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Message: 3875 - Contents - Hide Contents

Date: Thu, 14 Feb 2002 11:06 +0

Subject: Re: ^ and ** (was: h72 = h31 + h41)

From: graham@xxxxxxxxxx.xx.xx

In-Reply-To: <a4fqll+7o7l@xxxxxxx.xxx>
paulerlich wrote:

> let's keep ^ for exponentiation and use > /\ for wedge product
Yuck! I've got my Python library to overload ^ for wedge products. As ** is already exponentiation there's no conflict. And as ** for exponentiation goes right back to Fortran, it's not like we're breaking any standards anyway. I don't know of any languages that can overload /\. Graham
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Message: 3876 - Contents - Hide Contents

Date: Thu, 14 Feb 2002 11:06:48

Subject: Some 58-et reduced bases

From: genewardsmith

Here are the 11 and 13 limit MT reduced bases for the 58 et; perhaps some 58-et scale possibilities can be explored with their help.

11-limit: <126/125, 176/175, 243/242, 896/891>

13-limit: <126/125, 144/143, 176/175, 196/195, 354/363>


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Message: 3877 - Contents - Hide Contents

Date: Thu, 14 Feb 2002 07:03:59

Subject: Re: h72 = h31 + h41

From: genewardsmith

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

> but i've seen ^ used in connection with wedgies, so my > question is still not really answered: do we have to use ** > now to represent "raise to the power of"? apparently, > whatever ^ is being used for, it's something else other > than that.
The "^" symbol is well-established as a notation both for exponentiation and wedge product; I would use it for either myself so long as there seemed no potential for confusion. Fortran gave us "**" for exponentiation also, which is fine, and not used for anything else to my knowledge.
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Message: 3878 - Contents - Hide Contents

Date: Thu, 14 Feb 2002 19:11:53

Subject: Re: Some 58-et reduced bases

From: genewardsmith

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

> 13-limit: <126/125, 144/143, 176/175, 196/195, 354/363>
<126/125, 144/143, 176/175, 196/195, 364/363>
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Message: 3879 - Contents - Hide Contents

Date: Thu, 14 Feb 2002 20:38:27

Subject: Re: ^ and ** (was: h72 = h31 + h41)

From: paulerlich

--- In tuning-math@y..., graham@m... wrote:
> In-Reply-To: <a4fqll+7o7l@e...> > paulerlich wrote: >
>> let's keep ^ for exponentiation and use >> /\ for wedge product >
> Yuck! I've got my Python library to overload ^ for wedge
products. As **
> is already exponentiation there's no conflict. And as ** for > exponentiation goes right back to Fortran, it's not like we're breaking > any standards anyway. I don't know of any languages that can overload /\.
how about the english language? i thought that's what we were talking about.
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Message: 3880 - Contents - Hide Contents

Date: Fri, 15 Feb 2002 21:38:01

Subject: Proposed definitions--convex, block

From: genewardsmith

A scale is *convex* if every octave equivalence class contained in the convex hull
Convex Hull -- from MathWorld * [with cont.] 
of the classes of the scale is itself a class of the scale.

A scale is a *block* iff it is epimorphic and convex.


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Message: 3881 - Contents - Hide Contents

Date: Fri, 15 Feb 2002 14:30:09

Subject: Re: Proposed definitions--convex, block

From: monz

> From: genewardsmith <genewardsmith@xxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Friday, February 15, 2002 1:38 PM > Subject: [tuning-math] Proposed definitions--convex, block > > > A scale is *convex* if every octave equivalence class contained in the convex hull > Convex Hull -- from MathWorld * [with cont.] > of the classes of the scale is itself a class of the scale. > > A scale is a *block* iff it is epimorphic and convex.
when you say "block", do you mean "periodicity-block"? -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
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Message: 3882 - Contents - Hide Contents

Date: Fri, 15 Feb 2002 22:32:03

Subject: Some 11-limit bases and associated temperament bases

From: genewardsmith

Here are 11-limit bases for 22,31,41,46,58, and 72. Associated to these
are four linear temperaments, which can also be regarded as forming a
basis for the et; these are obtained by leaving out one of the commas.
These are by no means necessarily the four best linear temperaments
for that et, but they are interesting and do define it.

In the case of the 22-et, we get two versions of pajara; these are the
same for 22, but the [2,-4,-4,-12,-11,-12,-26,2,-14,-20] version is
consistent with h10-v11, h12, h34+v7, while the
[2,-4,-4,10,-11,-12,9,2,37,42] version is consistent with h10, 
h12-v11, and h34+v7-v11. The temperament consisting of two chains of
9/7 a half-ocatave apart still looks good for the 22-et, and I wonder
if Paul has ever had occasion to try it. A scale of 8 or 14 tones
suggests itself for this.

The temperament with a compromise 12/11 or 11/10 generator in the
31-et seems to go with the discussion of Arabic, though it is quite
different, and the 11/7 generator system looks worth exploring.

The 333333333333334 MOS of the 46-et suggested by the temperament with
a 77.963 cent generator looks interesting, if a little melodically
bland.

The temperament consisting of two chains of major thirds a half-octave
apart seems like an interesting 72-et alternative.

22: [50/49, 55/54, 64/63, 99/98]


[99/98, 64/63, 55/54]

wedgie   [1, 9, -2, -6, 12, -6, -13, -30, -45, -10]

map   [[0, -1, -9, 2, 6], [1, 2, 6, 2, 1]]

generators   [490.8051577, 1200.]

bad   344.0525564   rms   12.62392639   g   7.265377779


[99/98, 64/63, 50/49]

wedgie   [2, -4, -4, -12, -11, -12, -26, 2, -14, -20]

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

generators   [492.8941324, 600.]

bad   328.2749546   rms   9.552922731   g   8.349508111


[99/98, 55/54, 50/49]

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

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

generators   [434.9412914, 600.]

bad   238.7261365   rms   11.89273381   g   6.047431569


[64/63, 55/54, 50/49]

wedgie   [2, -4, -4, 10, -11, -12, 9, 2, 37, 42]

map   [[0, -1, 2, 2, -5], [2, 4, 3, 4, 11]]

generators   [490.1172150, 600.]

bad   373.4699058   rms   12.26714783   g   7.764387569



31: [81/80, 99/98, 121/120, 126/125]


[126/125, 121/120, 99/98]

wedgie   [11, 13, 17, 12, -5, -4, -19, 3, -17, -25]

map   [[0, -11, -13, -17, -12], [1, 3, 4, 5, 5]]

generators   [154.5139994, 1200.]

bad   299.8803381   rms   6.113337322   g   10.33717287


[126/125, 121/120, 81/80]

wedgie   [2, 8, 20, 5, 8, 26, 1, 24, -16, -55]

map   [[0, 2, 8, 20, 5], [1, 1, 0, -3, 2]]

generators   [348.1847713, 1200.]

bad   331.9892755   rms   6.645961894   g   10.45056389


[126/125, 99/98, 81/80]

wedgie   [1, 4, 10, 18, 4, 13, 25, 12, 28, 16]

map   [[0, -1, -4, -10, -18], [1, 2, 4, 7, 11]]

generators   [502.9994276, 1200.]

bad   313.6023849   rms   6.584357338   g   10.15592719


[121/120, 99/98, 81/80]

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

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

generators   [425.8501579, 1200.]

bad   218.9540099   rms   6.965622568   g   7.914724072



41: [100/99, 225/224, 243/242, 385/384]


[385/384, 243/242, 225/224]

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

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

generators   [116.6722644, 1200.]

bad   125.5016755   rms   1.901465778   g   12.35198075


[385/384, 243/242, 100/99]

wedgie   [4, 9, -15, 10, 5, -35, 2, -60, -8, 80]

map   [[0, 4, 9, -15, 10], [1, 1, 1, 5, 2]]

generators   [175.4823391, 1200.]

bad   394.7928774   rms   5.007159926   g   13.74253044


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

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

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

generators   [380.7138125, 1200.]

bad   242.7224832   rms   4.730404304   g   10.62006188


[243/242, 225/224, 100/99]

wedgie   [4, 9, 26, 10, 5, 30, 2, 35, -8, -62]

map   [[0, 4, 9, 26, 10], [1, 1, 1, -1, 2]]

generators   [175.7323939, 1200.]

bad   362.5442426   rms   5.060204371   g   12.97525116



46: [121/120, 126/125, 176/175, 245/243]


[176/175, 126/125, 245/243]

wedgie   [7, 9, 13, 31, -2, 1, 25, 5, 41, 42]

map   [[0, 7, 9, 13, 31], [1, -1, -1, -2, -8]]

generators   [443.6497958, 1200.]

bad   461.2544269   rms   4.975617526   g   15.14454169


[176/175, 126/125, 121/120]

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

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

generators   [77.93200627, 1200.]

bad   223.3668950   rms   4.418576095   g   10.52547929


[176/175, 245/243, 121/120]

wedgie   [4, -8, 14, -2, -22, 11, -17, 55, 23, -54]

map   [[0, -2, 4, -7, 1], [2, 5, 1, 12, 6]]

generators   [547.4864711, 600.]

bad   300.7076870   rms   5.273952598   g   11.31370850


[126/125, 245/243, 121/120]

wedgie   [14, 18, 26, 16, -4, 2, -23, 10, -25, -45]

map   [[0, 7, 9, 13, 8], [2, -2, -2, -4, 1]]

generators   [443.4105637, 600.]

bad   464.7137988   rms   5.592402087   g   14.18248417



58: [126/125, 176/175, 243/242, 896/891]


[243/242, 896/891, 176/175]

wedgie   [4, -8, 26, 10, -22, 30, 2, 83, 51, -62]

map   [[0, -2, 4, -13, -5], [2, 4, 3, 11, 9]]

generators   [248.2547452, 600.]

bad   458.2298129   rms   4.200153836   g   16.69901622


[243/242, 896/891, 126/125]

wedgie   [6, 17, 39, 15, 13, 45, 3, 43, -24, -93]

map   [[0, 6, 17, 39, 15], [1, -1, -5, -14, -3]]

generators   [517.1519478, 1200.]

bad   541.0798888   rms   3.824357079   g   19.51739151


[243/242, 176/175, 126/125]

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

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

generators   [310.1470775, 1200.]

bad   240.3020098   rms   3.316530343   g   13.06303399


[896/891, 176/175, 126/125]

wedgie   [2, -4, -16, -24, -11, -31, -45, -26, -42, -12]

map   [[0, -1, 2, 8, 12], [2, 4, 3, -1, -3]]

generators   [496.2164639, 600.]

bad   336.4543492   rms   3.182069339   g   16.38814903



72: [225/224, 243/242, 385/384, 4000/3993]


[4000/3993, 385/384, 243/242]

wedgie   [6, 17, -26, 15, 13, -58, 3, -108, -24, 132]

map   [[0, 6, 17, -26, 15], [1, -1, -5, 14, -3]]

generators   [516.6837803, 1200.]

bad   353.9145617   rms   1.860428748   g   23.31155446


[4000/3993, 385/384, 225/224]

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

map   [[0, -6, 1, -10, 3], [2, 7, 4, 12, 5]]

generators   [383.2159771, 600.]

bad   195.0280472   rms   1.584514409   g   17.95231779


[4000/3993, 243/242, 225/224]

wedgie   [6, 17, 46, 15, 13, 56, 3, 59, -24, -117]

map   [[0, 6, 17, 46, 15], [1, -1, -5, -17, -3]]

generators   [516.6897696, 1200.]

bad   347.1579268   rms   1.824911029   g   23.31155446


[385/384, 243/242, 225/224]

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

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

generators   [116.6722644, 1200.]

bad   125.5016755   rms   1.901465778   g   12.35198075


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Message: 3883 - Contents - Hide Contents

Date: Fri, 15 Feb 2002 22:33:50

Subject: Re: Proposed definitions--convex, block

From: genewardsmith

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

> when you say "block", do you mean "periodicity-block"?
Right--you can prove convexity entails a regular covering, by the way.
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Message: 3884 - Contents - Hide Contents

Date: Fri, 15 Feb 2002 23:02:50

Subject: Re: Some 11-limit bases and associated temperament bases

From: paulerlich

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> Here are 11-limit bases for 22,31,41,46,58, and 72. Associated to
these are four linear temperaments, which can also be regarded as forming a basis for the et; these are obtained by leaving out one of the commas. These are by no means necessarily the four best linear temperaments for that et, but they are interesting and do define it.
> > In the case of the 22-et, we get two versions of pajara; these are
the same for 22, but the [2,-4,-4,-12,-11,-12,-26,2,-14,-20] version is consistent with h10-v11, h12, h34+v7, while the
> [2,-4,-4,10,-11,-12,9,2,37,42] version is consistent with h10, > h12-v11, and h34+v7-v11.
you're saying there are two ways of extending pajara to the 11-limit, right?
>The temperament consisting of two chains of 9/7 a half-ocatave apart >still looks good for the 22-et, and I wonder if Paul has ever had >occasion to try it.
it's on my list, for when i get together with ara (of pajara) -- i'd like to eventually try out all the good 'linear temperaments' in 22 and 31. 11-limit harmony, at least otonal with lots of notes, works great in these equal temperaments.
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Message: 3885 - Contents - Hide Contents

Date: Fri, 15 Feb 2002 23:06:14

Subject: Re: Proposed definitions--convex, block

From: paulerlich

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >
>> when you say "block", do you mean "periodicity-block"? > > Right
in other words, gene is proposing a strict limitation on how far he'll allow you to transpose notes of a fokker periodicity block by its unison vectors, before the periodicity block ceases to look 'blocky' anymore. i believe gene also has a related definition of 'semiblock', which would include things like my pentachordal decatonics. right, gene?
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Message: 3886 - Contents - Hide Contents

Date: Fri, 15 Feb 2002 15:48:55

Subject: Re: Proposed definitions--convex, block

From: monz

> From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Friday, February 15, 2002 3:06 PM > Subject: [tuning-math] Re: Proposed definitions--convex, block > > > --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
>> --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>
>>> when you say "block", do you mean "periodicity-block"? >> >> Right >
> in other words, gene is proposing a strict limitation on how far > he'll allow you to transpose notes of a fokker periodicity block by > its unison vectors, before the periodicity block ceases to > look 'blocky' anymore. i believe gene also has a related definition > of 'semiblock', which would include things like my pentachordal > decatonics. right, gene?
cool -- thanks for the explanation, paul. before we get into semiblocks, can i have a few examples of "convex" (and nonconvex?) for the Dictionary entries? thanks. Internet Express - Quality, Affordable Dial Up... * [with cont.] (Wayb.) -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
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Message: 3887 - Contents - Hide Contents

Date: Sat, 16 Feb 2002 21:54:26

Subject: Polyhexes and scales

From: genewardsmith

Here is a web page on polyhexes, which are best understood by looking at them:

Polyhexes * [with cont.]  (Wayb.)

Polyhexes
are relevant to 5-limit connected scales, since if we surround each
note-class on the lattice with a hexagon, we get a polyhex. Since
tempered scales are often images under a mapping of 
5-limit just scales, this is also relevant to large numbers of
tempered scales, in particular those associated to a planar
temperament.

Since a scale has an orientation (pointing along the 3/2 axis, the 5/3
axis, or the 8/5 axis being distinct), there are even more of them
than there are polyhexes. On the other hand, if we add conditions,
such as being epimorphic, we cut that down again. If we take hn for
some n in the 5-limit, we can color the lattice with n different
colors. An n-polyhex lying in such a way that it covers all of the
colors corresponds to an epimorphic and connected scale. It might be
interesting to see what these colored lattices look like, if Joe or
Paul want to take a shot at it.

The number of polyhexes for each n for the first few n are

1: 1
2: 1
3: 3
4: 7
5: 22
6: 82
7: 333
8: 1448
9: 6572
10: 30490
11: 143552

This is according to information on the page

AOL Hometown * [with cont.]  (Wayb.)


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Message: 3888 - Contents - Hide Contents

Date: Sat, 16 Feb 2002 00:13:27

Subject: hi monz

From: paulerlich

hi monz -- check the main tuning list. i made a better graph for your 
et page.

the new graph suggests a line through 27, 61, 34, 75, 41, and 48 that 
we missed out on before. i'm sure gene or graham could quickly tell 
us the comma that this corresponds to . . .


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Message: 3889 - Contents - Hide Contents

Date: Sat, 16 Feb 2002 01:03:59

Subject: Re: hi monz

From: genewardsmith

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> the new graph suggests a line through 27, 61, 34, 75, 41, and 48 that > we missed out on before. i'm sure gene or graham could quickly tell > us the comma that this corresponds to . . .
20000/19683, the minimal diesis.
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Message: 3890 - Contents - Hide Contents

Date: Sat, 16 Feb 2002 01:23:49

Subject: Re: Some 11-limit bases and associated temperament bases

From: genewardsmith

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> you're saying there are two ways of extending pajara to the 11-limit, > right?
I'm saying there are two ways, which are the same for 22-et but different for 34-et.
> it's on my list, for when i get together with ara (of pajara) --
I thought pajara was named for Ethiopian bread. Of course, there's always the possibility it's named for Eliseo Pajaro. :)
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Message: 3891 - Contents - Hide Contents

Date: Sat, 16 Feb 2002 01:28:43

Subject: Re: Proposed definitions--convex, block

From: genewardsmith

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> in other words, gene is proposing a strict limitation on how far > he'll allow you to transpose notes of a fokker periodicity block by > its unison vectors, before the periodicity block ceases to > look 'blocky' anymore.
Actually, I'm not. I'm simply saying the blocky shape can't have concavities. i believe gene also has a related definition
> of 'semiblock', which would include things like my pentachordal > decatonics. right, gene?
Except that under this definition, which allows a block to be something much more general than a Fokker block, the pentachordal decatonics could easily be blocks--Genesis Minus turned out to be, so why not pentachordal decatonics? I could calculate the convex hull and check.
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Message: 3892 - Contents - Hide Contents

Date: Sun, 17 Feb 2002 22:19:44

Subject: Re: Some 11-limit bases and associated temperament bases

From: paulerlich

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> 
wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: >
>> you're saying there are two ways of extending pajara to the 11-limit, >> right? >
> I'm saying there are two ways, which are the same for 22-et but
different for 34-et.
>
>> it's on my list, for when i get together with ara (of pajara) -- >
> I thought pajara was named for Ethiopian bread.
no, that's injera, my name for the {81:80, 50:49} linear temperament . . .
>Of course, there's always the possibility it's named for Eliseo >Pajaro
don't know who that is, but pajaro is spanish for bird . . .
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Message: 3893 - Contents - Hide Contents

Date: Sun, 17 Feb 2002 10:57:46

Subject: Re: twintone, paultone

From: clumma

I like to re-visit this:

>>> The "problem" occurs when modulating from the best approx. of >>> one chord to the best approx. of another, and thereby creating >>> anomalous (as in, non-existent in JI) commas. >>
>> That won't happen if you confine yourself to a regular >> temperamemt /.../
I don't understand. Let's call 24-et a linear temperament of a chain of 24 near 7:4's (19 steps each). Now play the following: 5:7:8 -> 7:8:10 Ignoring consistency and using the lowest-rms approximations, that will be: 0,12,16 -> 12,17,24 The comma from 16 to 17 steps doesn't exist in JI. So a chain of 950-cent intervals must not be a "regular" temperament. Why? -Carl
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Message: 3894 - Contents - Hide Contents

Date: Sun, 17 Feb 2002 22:23:34

Subject: Re: twintone, paultone

From: genewardsmith

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

> I don't understand. Let's call 24-et a linear temperament of a > chain of 24 near 7:4's (19 steps each). Now play the following:
OK. You are now looking at the linear temperament "Lumma", with MT reduced basis <49/48, 81/80> and wedgie [2,8,1,-20,4,8]. If you care about the thirds, you probably want to do this in 19-et; otherwise 24-et is a good choice.
> 5:7:8 -> 7:8:10
"Lumma" maps this to [8,-4]:[1,2]:[0,3] -> [1,2]:[0,3]:[8,-3]
> Ignoring consistency and using the lowest-rms approximations, that > will be:
You're not allowed to do this, since you are using the regular temperament Lumma and must use what it gives you.
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Message: 3895 - Contents - Hide Contents

Date: Sun, 17 Feb 2002 22:23:23

Subject: Re: twintone, paultone

From: paulerlich

--- In tuning-math@y..., "clumma" <carl@l...> wrote:
> I like to re-visit this: >
>>>> The "problem" occurs when modulating from the best approx. of >>>> one chord to the best approx. of another, and thereby creating >>>> anomalous (as in, non-existent in JI) commas. >>>
>>> That won't happen if you confine yourself to a regular >>> temperamemt /.../ >
> I don't understand. Let's call 24-et a linear temperament of a > chain of 24 near 7:4's (19 steps each). Now play the following: > > 5:7:8 -> 7:8:10 > > Ignoring consistency and using the lowest-rms approximations, that > will be: > > 0,12,16 -> 12,17,24 > > The comma from 16 to 17 steps doesn't exist in JI. So a chain of > 950-cent intervals must not be a "regular" temperament. Why?
a regular temperament includes, in its specification, the interval that each ji consonance is mapped to.
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Message: 3896 - Contents - Hide Contents

Date: Sun, 17 Feb 2002 22:58:08

Subject: Re: twintone, paultone

From: Carl Lumma

>> >he anomalous comma problem wouldn't be grounds for this. But >> the fact that consistency serves as a badness measure might be. >
>I still don't know what the problem is. If you go to a high enough n, >any n-et will have more than one good mapping. Is that still a problem?
It's not a problem unless you're counting on the best rms approximations a given et has to offer. All consistency means at high n is that the best approximations all in the et all fall in the same mapping. -Carl
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Message: 3897 - Contents - Hide Contents

Date: Sun, 17 Feb 2002 23:02:13

Subject: Re: Maximal consistent sets of odds for ETs

From: Carl Lumma

>> > think we are astoundingly ignorant of the rational identities >> available in ETs where 3's or 5's are not included.
There's some truth to this.
>> Who can generate a list for all the ETs up to 2000, giving for >> each ET the maximal sets of mutually-consistently-approximated >> odd numbers up to 35? >
>didn't carl write a program to do this or something very much >like it?
Given a list of chords and an et (I could step through all ets up to 2000, but I think 282 would be plenty enough), I can return: () The portion of chords in the input list that are not consistently represented (I can also return the actual failing or passing chords, but these are often abundant) in the et. () The identities which are present in any of the input chords but in none of the passing chords. () The identities which are present in any of the input chords and in all of the failing chords. I don't think any of this gets you just what you want, Dave... I suppose I could add code that would create as the input list all 17-tone subsets of the 35-limit otonality (an 18-ad), and if none of them pass, all 16-note subsets, and so on. In many cases there may be multiple "maximal" sets (sets of the same card that pass). How would you like me to proceed? -Carl
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Message: 3898 - Contents - Hide Contents

Date: Mon, 18 Feb 2002 03:36:18

Subject: Pelog vs Pelogic

From: dkeenanuqnetau

Graham, I think that the entry on your excellent catalog page should 
not be titled "Pelog". It's highly debatable whether this 5-limit 
temperament really has anything to do with the raison detre of Pelog. 
Pelogic is what we've been calling it on this list, although it could 
also be called "Pelogish". Easier to pronounce while keeping the hard 
"g".


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Message: 3899 - Contents - Hide Contents

Date: Mon, 18 Feb 2002 04:14:42

Subject: Maximal consistent sets of odds for ETs

From: dkeenanuqnetau

A similar message was posted to the tuning list:

I think we are astoundingly ignorant of the rational identities 
available in ETs where 3's or 5's are not included.

Who can generate a list for all the ETs up to 2000, giving for each ET 
the maximal sets of mutually-consistently-approximated odd numbers up 
to 35? And then give the reverse-lookup version of that list, where 
each maximal set has the corresponding list of ETs after it.

So for example, if it were limited to odds up to 13,
20-tET would show 1:3:11:13 and 1:3:7:11, but not 1:3:7 (because it's 
included in 1:3:7:11) and not 1:3:7:11:13 (because they are not all 
mutually consistent). 11-tET would show 1:3:11 and 1:7:9:11.

Likewise, in the reverse list, 20-tET would not appear next to 1:3:7 
because it has a larger consistent set containing that. It would 
appear in only two places. 

Remember I'm assuming a limit of 13 for these example and would really 
want to go up to 35.

Sets would only be given in lowest terms, e.g. not 3:9:27 as well as 
1:3:9.


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