Tuning-Math Digests messages 3950 - 3974

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Message: 3950

Date: Fri, 22 Feb 2002 19:55:31

Subject: Re: monz's et graph (from my lumma.gif)

From: paulerlich

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., Carl Lumma <carl@l...> wrote:
> 
> > The generator could be 98 cents... 6/73 gives MOS of 61, 49, 37, 
25,
> > 13, and 12 according to Scala.
> 
> Right--the comma is 262144/253125, and the rms generator 98.317 
cents.

so monz should have a [18 -4 -5] label on the 12-73-61-49-37 line.


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Message: 3951

Date: Fri, 22 Feb 2002 00:58:02

Subject: Re: monz's et graph (from my lumma.gif)

From: Carl Lumma

>He also gives the 5-limit comma for this series as [-4 -5].

And shows a couple of series that don't have lines on monz's
chart:

"""
(3 4) : 28 47 19 48 29
(-2 7) : 26 29 32
"""

-Carl


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Message: 3952

Date: Fri, 22 Feb 2002 19:56:59

Subject: Re: handy text breakdown of monz's chart

From: paulerlich

--- In tuning-math@y..., Carl Lumma <carl@l...> wrote:

> Lines
> ------
>        (no name) 250:243 [-5 3] 29, 22, 59, 37, 15

that's porcupine, as in miller's mizarian porcupine overture.


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Message: 3953

Date: Fri, 22 Feb 2002 01:39:17

Subject: handy text breakdown of monz's chart

From: Carl Lumma

Lines
------
       (no name) 250:243 [-5 3] 29, 22, 59, 37, 15
  diaschismic 2048:2025 [-4 -2] 22, 78, 56, 90, 34, 80, 46, 58, 70, 12
            (no name) ? [-8 -7] 59, 71, 83, 95, 12
(no name) 262144:253125 [-4 -5] 25, 37, 49, 61, 73, 12
          diesic 128:125 [0 -3] 15, 42, 27, 39, 12
     schismic 32805:32768 [8 1] 29, 41, 94, 53, 65, 77, 89, 12
         magic 3125:3072 [-1 5] 22, 63, 41, 60, 79, 19, 35
              kleismic ? [-5 6] 15, 49, 83, 34, 87, 53, 72, 91, 19, 23
             (no name) ? [-9 4] 48, 41, 75, 34, 95, 61, 27
             (no name) ? [5 -9] 47, 31, 77, 46, 61, 15
             (no name) ? [8 -5] 23, 35, 47, 12
           wuerschmidt ? [1 -8] 28, 31, 96, 65, 99, 34, 71, 37
               orwell ? [-8 -7] 31, 84, 53, 75, 97, 22
             (no name) ? [4 -4] 28, 40, 52, 64, 12
              (no name) ? [3 4] 29, 48, 19, 47, 28

Intersections (by eye)
----------------------
12 - 8
31 - 5
22, 37, 15, 34, 19 - 4
29, 41, 75, 61, 53, 72, 23 - 3

-Carl


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Message: 3954

Date: Fri, 22 Feb 2002 12:06:50

Subject: Re: handy text breakdown of monz's chart

From: monz

paul, i think your "favorite page on the internet"
just got even better!  see my latest post to the tuning list.



-monz


 



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Message: 3955

Date: Fri, 22 Feb 2002 10:05:46

Subject: Re: handy text breakdown of monz's chart

From: genewardsmith

--- In tuning-math@y..., Carl Lumma <carl@l...> wrote:
> Lines
> ------
>        (no name) 250:243 [-5 3] 29, 22, 59, 37, 15

The comma is the maximal diesis, but Paul threated to die sick if we used all these diesis names at once for temperaments.


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Message: 3956

Date: Fri, 22 Feb 2002 21:02:34

Subject: spin cycle

From: jpehrson2

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

Yahoo groups: /tuning-math/message/3366 *


> Chapt. One, IV.4. Group Theory *


***Well, this was an interesting read, but I'm dubious at best.  In 
order to be convinced, I'd like to see more Mozart examples 
on "regular staff notation" like a "real" music theory article.  It's 
quite possible that the examples were selected which just *happened* 
to prove the "formula."

I'd need to see vastly more instances of this so-called 
Mozart "formula" before I'd believe it really exists and that Mozart 
used it to generate his works...

Besides, due to the nature of the way themes are *normally* generated 
in "common practice" Western music, with small units of 2, 4, 
expanding to 8, 16, etc., almost *all* themes could be expressed by 
some kind of common "compound," yes?  It's saying more about the 
generation of themes in common practice (i.e. dead white male) 
Western music than anything else, so it seems.

Regarding the Beethoven, that seems even *more* specious.  I don't 
know much about "Group Theory" but I can guess that it's expansive 
enough that you could included *just about anything* if you angle it 
the right way.

This seems like a "spin cycle" worthy of the greatest of washing 
machines...

JP


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Message: 3957

Date: Fri, 22 Feb 2002 12:24 +0

Subject: Re: comments sought

From: graham@xxxxxxxxxx.xx.xx

In-Reply-To: <003501c1bb52$449655e0$af48620c@xxx.xxx.xxx>
monz wrote:

> in my MIDI rendition of the beginning of the piece in 55edo
> Mozart's tuning: 55-EDO,  (c) 2001 by Joseph L. Monzo *
> i was careful to tune the sharps and flats differently to
> reflect Mozart's notation, which had to be done by hand
> because none of the programs i know of (Manuel's Scala,
> Graham's Midiconv, John deLaubenfels adaptune) can retune
> to more than 12 tones per octave.

What do you mean?  Midiconv doesn't have any 12-fetishism!  I don't think 
Scale does either.


                     Graham


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Message: 3958

Date: Fri, 22 Feb 2002 14:20:50

Subject: Re: comments sought

From: manuel.op.de.coul@xxxxxxxxxxx.xxx

>What do you mean?  Midiconv doesn't have any 12-fetishism! 
>I don't think Scala does either.

They don't, but Joe means the MIDI 12-fetishism. If there's
a F# and a Gb in the score, we aren't able to guess which
because they have the same note number in the MIDI-file.
Still doing _all_ notes by hand what Joe does is a waste
of time.

Manuel


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Message: 3959

Date: Fri, 22 Feb 2002 01:41:32

Subject: Re: magma

From: paulerlich

--- In tuning-math@y..., Carl Lumma <carl@l...> wrote:
> 
> >carl, why won't you answer us on the tuning list? we're asking 
about 
> >the cd you made for me of various a capella groups. could you 
discuss 
> >them please on tuning?
> 
> Gee, I thought I did respond... here it is: 34636.

i actually posted that message a long time ago -- look at the date in 
that message. the list is blowing its nose again.

> BTW Gene, those ad blocking services work by filtering all your
> http traffic through their server.

are you serious?? someone tell me this isn't so.


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Message: 3961

Date: Tue, 26 Feb 2002 23:25:37

Subject: Re: Past Paul Post

From: paulerlich

--- In tuning-math@y..., "Orphon Soul, Inc." <tuning@o...> wrote:
> Mr Erlich... Sometime in the last week or so you asked me something 
about
> the ³cradling² technique Iıd coined, in relation to something you 
do... Iım
> sorry Iım lost here in catching up posts I might have even deleted 
it by
> mistake but I havenıt been able to find it and I donıt remember 
what list it
> was even on.
> 
> Might you reask, sir?
> 
> mj

i have no clue.


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Message: 3962

Date: Fri, 01 Mar 2002 05:27:32

Subject: blast from the past re: microtempering partch

From: paulerlich

sent from me to dave keenan Thu 1/13/00 5:26 PM . . .

You wrote,

>In earlier messages I gave the following simultaneous distribution 
of the
>224:225 and 384:385 commas or "unison vectors".

Just for fun, here are the numbers of notes given by various pairs of 
11-limit subdiaschismic unison vectors, keeping 224:225 and 384:385 
as the other two unison vectors:

98:99, 99:100 -- 22 notes
98:99, 125:126 -- 31 notes
98:99, 242:243 -- 31 notes
98:99, 243:245 -- 22 notes
98:99, 440:441 -- 31 notes
98:99, 891:896 -- 22 notes
98:99, 1024:1029 -- 31 notes
98:99, 1323:1331 -- 31 notes
98:99, 2400:2401 -- 31 notes
98:99, 3024:3025 -- 31 notes
98:99, 3993:4000 -- 22 notes
98:99, 4374:4375 -- 53 notes
98:99, 9800:9801 -- 22 notes
99:100, 120:121 -- 22 notes
99:100, 125:126 -- 19 notes
99:100, 175:176 -- 22 notes
99:100, 242:243 -- 41 notes
99:100, 440:441 -- 41 notes
99:100, 1024:1029 -- 41 notes
99:100, 1323:1331 -- 63 notes
99:100, 2400:2401 -- 41 notes
99:100, 3024:3025 -- 41 notes
99:100, 3993:4000 -- 22 notes
99:100, 4374:4375 -- 19 notes
99:100, 5625:5632 -- 22 notes
99:100, 9800:9801 -- 22 notes
120:121, 125:126 -- 31 notes
120:121, 242:243 -- 31 notes
120:121, 243:245 -- 22 notes
120:121, 440:441 -- 31 notes
120:121, 891:896 -- 22 notes
120:121, 1024:1029 -- 31 notes
120:121, 1323:1331 -- 31 notes
120:121, 2400:2401 -- 31 notes
120:121, 3024:3025 -- 31 notes
120:121, 3993:4000 -- 22 notes
120:121, 4374:4375 -- 53 notes
120:121, 9800:9801 -- 22 notes
125:126, 175:176 -- 31 notes
125:126, 242:243 -- 31 notes
125:126, 243:245 -- 19 notes
125:126, 441:440 -- 31 notes
125:126, 891:896 -- 19 notes
125:126, 1024:1029 -- 31 notes
125:126, 1323:1331 -- 62 notes
125:126. 2400:2401 -- 31 notes
125:126, 3024:3025 -- 31 notes
125:126, 3993:4000 -- 50 notes
125:126, 4374:4375 -- 19 notes
125:126, 5625:5632 -- 31 notes
125:126, 9800:9801 -- 50 notes
175:176, 243:245 -- 22 notes
175:176, 891:896 -- 22 notes
175:176, 1024:1029 -- 31 notes
175:176, 1323:1331 -- 31 notes
175:176, 2400:2401 -- 31 notes
175:176, 3024:3025 -- 31 notes
175:176, 3993:4000 -- 22 notes
175:176, 9800:9801 -- 22 notes
242:243, 243:245 -- 41 notes
242:243, 891:896 -- 41 notes
242:243, 1323:1331 -- 31 notes
242:243, 3993:4000 -- 72 notes
242:243, 4374:4375 -- 72 notes
242:243, 5625:5632 -- 31 notes
242:243, 9800:9801 -- 72 notes
243:245, 440:441 -- 41 notes
243:245, 1024:1029 -- 41 notes
243:245, 1323:1331 -- 63 notes
243:245, 2400:2401 -- 41 notes
243:245, 3024:3025 -- 41 notes
243:245, 3993:4000 -- 22 notes
243:245, 4374:4375 -- 19 notes
243:245, 5625:5632 -- 22 notes
243:243, 9800:9801 -- 72 notes
440:441, 891:896 -- 41 notes
440:441, 1323:1331 -- 31 notes
440:441, 3993:4000 -- 72 notes
440:441, 4374:4375 -- 72 notes
440:441, 5625:5632 -- 31 notes
440:441, 9800:9801 -- 72 notes
891:896, 1024:1029 -- 41 notes
891:896, 1323:1331 -- 63 notes
891:896, 2400:2401 -- 41 notes
891:896, 3024:3025 -- 41 notes
891:896, 3993:4000 -- 22 notes
891:896, 4374:4375 -- 19 notes
891:896, 5625:5632 -- 22 notes
891:896, 9800:9801 -- 22 notes
1024:1029, 1323:1331 -- 31 notes
1024:1029, 3993:4000 -- 72 notes
1024:1029, 4374:4375 -- 72 notes
1024:1029, 5625:5632 -- 31 notes
1024:1029, 9800:9801 -- 72 notes
1323:1331, 2400:2401 -- 31 notes
1323:1331, 3024:3025 -- 31 notes
1323:1331, 3993:4000 -- 94 notes
1323:1331, 4374:4375 -- 125 notes
1323:1331, 5625:5632 -- 31 notes
1323:1331, 9800:9801 -- 94 notes
2400:2401, 3993:4000 -- 72 notes
2400:2401, 4374:4375 -- 72 notes
2400:2401, 5625:5632 -- 31 notes
2400:2401, 9801:9800 -- 72 notes
3024:3025, 3993:4000 -- 72 notes
3024:3025, 4374:4375 -- 72 notes
3024:3025, 5625:5632 -- 31 notes
3024:3025, 9800:9801 -- 72 notes
3993:4000, 4374:4375 -- 72 notes
3993:4000, 5625:5632 -- 22 notes
4374:4375, 5625:5632 -- 53 notes
4374:4375, 9800:9801 -- 72 notes
5625:5632, 9800:9801 -- 22 notes


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Message: 3963

Date: Fri, 01 Mar 2002 09:36:42

Subject: Re: blast from the past re: microtempering partch

From: paulerlich

we haven't heard much about these 63-tone systems -- any comments on 
them, gene?


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Message: 3964

Date: Fri, 01 Mar 2002 08:24:32

Subject: Re: blast from the past re: microtempering partch

From: genewardsmith

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

> Just for fun, here are the numbers of notes given by various pairs of 
> 11-limit subdiaschismic unison vectors, keeping 224:225 and 384:385 
> as the other two unison vectors:

Where did this list of 11-limit commas come from?


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Message: 3965

Date: Fri, 01 Mar 2002 10:30:05

Subject: Re: blast from the past re: microtempering partch

From: genewardsmith

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

> we haven't heard much about these 63-tone systems -- any comments on 
> them,
gene?

If you wedge together 100/99^225/224^385/384 or 100/99^225/224^540/539
or 100/99^385/384^540/539 you get the 
11-limit version of magic, but there doesn't seem to be any reason to
prefer the 63 version over the 41. However, it extends to a different
and more accurate 13-limit version of magic in the 63 form.


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Message: 3966

Date: Fri, 01 Mar 2002 08:54:44

Subject: Re: blast from the past re: microtempering partch

From: paulerlich

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> 
> > Just for fun, here are the numbers of notes given by various 
pairs of 
> > 11-limit subdiaschismic unison vectors, keeping 224:225 and 
384:385 
> > as the other two unison vectors:
> 
> Where did this list of 11-limit commas come from?

i'll be darned if i remember. any glaring omissions?


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Message: 3967

Date: Fri, 01 Mar 2002 10:47:15

Subject: Re: blast from the past re: microtempering partch

From: genewardsmith

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

I do get something other than magic for some combinations; for
instance 100/99^225/224^1331/1323 gives a temperament with
diesis-sized steps: 1/31, 2/63, 3/94 or 4/125.


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Message: 3968

Date: Fri, 01 Mar 2002 10:48:57

Subject: Re: blast from the past re: microtempering partch

From: paulerlich

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> 
> > we haven't heard much about these 63-tone systems -- any comments 
on 
> > them, gene?
> 
> If you wedge together 100/99^225/224^385/384 or 
100/99^225/224^540/539 or 100/99^385/384^540/539 you get the 
> 11-limit version of magic, but there doesn't seem to be any reason 
to prefer the 63 version over the 41. However, it extends to a 
different and more accurate 13-limit version of magic in the 63 form.

cool!

here's one of the 63-tone fokker periodicity blocks:


        cents        numerator   denominator
            0            1            1
       14.367          121          120
       38.906           45           44
       53.273           33           32
       80.537           22           21
       84.467           21           20
       119.44           15           14
       123.37          189          176
       150.64           12           11
          165           11           10
       189.54          135          121
       203.91            9            8
        235.1           63           55
       235.68           55           48 
       274.58           75           64
       284.45           33           28
       305.78          105           88
       315.64            6            5
       347.41           11            9
       354.55           27           22
       386.31            5            4
       396.18           44           35
       417.51           14           11
       435.08            9            7
       466.85           55           42
       470.78           21           16
       498.04            4            3
       509.69          945          704
       536.95           15           11
       551.32           11            8
       568.15          168          121
       590.22           45           32
       617.49           10            7
       621.42           63           44
       648.68           16           11
       670.76          165          112
       687.59          180          121
       701.96            3            2
       716.32          121           80
       733.15           84           55
       755.23           99           64
       782.49           11            7
       786.42           63           40
       813.69            8            5
       835.76          363          224
       852.59           18           11
       866.96           33           20
       894.22          176          105
       905.87           27           16
       933.13           12            7
       937.06          189          110
       968.83            7            4
        986.4           99           56
       1007.7          315          176
       1017.6            9            5
       1049.4           11            6
       1056.5           81           44
       1088.3           15            8
       1098.1           66           35
       1119.5           21           11
       1129.3           48           25
       1168.2          108           55
       1168.8           55           28

it looks pretty 'fishy' . . .


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Message: 3969

Date: Sun, 03 Mar 2002 14:00:34

Subject: maps, uvs

From: Carl Lumma

Can someone post a general method for transforming a
list of unison vectors into a map and vice versa?

-Carl


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Message: 3970

Date: Sun, 3 Mar 2002 22:36 +00

Subject: Re: maps, uvs

From: graham@xxxxxxxxxx.xx.xx

Carl Lumma wrote:

> Can someone post a general method for transforming a
> list of unison vectors into a map and vice versa?

See

<Automatically generated temperaments *>

and work out the code.  I can see the value in a longer explanation of 
that, but I haven't done it and certainly won't at this time of night.

The basic method is to take the wedge product of the vectors, and the 
octave-equivalent part is the mapping by generator.  Wedge it with some 
chromatic unison vector and you get an example ET mapping.  You can 
combine the two to get either kind of mapping I print out, but you'll have 
to check the code to see how.

There's also <Unison vector to MOS script *> that uses matrix 
operations instead of wedge products.  There, you put the octave at the 
top, the chromatic UV second, and the commatic UVs below in a matrix.  
Take the adjoint, and the left hand column is your example ET and the next 
column is the mapping by generator.  They're both mathematically 
equivalent to the same things you get from wedge products.


                      Graham


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Message: 3971

Date: Sun, 03 Mar 2002 22:38:27

Subject: Re: maps, uvs

From: genewardsmith

--- In tuning-math@y..., Carl Lumma <carl@l...> wrote:

> Can someone post a general method for transforming a
> list
of unison vectors into a map and vice versa?

(1) Wedge the unisons together to get a wedgie

(2) If you have a linear temperament wedgie, wedge this with 2 to
get the map to steps of the generator, and then solve the linear
equations to get corresponding octaves for the generator map

(3) If you aren't dealing with a linear temperament, you need to find
a basis for the generators--wedging with the elements of this basis
will give the map

(4) I've started writing up a paper on the mathematics of temperament,
so I should have this explained in detail. When it's ready I'd like
some comments on it!


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Message: 3972

Date: Sun, 3 Mar 2002 16:34:23

Subject: Re: maps, uvs

From: monz

hi Gene and Carl,


> From: Carl Lumma <carl@xxxxx.xxx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Sunday, March 03, 2002 2:00 PM
> Subject: [tuning-math] maps, uvs
>
>
> Can someone post a general method for transforming a
> list of unison vectors into a map and vice versa?
> 
> -Carl



see the second section of:

Tuning Dictionary, "periodicity block"
Definitions of tuning terms: periodicity block, (c) 1998 by Joe Monzo *


where i quote Gene's method of finding a notation
which maps to JI pitches.  (BTW, there were errors
in this originally ... did i get them all out, Gene?)


to find the mapping to EDOs, put the unison-vectors
in vector form into a matrix, then calculate the
determinant and the inverse.  if the inverse is
unimodular (= has a determinant = 1), then it gives
the mapping to EDOs, the cardinality of which (i.e.,
mapping of prime-factor 2) is in the top row.  see:

Tuning Dictionary, "matrix"
Internet Express - Quality, Affordable Dial Up, DSL, T-1, Domain Hosting, Dedicated Servers and Colocation *



-monz



 



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Message: 3973

Date: Sun, 3 Mar 2002 17:05:49

Subject: Re: maps, uvs

From: monz

> From: monz <joemonz@xxxxx.xxx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Sunday, March 03, 2002 4:34 PM
> Subject: Re: [tuning-math] maps, uvs
>
>
> to find the mapping to EDOs, put the unison-vectors
> in vector form into a matrix, then calculate the
> determinant and the inverse.  if the inverse is
> unimodular (= has a determinant = 1), then it gives
> the mapping to EDOs, the cardinality of which (i.e.,
> mapping of prime-factor 2) is in the top row.  see:
> 
> Tuning Dictionary, "matrix"
> Internet Express - Quality, Affordable Dial Up, DSL, T-1, Domain Hosting, Dedicated Servers and Colocation *


oops ... the matrix is unimodular if the determinant
is +1 or -1.



-monz



 



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Get your free @yahoo.com address at Yahoo! Mail Setup *


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Message: 3974

Date: Sun, 03 Mar 2002 17:15:09

Subject: listing linear temperaments

From: Carl Lumma

//This is long.  May I humbly suggest we do it up old-school, like
//in the Classic onelist years, and reply to everything until we
//agree on everything?  Let's get something export-quality!  Dave
//Keenan, activate your magic power ring, Voltron is needed once
//again!  Monz, break out the colored chalks!  Paul, I totally
//understand you wanting to take a break, and I've always been
//behind a book from you, but why not finish the paper on linear
//temperamenst first?

A paper.  I think it's a great idea.  And, the 569 of us who don't
have a computer set up to do calculations on linear temperaments
need a list!

Graham's catalog, "The grooviest 7-limit temperaments", Monzo's lines,
and Herman Miller's "Carl's favorite page on the internet" Warped
Canons page are huge, huge, huge.  But wouldn't it be cool to really
get the goat?


---------------------------
Selection criteria
(1) Badness
---------------------------


~~~~~~~~~~~~~~~~
(1a) Gene's list
~~~~~~~~~~~~~~~~

Paul wrote...
>Gene, who's way, way ahead of any other theorist on this list (and
>possibly anywhere) has (like Dave Keenan and Graham Breed before
>him) completed a comprehensive search for linear temperaments for
>7-limit music. He proposed a 'badness' measure defined as:
>
>step^3 cent
>
>where step is a measure of the typical number of notes in a scale
>for this temperament (given any desired degree of harmonic depth),
>and cent is a measure of the deviation from JI 'consonances' in
>cents.

Is this still state-of-the-art-badness?  I seem to remember
something about different exponents for each prime/odd (?) identity,
taken from coefficients of Diophantine equations, or some such?
Don't need details, just want to know if we need a new top 20.

Paul wrote...
>He then ranked his 505 temperaments by 'goodness'. The familiar
>ones don't come in until later, so bear with me . . .

Gene, you initially stopped after listing 20.  Did you ever list
the requested next few needed to uncover meantone?  Are you still
happy with your list?

~~~~~~~~~~~~~~~~~~~~~
(1b) The slippery six
~~~~~~~~~~~~~~~~~~~~~

I wasn't reading the tuning-math very closely back then, but Gene,
your top 20 is generated by starting with some large number of
ets and then seeing what temperaments they share, sort of like a
more precise version of looking for lines on Herman Miller's / Paul's
charts, right?  But you found that some temperaments only hit a
single et up to your cutoff -- those were the slippery six, right?
Do we have a general solution to this problem -- making the cutoff
really high, an entirely different method, etc.?

Speaking of this method, nobody ever answered this:

Carl wrote...
>More to the point, every line on this plane is a linear temperament,
>right?  So what makes low-numbered (less than 100) equal
>temperaments cluster on some of them?

What makes some linear temperaments belong to more than one et, out
of ets as high as some given number?  They would have to share a
common generator...  Is sharing a common generator related to the
un-even distribution of the rationals on the number line (such as
makes harmonic entropy work)?

Carl wrote...
>Intersections (by eye)
>----------------------
>12 - 8
>31 - 5
>22, 37, 15, 34, 19 - 4
>29, 41, 75, 61, 53, 72, 23 - 3

Why are some of the 'best' ets (ones that have gotten so much
attention on these lists for so many different reasons, for
so long) here?  Is it because we've often defined "best" as
"consistent", and where two lines cross the same tuning is being
reached two different ways (via two different maps), which
requires consistency?


---------------------------
Selection criteria
(2) Maps and commas
--------------------------


A map uniquely defines a linear temperament?  Or do you also
need period?  Looking at Graham's catalog, I'm not sure how to
use maps with non-octave periods.

Carl wrote...
>I say the most powerful maps are the ones with the smallest
>numbers in them.  Sum of abs value would work.

Or, maybe the sum of the abs values of the max and min numbers in
the map, for a given limit (or divided by the card of the map, if
you want to compare across limits).  Which is better?

There's definitely some overlap with badness here, but by not
considering the quality of the approximations, doesn't this tell
us more about the abstract musical-theoretic properties of a
temperament?

Carl wrote...
>Finally, re the jumping jacks / ideal comma question... what's the
>question?  How are we defining "most powerful" comma?

?

Carl wrote...
>What's the relationship between a comma vanishing and a map?

?


-------------------------
The contents of the list
-------------------------

Paul wrote...
>>Generators on the table

Carl wrote...
>Yes, I completely agree. Who can furnish rms optimums?

Paul wrote...
>i bet gene can do this in a jiffy. maybe graham too.
>and oh, we need the period as well as the generator.

I completely agree.

Paul wrote...
>actually, gene already did this back in december.

I looked.  My eyes!  The searching did nothing!

Paul wrote...
>i'm making a graph that includes these as well as the ets.
>
>well, i tried to, but the points get too crowded near the
>center for me to label them.

F the graph.  Let's have a list!

Paul wrote...
>but it's easy to see the optimum point on the graph on monz's
>page already. simply look at the line representing the
>temperament you're interested in, and the point on that line
>that comes closest to the center ('origin') of the graph is
>the optimal one. so optimal meantone is near 50-equal, and
>optimal magic (in 5-limit at least) is near 60-equal, etc.

I figured as much.  But what if the nearest et below 100 is
off the optimum some?  Why not do it right?

Paul wrote...
>Topping off Gene's list are some very funky simple temperaments,
/.../
>For these, I quote the simplest pair of unison vectors:
>
> (1) <21/20,27/25>
> (2) <8/7,15/14>
> (3) <9/8,15/14>
> (4) <25/24,49/48>
> (5) <15/14,25/24>
> (6) <21/20,25/24>
> (7) <15/14,35/32>
> (8) <7/6,16/15>
> (9) <16/15,21/20>

Can we get a list with optimum generator, et series, commas, maps,
periods for these (and the rest of the top 20)?  Are any of the
"Monzo's lines" temperaments in here?

-Carl


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