tempered out; if we take the
{2,7,11,17} subtemperament, 4928/4913 is tempered out also. The
11-note MOS, not a very complex object, has a lot of {2,7,11,17}
harmony to play with, and would surely be exotic in sound. If we now
add the 13, we are tempering out 833/832, and could move up to a more
complex scale to take advantage of it, such as the 24 note MOS.
Finally if we go all the way to the full 17-limit, we could use, for
instance, the 57 note MOS, and temper out 273/272, 351/350, 441/440,
561/560 and 847/845.

hi Gene,

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
>
> > in any case, i don't want to keep your entire essay
> > on my "icon" page, since it's about much more than only
> > icons ... please help me with the editing of that definition.
>
> I proposed the name "icon", since it defines an image,
> but no one ever uses it. It would make more sense to have
> a definition for the "prime mapping" or "mapping to primes"
> of a temperament, which is what people actually say.
>
> Prime mapping
>
> For a regular temperament of rank n, or in other words of
> "dimension" n-1, the prime mapping is a list of n vals
> which defines how the prime intervals are to be expressed
> in terms of the particular set of generators chosen.
>
> For instance, [<1 2 4 7}, <0 -1 -4 -10|] is a prime
> mapping for 7-limit meantone; in terms of the two generators
> P and G, it says (expressed additively) that a 2 is given
> by P, a 3 by 2P-G, a 5 by 4P-4G, and a 7 by 7P-10G; so the
> generators are an octave and a fourth.
>
> We could equally well express meantone in terms of 12P+7G
> and 19P+8G, in which case the prime mapping would be given
> by [<12 19 28 34|, <19 30 44 53|]. Either system could be
> useful; using the second one any meantone tuning between
> 12 and 19 is expressed as a linear combination using
> non-negative integer coefficients of the 12 and 19 vals.

this is great, exactly what i needed.

in fact, i was already planning an Encyclopaedia page
for "map" or "mapping", so this will go there.

... i'll still keep a page for "icon" too, referring
to the "map" page, since it's a term that's published
online in your work.

thanks a bunch!

-monz

hi Gene,

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

> For instance, [<1 2 4 7}, <0 -1 -4 -10|] is a prime
> mapping for 7-limit meantone; in terms of the two
> generators P and G, it says (expressed additively)
> that a 2 is given by P, a 3 by 2P-G, a 5 by 4P-4G,
> and a 7 by 7P-10G;

so far i follow you completely ...

> so the generators are an octave and a fourth.

please show me that math that explains that.

> We could equally well express meantone in terms of 12P+7G
> and 19P+8G, in which case the prime mapping would be given
> by [<12 19 28 34|, <19 30 44 53|].

again, i understand the mapping, but not how you find
the size of P and G.

> Either system could be useful; using the second one
> any meantone tuning between 12 and 19 is expressed as
> a linear combination using non-negative integer
> coefficients of the 12 and 19 vals.

you lost me here.

-monz

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:

> > For instance, [<1 2 4 7|, <0 -1 -4 -10|] is a prime
> > mapping for 7-limit meantone; in terms of the two
> > generators P and G, it says (expressed additively)
> > that a 2 is given by P, a 3 by 2P-G, a 5 by 4P-4G,
> > and a 7 by 7P-10G;

> > so the generators are an octave and a fourth.

> please show me that math that explains that.

We have

Octave = P
Twelvth = 2P - G,

Solving the linear equations for P and G gives

P = Octave
G = 2 Octave - Twelvth

>
>
>
> > We could equally well express meantone in terms of 12P+7G
> > and 19P+8G, in which case the prime mapping would be given
> > by [<12 19 28 34|, <19 30 44 53|].
>
>
> again, i understand the mapping, but not how you find
> the size of P and G.

P and G I am assuming are the P and G associated to the original prime
mapping I gave, which is to say, the corresponding vals are
p = <1 2 4 7| and g = <0 -1 -4 -10|. The vals are associated to the
generators in terms of cents, so by an abuse of language you could
identify them. If you don't want to be that abusive, if P is a tuning
for the octave, say 1200 cents, and G is a tuning for the fourth, say
504 cents, thenany 7-limit number q is mapped by this tuning of
meantone (actually 50 equal) to p(q)P + g(q)G cents.

> > Either system could be useful; using the second one
> > any meantone tuning between 12 and 19 is expressed as
> > a linear combination using non-negative integer
> > coefficients of the 12 and 19 vals.

> you lost me here.

I meant any equal tuning if that was the problem; 12 and 19 are
relatively prime, so any integer N can be written as a sum 12a + 19b =
N for some integers a and b. If we want a meantone val, then we pick a
and b positive, which if N is small enough will give us at most one
solution. For example, 16*12+13*19 and 3*12+19 both equal 55, but the
latter has no negative coefficients. The only positive solution is
3*12+19, so the correct val for 55-et meantone is 3<12 19 28 34| +
<19 30 44 53|, and we see 55 is on the 12 side of things. For larger N
we eventually will get more than one positive solution, and so more
than one possible meantone. For instance, for 270 we still only have
one positive solution, 13*12 + 9*19, but for 494 we have the contorted
solutions 26*19 and 38*12+2, as well as 19*12+4*19.