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Message: 6625 - Contents - Hide Contents

Date: Sat, 22 Mar 2003 19:20:20

Subject: Re: Quartaminorthirds and muggles

From: Carl Lumma

> could i put in a request -- when you look at these, try to find 
> omnitetrachordal variants to the basic MOS scales . . .

Paul,


1. Omnitetrachordal just means the 4/3 doesn't have to be
cut in *four*, right?  The symmetry still has to be 3:2,
right?

2. How might one find such variants?

-Carl

Message: 6626 - Contents - Hide Contents

Date: Sat, 22 Mar 2003 19:21:04

Subject: Re: Beatles[17] and squares[17]

From: Carl Lumma

> what can we say about the conditions leading to this kind of
> uniformity of the tetradic materials as a function of "root"
> scale degree number?

Paul,


Could you explain what you're asking here?  I'm not tracking
you.

-C.

Message: 6627 - Contents - Hide Contents

Date: Sat, 22 Mar 2003 23:26:13

Subject: Re: T[n] where n is small

From: Carl Lumma

>> > don't know how many lines of maple you do this with,
>> but if they're few you can post them here and I can
>> either translate to Scheme or run them in maple myself.
>

>I could send you some Maple code I've written for various
>things if you have access to Maple.


Not sure how much of the stuff you do I can grok, but I do
have Maple, so send away!

-Carl

Message: 6628 - Contents - Hide Contents

Date: Sat, 22 Mar 2003 00:30:28

Subject: Blackjack and 36/35

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:


> keep it up, gene! i think one of the more important chromatic unison 
> vectors kalle and carl were interested in was 49/48.


I think 15/14, 21/20 and 49/48 all need to be investigated, but I'm
still sorting things out, as I've just found our old friend torsion
makes the situation for schismic[24] more complicated.

Here is Blackjack, where we see the wolf fifths continue to be the
flys in the ointment in these chroma-36/35 systems:

[6, -7, -2, -25, -20, 15] 

[1, 49/48, 15/14, 35/32, 8/7, 7/6, 49/40, 5/4, 21/16, 4/3, 7/5, 10/7,
3/2, 32/21, 8/5, 49/30, 12/7, 7/4, 64/35, 15/8, 49/25]

[1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 12/7]
[1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 12/7]
[1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 12/7]
[1, 9/7, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
[1, 9/7, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
[1, 9/7, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
[1, 5/4, 35/24, 7/4] [1, 7/6, 35/24, 5/3]
[1, 9/7, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
[1, 5/4, 35/24, 7/4] [1, 7/6, 35/24, 5/3]
[1, 9/7, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
[1, 5/4, 35/24, 7/4] [1, 7/6, 35/24, 5/3]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
[1, 5/4, 35/24, 7/4] [1, 7/6, 35/24, 5/3]
[1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 12/7]
[1, 5/4, 35/24, 7/4] [1, 7/6, 35/24, 5/3]
[1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 12/7]
[1, 5/4, 35/24, 7/4] [1, 7/6, 35/24, 5/3]

Message: 6629 - Contents - Hide Contents

Date: Sat, 22 Mar 2003 22:11:00

Subject: Heavy duty hemis

From: Gene Ward Smith

The "Hemis" in question are some high-complexity temperaments with
generators half of a consonance, with scales for which 36/35 is a
chroma. As expected, hemififths is the star of the show because it has
only two wolf fifths out of the 34. When checking it out, bear in mind
it is a highly accurate temperament with an rms error of 0.585 cents.

Hemififths[34]

[2, 25, 13, 35, 15, -40]
[1, 28/27, 21/20, 15/14, 49/45, 9/8, 8/7, 7/6, 32/27, 49/40, 56/45,
80/63, 9/7, 21/16, 4/3, 112/81, 7/5, 10/7, 81/56, 3/2, 32/21, 14/9,
63/40, 45/28, 49/30, 27/16, 12/7, 7/4, 16/9, 90/49, 28/15, 40/21,
27/14, 63/32]
[56/45, 32/21, 28/15, 8/7, 7/5, 12/7, 21/20, 9/7, 63/40, 27/14, 32/27,
81/56, 16/9, 49/45, 4/3, 49/30, 1, 49/40, 3/2, 90/49, 9/8, 112/81,
27/16, 28/27, 80/63, 14/9, 40/21, 7/6, 10/7, 7/4, 15/14, 21/16, 45/28,
63/32]

Circle of generators, 49/40 generator

[1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
[1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
[1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
[1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
[1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
[1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
[1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
[1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
[1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
[1, 9/7, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
[1, 9/7, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
[1, 9/7, 3/2, 7/4] [1, 7/6, 3/2, 12/7]
[1, 9/7, 3/2, 7/4] [1, 7/6, 3/2, 12/7]
[1, 9/7, 3/2, 7/4] [1, 7/6, 3/2, 12/7]
[1, 9/7, 3/2, 7/4] [1, 7/6, 3/2, 12/7]
[1, 9/7, 3/2, 7/4] [1, 7/6, 3/2, 12/7]
[1, 9/7, 3/2, 7/4] [1, 7/6, 3/2, 12/7]
[1, 9/7, 3/2, 7/4] [1, 7/6, 3/2, 12/7]
[1, 9/7, 3/2, 7/4] [1, 7/6, 3/2, 12/7]
[1, 9/7, 3/2, 7/4] [1, 7/6, 3/2, 12/7]
[1, 9/7, 3/2, 7/4] [1, 7/6, 3/2, 12/7]
[1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 12/7]
[1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 12/7]
[1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
[1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
[1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
[1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
[1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
[1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
[1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
[1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
[1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
[1, 9/7, 54/35, 9/5] [1, 6/5, 54/35, 12/7]
[1, 9/7, 54/35, 9/5] [1, 6/5, 54/35, 12/7]


Hemiweurschmidt[25]

[16, 2, 5, -34, -37, 6]
[1, 49/48, 25/24, 35/32, 28/25, 8/7, 7/6, 49/40, 5/4, 32/25, 64/49,
48/35, 7/5, 10/7, 35/24, 49/32, 25/16, 8/5, 49/30, 12/7, 7/4, 25/14,
64/35, 48/25, 49/25]
[25/24, 7/6, 64/49, 35/24, 49/30, 64/35, 49/48, 8/7, 32/25, 10/7, 8/5,
25/14, 1, 28/25, 5/4, 7/5, 25/16, 7/4, 49/25, 35/32, 49/40, 48/35,
49/32, 12/7, 48/25]

Generator circle for 28/25

[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
[1, 5/4, 35/24, 7/4] [1, 6/5, 35/24, 12/7]
[1, 5/4, 35/24, 7/4] [1, 6/5, 35/24, 12/7]
[1, 5/4, 35/24, 7/4] [1, 7/6, 35/24, 12/7]
[1, 5/4, 35/24, 7/4] [1, 7/6, 35/24, 12/7]
[1, 5/4, 35/24, 7/4] [1, 7/6, 35/24, 12/7]
[1, 5/4, 35/24, 7/4] [1, 7/6, 35/24, 5/3]
[1, 5/4, 35/24, 7/4] [1, 7/6, 35/24, 5/3]
[1, 5/4, 35/24, 7/4] [1, 7/6, 35/24, 5/3]
[1, 5/4, 35/24, 7/4] [1, 7/6, 35/24, 5/3]
[1, 5/4, 35/24, 7/4] [1, 7/6, 35/24, 5/3]
[1, 5/4, 35/24, 7/4] [1, 7/6, 35/24, 5/3]
[1, 5/4, 35/24, 128/75] [1, 7/6, 35/24, 5/3]
[1, 5/4, 35/24, 128/75] [1, 7/6, 35/24, 5/3]
[1, 5/4, 35/24, 128/75] [1, 7/6, 35/24, 5/3]
[1, 128/105, 35/24, 128/75] [1, 7/6, 35/24, 5/3]
[1, 128/105, 35/24, 128/75] [1, 7/6, 35/24, 5/3]


Hemithirds[37]

[15, -2, -5, -38, -50, -6]
[1, 50/49, 21/20, 16/15, 15/14, 35/32, 28/25, 8/7, 75/64, 25/21, 6/5,
49/40, 5/4, 32/25, 21/16, 4/3, 75/56, 175/128, 7/5, 10/7, 147/100,
112/75, 3/2, 32/21, 25/16, 8/5, 80/49, 5/3, 42/25, 128/75, 7/4, 25/14,
64/35, 28/15, 15/8, 40/21, 49/25]
[15/14, 6/5, 75/56, 3/2, 42/25, 15/8, 21/20, 75/64, 21/16, 147/100,
80/49, 64/35, 50/49, 8/7, 32/25, 10/7, 8/5, 25/14, 1, 28/25, 5/4, 7/5,
25/16, 7/4, 49/25, 35/32, 49/40, 175/128, 32/21, 128/75, 40/21, 16/15,
25/21, 4/3, 112/75, 5/3, 28/15]

Circle of generators, 28/25 generator

[1, 5/4, 35/24, 7/4] [1, 7/6, 35/24, 5/3]
[1, 5/4, 35/24, 7/4] [1, 7/6, 35/24, 5/3]
[1, 5/4, 35/24, 7/4] [1, 7/6, 35/24, 5/3]
[1, 5/4, 35/24, 7/4] [1, 7/6, 35/24, 5/3]
[1, 5/4, 35/24, 7/4] [1, 7/6, 35/24, 5/3]
[1, 5/4, 35/24, 7/4] [1, 7/6, 35/24, 5/3]
[1, 5/4, 35/24, 7/4] [1, 7/6, 35/24, 5/3]
[1, 5/4, 35/24, 7/4] [1, 7/6, 35/24, 5/3]
[1, 5/4, 35/24, 7/4] [1, 7/6, 35/24, 5/3]
[1, 5/4, 35/24, 7/4] [1, 7/6, 35/24, 5/3]
[1, 5/4, 35/24, 7/4] [1, 7/6, 35/24, 5/3]
[1, 5/4, 35/24, 7/4] [1, 7/6, 35/24, 5/3]
[1, 5/4, 35/24, 7/4] [1, 7/6, 35/24, 5/3]
[1, 5/4, 35/24, 7/4] [1, 7/6, 35/24, 5/3]
[1, 5/4, 35/24, 7/4] [1, 7/6, 35/24, 5/3]
[1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
[1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 5/3]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 5/3]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 5/3]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
[1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
[1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7]


Hemikleismic[23]

[12, 10, -9, -12, -48, -49]
[1, 25/24, 21/20, 35/32, 8/7, 125/108, 6/5, 5/4, 63/50, 21/16, 48/35,
25/18, 36/25, 35/24, 32/21, 63/40, 8/5, 5/3, 125/72, 7/4, 64/35,
40/21, 48/25]
[35/24, 8/5, 7/4, 48/25, 21/20, 125/108, 63/50, 25/18, 32/21, 5/3,
64/35, 1, 35/32, 6/5, 21/16, 36/25, 63/40, 125/72, 40/21, 25/24, 8/7,
5/4, 48/35]

Generator circle for 35/32 generator

[1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3]
[1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3]
[1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3]
[1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3]
[1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3]
[1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3]
[1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 5/3]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 5/3]
[1, 5/4, 35/24, 7/4] [1, 6/5, 35/24, 5/3]
[1, 5/4, 35/24, 7/4] [1, 6/5, 35/24, 5/3]
[1, 128/105, 35/24, 7/4] [1, 6/5, 35/24, 5/3]
[1, 128/105, 35/24, 7/4] [1, 6/5, 35/24, 5/3]
[1, 128/105, 35/24, 7/4] [1, 6/5, 35/24, 5/3]
[1, 128/105, 35/24, 7/4] [1, 6/5, 35/24, 5/3]
[1, 128/105, 35/24, 7/4] [1, 6/5, 35/24, 5/3]
[1, 128/105, 35/24, 7/4] [1, 6/5, 35/24, 5/3]
[1, 128/105, 35/24, 7/4] [1, 6/5, 35/24, 5/3]
[1, 128/105, 35/24, 7/4] [1, 6/5, 35/24, 5/3]
[1, 128/105, 35/24, 7/4] [1, 7/6, 35/24, 5/3]
[1, 128/105, 35/24, 7/4] [1, 7/6, 35/24, 5/3]

Message: 6630 - Contents - Hide Contents

Date: Sat, 22 Mar 2003 00:46:21

Subject: Re: Blackjack and 36/35

From: wallyesterpaulrus

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

> --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
> 

>> keep it up, gene! i think one of the more important chromatic 
unison 
>> vectors kalle and carl were interested in was 49/48.
> 

> I think 15/14, 21/20 and 49/48 all need to be investigated, but I'm
> still sorting things out, as I've just found our old friend torsion
> makes the situation for schismic[24] more complicated.


*now* do you understand why i said that NMOS scales have something to 
do with torsion?? :)

Message: 6631 - Contents - Hide Contents

Date: Sat, 22 Mar 2003 22:14:26

Subject: Re: More on MOS/temperaments

From: Gene Ward Smith

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

>>> And recall, this is only one comma!
>> 

>> i'm not sure what you mean to imply by that -- and of course this 
>> chroma is only defined plus or minus any arbitrary number of
>> commatic unison vectors.
> 

> Maybe Gene's alluding to the planar and higher cases.


No, I've just been looking, so far, at 36/35 chroma cases. If you want
major triads and minor triads to convert, you could look at 25/24; if
you want other fun stuff, including asses, 15/14 or 21/20. 49/48 is
also worth exploring.

Message: 6632 - Contents - Hide Contents

Date: Sat, 22 Mar 2003 01:47:53

Subject: Diaschismic[24] with 36/35 chroma

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:

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


>> I think 15/14, 21/20 and 49/48 all need to be investigated, but I'm
>> still sorting things out, as I've just found our old friend torsion
>> makes the situation for schismic[24] more complicated.
> 

> *now* do you understand why i said that NMOS scales have something to 
> do with torsion?? :)


It's just revelations left and right. It hadn't bit me before, but it
does have a bite, so you are right.

Not only is schismic[24] a bit of a pain, I found that for
diaschismic[24] I needed to go around the circle of generators, since
the scale didn't come out with the right order. Here it is; the list
of tetrads goes around a circle of 12 16/15 generators; of course
there are 12 more tetrads just like these.


Diaschismic

[2, -4, -16, -11, -31, -26] 

[1, 64/63, 21/20, 16/15, 10/9, 9/8, 32/27, 6/5, 5/4, 32/25, 4/3,
27/20, 45/32, 10/7, 40/27, 3/2, 25/16, 8/5, 5/3, 27/16, 16/9, 9/5,
15/8, 40/21]

[1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
[1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3]
[1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3]
[1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3]
[1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3]
[1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 5/3]
[1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
[1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
[1, 5/4, 35/24, 7/4] [1, 7/6, 35/24, 5/3]

Message: 6633 - Contents - Hide Contents

Date: Sat, 22 Mar 2003 22:26:56

Subject: Re: Beatles[17] and squares[17]

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:

> what are the generators of these scales? what can we say about the 
> conditions leading to this kind of uniformity of the tetradic 
> materials as a function of "root" scale degree number?


This uniformity is connected to the fact that 36/35 is a chroma. As
for generators:


>> Beatles

>> Tetrads in generator circle order (49/40 generator)

>> Squares

>> Tetrads in generator circle order (9/7 generator)

Message: 6634 - Contents - Hide Contents

Date: Sat, 22 Mar 2003 02:53:34

Subject: Quartaminorthirds and muggles

From: Gene Ward Smith

Here are quartaminorthirds[16] and muggles[16], more 36/35-chroma systems.

Quartaminorthirds[16]

[9, 5, -3, -13, -30, -21] 
[1, 21/20, 35/32, 8/7, 6/5, 5/4, 21/16, 48/35, 10/7, 35/24, 32/21,
8/5, 5/3, 7/4, 64/35, 40/21]

[1, 5/4, 35/24, 7/4] [1, 6/5, 35/24, 5/3]
[1, 5/4, 35/24, 7/4] [1, 6/5, 35/24, 5/3]
[1, 5/4, 35/24, 7/4] [1, 6/5, 35/24, 5/3]
[1, 5/4, 35/24, 7/4] [1, 6/5, 35/24, 5/3]
[1, 49/40, 35/24, 7/4] [1, 6/5, 35/24, 5/3]
[1, 49/40, 35/24, 7/4] [1, 7/6, 35/24, 5/3]
[1, 49/40, 35/24, 7/4] [1, 7/6, 35/24, 5/3]
[1, 49/40, 35/24, 7/4] [1, 7/6, 35/24, 5/3]
[1, 49/40, 35/24, 7/4] [1, 7/6, 35/24, 5/3]
[1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 5/3]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 5/3]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 5/3]


Muggles[16]

[5, 1, -7, -10, -25, -19] 
[1, 21/20, 16/15, 8/7, 6/5, 5/4, 21/16, 4/3, 10/7, 3/2, 25/16, 8/5,
5/3, 7/4, 15/8, 40/21]

[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 5/3]
[1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3]
[1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
[1, 5/4, 35/24, 7/4] [1, 7/6, 35/24, 5/3]
[1, 5/4, 35/24, 7/4] [1, 6/5, 35/24, 5/3]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 5/3]
[1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3]
[1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
[1, 32/27, 35/24, 7/4] [1, 7/6, 35/24, 5/3]
[1, 5/4, 35/24, 7/4] [1, 7/6, 35/24, 5/3]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 5/3]
[1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3]
[1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
[1, 5/4, 35/24, 7/4] [1, 7/6, 35/24, 5/3]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 5/3]

Message: 6635 - Contents - Hide Contents

Date: Sat, 22 Mar 2003 22:37:09

Subject: Re: Quartaminorthirds and muggles

From: wallyesterpaulrus

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

>> could i put in a request -- when you look at these, try to find 
>> omnitetrachordal variants to the basic MOS scales . . .
> 
> Paul,
> 

> 1. Omnitetrachordal just means the 4/3 doesn't have to be
> cut in *four*, right?


are you sure you're phrasing that question correctly?


> The symmetry still has to be 3:2,
> right?


as opposed to 4:3? i think it's more 3:2 than 4:3.


> 
> 2. How might one find such variants?


i don't know a general rule, but interestingly even scales with 3 
steps sizes can be omnitetrachordal (we found a 22-tone example in 
the shrutar discussion).

Message: 6636 - Contents - Hide Contents

Date: Sat, 22 Mar 2003 04:32:04

Subject: Re: Quartaminorthirds and muggles

From: wallyesterpaulrus

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

> Here are quartaminorthirds[16] and muggles[16], more 36/35-chroma 
systems.


could i put in a request -- when you look at these, try to find 
omnitetrachordal variants to the basic MOS scales . . . 

. . . and then, when they're all done, one could to the hyper-MOS 
scales of the second order: fokker periodicity blocks where two 
unison vectors are not tempered out. and the omnitetrachordal 
variants of those . . .

Message: 6637 - Contents - Hide Contents

Date: Sat, 22 Mar 2003 22:43:14

Subject: Re: Beatles[17] and squares[17]

From: wallyesterpaulrus

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

> --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:

>> what are the generators of these scales? what can we say about 
the 
>> conditions leading to this kind of uniformity of the tetradic 
>> materials as a function of "root" scale degree number?
> 

> This uniformity is connected to the fact that 36/35 is a chroma. As
> for generators:
> 
>>> Beatles
> 

>>> Tetrads in generator circle order (49/40 generator)
> 
>>> Squares
> 
>>> Tetrads in generator circle order (9/7 generator)


oh, the uniformity is because you put the tetrads in generator circle 
order, isn't it? what does 36/35 have to do with it? i thought you 
were giving the tetrads in scale order. perhaps in fifths order would 
be best of all.

Message: 6638 - Contents - Hide Contents

Date: Sat, 22 Mar 2003 05:03:55

Subject: Beatles[17] and squares[17]

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:


> could i put in a request -- when you look at these, try to find 
> omnitetrachordal variants to the basic MOS scales . . . 

Yipe!

> . . . and then, when they're all done, one could to the hyper-MOS 
> scales of the second order: fokker periodicity blocks where two 
> unison vectors are not tempered out. and the omnitetrachordal 
> variants of those . . .


Double yipe. I'll have to think about this.

Beatles

[2, -9, -4, -19, -12, 16] 
[1, 16/15, 15/14, 8/7, 7/6, 49/40, 9/7, 4/3, 7/5, 10/7, 3/2, 14/9,
45/28, 12/7, 7/4, 28/15, 15/8] [14/9, 15/8, 7/6, 10/7, 7/4, 15/14,
4/3, 45/28, 1, 49/40, 3/2, 28/15, 8/7, 7/5, 12/7, 16/15, 9/7]

Tetrads in generator circle order (49/40 generator)

[1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
[1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
[1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
[1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
[1, 9/7, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
[1, 9/7, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
[1, 9/7, 3/2, 7/4] [1, 7/6, 3/2, 12/7]
[1, 9/7, 3/2, 7/4] [1, 7/6, 3/2, 12/7]
[1, 9/7, 3/2, 7/4] [1, 7/6, 3/2, 12/7]
[1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 12/7]
[1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 12/7]
[1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
[1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
[1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
[1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
[1, 5/4, 35/24, 7/4] [1, 7/6, 35/24, 5/3]
[1, 5/4, 35/24, 7/4] [1, 7/6, 35/24, 5/3]


Squares

[4, 16, 9, 16, 3, -24] 
[1, 21/20, 49/45, 9/8, 7/6, 49/40, 9/7, 4/3, 7/5, 10/7, 3/2, 14/9,
49/30, 12/7, 9/5, 49/27, 27/14] [9/8, 10/7, 49/27, 7/6, 3/2, 27/14,
49/40, 14/9, 1, 9/7, 49/30, 21/20, 4/3, 12/7, 49/45, 7/5, 9/5]

Tetrads in generator circle order (9/7 generator)

[1, 9/7, 32/21, 9/5] [1, 6/5, 32/21, 12/7]
[1, 9/7, 32/21, 9/5] [1, 6/5, 32/21, 12/7]
[1, 9/7, 32/21, 9/5] [1, 6/5, 32/21, 12/7]
[1, 9/7, 32/21, 9/5] [1, 6/5, 32/21, 12/7]
[1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
[1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 12/7]
[1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 12/7]
[1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 12/7]
[1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 12/7]
[1, 9/7, 3/2, 7/4] [1, 7/6, 3/2, 12/7]
[1, 9/7, 3/2, 7/4] [1, 7/6, 3/2, 12/7]
[1, 9/7, 3/2, 7/4] [1, 7/6, 3/2, 12/7]
[1, 9/7, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
[1, 9/7, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
[1, 9/7, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
[1, 9/7, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
[1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3]

Message: 6639 - Contents - Hide Contents

Date: Sat, 22 Mar 2003 15:01:07

Subject: Re: Quartaminorthirds and muggles

From: Carl Lumma

>> >. Omnitetrachordal just means the 4/3 doesn't have to be
>> cut in *four*, right?
>

>are you sure you're phrasing that question correctly?

Mmm...


>> The symmetry still has to be 3:2,
>> right?
>

>as opposed to 4:3? i think it's more 3:2 than 4:3.


As opposed to 5:4.  I always forget the meaning of omnitetrachordal.
I think, because the tetra's still in there.

-C.

Message: 6640 - Contents - Hide Contents

Date: Sat, 22 Mar 2003 06:22:08

Subject: Schismic[24] and Tritonic[33]

From: Gene Ward Smith

The right way to do these seems to be always to look at the circle of
generators, which gets us out of any sticky business. Schismic[24] is
a hell of a 7-limit system, we see once again. I haven't considered
writing in it because it seemed to well-explored, but that may only be
in theory. In any case, when looking at 36/35 chroma systems it's nice
to have 3/2 as a generator, since that limits the damage of the wolf
fifth.

Tritonic[33] is not quite in the same league, but as you can see it
could be interesting also.


Schismic[24]

[1, -8, -14, -15, -25, -10]
[1, 50/49, 21/20, 15/14, 10/9, 9/8, 25/21, 6/5, 5/4, 63/50, 4/3,
27/20, 7/5, 10/7, 40/27, 3/2, 63/40, 8/5, 5/3, 27/16, 16/9, 9/5, 15/8,
40/21]
[40/27, 10/9, 5/3, 5/4, 15/8, 7/5, 21/20, 63/40, 25/21, 16/9, 4/3, 1,
3/2, 9/8, 27/16, 63/50, 40/21, 10/7, 15/14, 8/5, 6/5, 9/5, 27/20, 50/49]

3/2 generator circle

[1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
[1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
[1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
[1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
[1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
[1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
[1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
[1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3]
[1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3]
[1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3]
[1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3]
[1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 5/3]
[1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
[1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
[1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
[1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
[1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
[1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
[1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
[1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
[1, 5/4, 35/24, 7/4] [1, 7/6, 35/24, 5/3]


Tritonic[33]

[5, -11, -12, -29, -33, 3]
[1, 50/49, 21/20, 15/14, 160/147, 9/8, 8/7, 147/125, 25/21, 6/5,
49/40, 5/4, 80/63, 21/16, 4/3, 200/147, 7/5, 10/7, 147/100, 3/2,
32/21, 63/40, 8/5, 80/49, 5/3, 27/16, 250/147, 7/4, 16/9, 147/80,
15/8, 40/21, 49/25]
[6/5, 27/16, 147/125, 80/49, 8/7, 8/5, 9/8, 63/40, 160/147, 32/21,
15/14, 3/2, 21/20, 147/100, 50/49, 10/7, 1, 7/5, 49/25, 200/147,
40/21, 4/3, 15/8, 21/16, 147/80, 80/63, 16/9, 5/4, 7/4, 49/40,
250/147, 25/21, 5/3]

7/5 generator circle

[1, 5/4, 35/24, 7/4] [1, 7/6, 35/24, 5/3]
[1, 5/4, 35/24, 7/4] [1, 7/6, 35/24, 5/3]
[1, 5/4, 35/24, 7/4] [1, 7/6, 35/24, 5/3]
[1, 5/4, 35/24, 7/4] [1, 7/6, 35/24, 5/3]
[1, 5/4, 35/24, 7/4] [1, 7/6, 35/24, 5/3]
[1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
[1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
[1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
[1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
[1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
[1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
[1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
[1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
[1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
[1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
[1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 5/3]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
[1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
[1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
[1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
[1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
[1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
[1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
[1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
[1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
[1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
[1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
[1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7]

Message: 6641 - Contents - Hide Contents

Date: Sat, 22 Mar 2003 15:14:26

Subject: this T[n] business

From: Carl Lumma

1. Paul, could you brief us on how the T[n] method compares
to what you and Kalle were using before?

2. The last version of this thread thread (see msg. 6017)
left off with how to identify important commas.  Gene's
mentioned square and triangular numbers as being better,
though I'm not sure why... though I imagine a high level of
non-primeness in general would be good, since it increases
chances of turning a simple interval into another simple
interval.  For the same reason, complexity (Tenney or
heuristic) might be good.

I suggested some measures which included size, but Paul's
probably right that size doesn't have anything to do with it.
Tangentially though, I asserted that Tenney complexity is
tainted wrt size, since smaller ratios tend to have bigger
numbers.

I got fed up with Excel and tested this assertion in Scheme.
Using all ratios with Tenney height less than 3000, I counted
how many approximated to each degree of 23-, 50-, and 100-et.
I didn't bother to plot the results, but a roughly equal
number of ratios seem to fall in all bins in each case,
whether I enforced octave equivalence or not.  So it seems my
assertion is wrong; simple ratios don't tend to be bigger.

-Carl

Message: 6642 - Contents - Hide Contents

Date: Sat, 22 Mar 2003 06:43:48

Subject: Re: Beatles[17] and squares[17]

From: wallyesterpaulrus

what are the generators of these scales? what can we say about the 
conditions leading to this kind of uniformity of the tetradic 
materials as a function of "root" scale degree number?

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

> --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
> 

>> could i put in a request -- when you look at these, try to find 
>> omnitetrachordal variants to the basic MOS scales . . . 
> 
> Yipe!
> 

>> . . . and then, when they're all done, one could to the hyper-MOS 
>> scales of the second order: fokker periodicity blocks where two 
>> unison vectors are not tempered out. and the omnitetrachordal 
>> variants of those . . .
> 

> Double yipe. I'll have to think about this.
> 
> Beatles
> 
> [2, -9, -4, -19, -12, 16] 
> [1, 16/15, 15/14, 8/7, 7/6, 49/40, 9/7, 4/3, 7/5, 10/7, 3/2, 14/9,
> 45/28, 12/7, 7/4, 28/15, 15/8] [14/9, 15/8, 7/6, 10/7, 7/4, 15/14,
> 4/3, 45/28, 1, 49/40, 3/2, 28/15, 8/7, 7/5, 12/7, 16/15, 9/7]
> 
> Tetrads in generator circle order (49/40 generator)
> 
> [1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
> [1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
> [1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
> [1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
> [1, 9/7, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
> [1, 9/7, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
> [1, 9/7, 3/2, 7/4] [1, 7/6, 3/2, 12/7]
> [1, 9/7, 3/2, 7/4] [1, 7/6, 3/2, 12/7]
> [1, 9/7, 3/2, 7/4] [1, 7/6, 3/2, 12/7]
> [1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 12/7]
> [1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 12/7]
> [1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
> [1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
> [1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
> [1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
> [1, 5/4, 35/24, 7/4] [1, 7/6, 35/24, 5/3]
> [1, 5/4, 35/24, 7/4] [1, 7/6, 35/24, 5/3]
> 
> 
> Squares
> 
> [4, 16, 9, 16, 3, -24] 
> [1, 21/20, 49/45, 9/8, 7/6, 49/40, 9/7, 4/3, 7/5, 10/7, 3/2, 14/9,
> 49/30, 12/7, 9/5, 49/27, 27/14] [9/8, 10/7, 49/27, 7/6, 3/2, 27/14,
> 49/40, 14/9, 1, 9/7, 49/30, 21/20, 4/3, 12/7, 49/45, 7/5, 9/5]
> 
> Tetrads in generator circle order (9/7 generator)
> 
> [1, 9/7, 32/21, 9/5] [1, 6/5, 32/21, 12/7]
> [1, 9/7, 32/21, 9/5] [1, 6/5, 32/21, 12/7]
> [1, 9/7, 32/21, 9/5] [1, 6/5, 32/21, 12/7]
> [1, 9/7, 32/21, 9/5] [1, 6/5, 32/21, 12/7]
> [1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
> [1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 12/7]
> [1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 12/7]
> [1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 12/7]
> [1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 12/7]
> [1, 9/7, 3/2, 7/4] [1, 7/6, 3/2, 12/7]
> [1, 9/7, 3/2, 7/4] [1, 7/6, 3/2, 12/7]
> [1, 9/7, 3/2, 7/4] [1, 7/6, 3/2, 12/7]
> [1, 9/7, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
> [1, 9/7, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
> [1, 9/7, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
> [1, 9/7, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
> [1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3]

Message: 6643 - Contents - Hide Contents

Date: Sat, 22 Mar 2003 23:28:49

Subject: Duodecimal[36]

From: Gene Ward Smith

A sort of ultimate in avoiding wolf fifths in 36/35-chroma systems is
achieved by duodecimal, the system with parallel 12-equals separated
by a comma which does double duty for 81/80 and 126/125. While
ostensibly a 7-limit system, if you look at the tetrads in this list
it looks suspiciously like a 5-limit system.

Duodecimal[36]

[0, 12, 24, 19, 38, 22]
[1, 81/80, 21/20, 200/189, 15/14, 10/9, 9/8, 500/441, 147/125, 25/21,
6/5, 5/4, 63/50, 80/63, 250/189, 4/3, 27/20, 7/5, 567/400, 10/7,
40/27, 3/2, 189/125, 63/40, 100/63, 8/5, 5/3, 27/16, 250/147, 441/250,
16/9, 9/5, 15/8, 189/100, 40/21, 125/63]
[[21/20, 63/40, 147/125, 441/250, 250/189, 125/63, 40/27, 10/9, 5/3,
5/4, 15/8, 7/5], [200/189, 100/63, 25/21, 16/9, 4/3, 1, 3/2, 9/8,
27/16, 63/50, 189/100, 567/400], [15/14, 8/5, 6/5, 9/5, 27/20, 81/80,
189/125, 500/441, 250/147, 80/63, 40/21, 10/7]]

Circle of fifths

[1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3]
[1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3]
[1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3]
[1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3]
[1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3]
[1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3]
[1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3]
[1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3]
[1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3]
[1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3]
[1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3]
[1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3]

I could repeat this twice more, but you get the idea--all tetrads are
alike.

Message: 6644 - Contents - Hide Contents

Date: Sat, 22 Mar 2003 23:35:00

Subject: Re: Beatles[17] and squares[17]

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:


> oh, the uniformity is because you put the tetrads in generator circle 
> order, isn't it? 


I was for a while, but it turns out it's better to stick with
generator circles.


>what does 36/35 have to do with it?


36/35 exchanges 7/6<-->6/5, 5/4<-->9/7, 5/3<-->12/7, 7/4<-->9/5 as we
move from one kind of tetrad to another. Unfortunately, it also
exchanges 35/24<-->3/2, leading to wolves.

Message: 6645 - Contents - Hide Contents

Date: Sat, 22 Mar 2003 23:57:14

Subject: Re: this T[n] business

From: wallyesterpaulrus

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

> 1. Paul, could you brief us on how the T[n] method compares
> to what you and Kalle were using before?


eh . . . i think it's all the same, except that we identified 
three particular chromatic unison vectors that were likely to produce 
the major tetrads and minor tetrads using the same pattern of scale 
steps -- gene is not focused on that property, but rather 
investigating one chromatic unison vector at a time amongst quite a 
few possibilities.


> 2. The last version of this thread thread (see msg. 6017)
> left off with how to identify important commas.  Gene's
> mentioned square and triangular numbers as being better,
> though I'm not sure why...

don't know

> though I imagine a high level of
> non-primeness in general would be good, since it increases
> chances of turning a simple interval into another simple
> interval.  For the same reason, complexity (Tenney or
> heuristic) might be good.


complexity would be bad (inverse complexity would be good), since 
simple intervals would tend to turn to complex intervals, and since 
you'd tend to get more notes in the scale.


> I suggested some measures which included size, but Paul's
> probably right that size doesn't have anything to do with it.
> Tangentially though, I asserted that Tenney complexity is
> tainted wrt size, since smaller ratios tend to have bigger
> numbers.


bigger numbers than what? than those in the ratios for larger 
intervals? nope, not unless you're holding something constant such as 
insisting on superparticularity.


> I got fed up with Excel and tested this assertion in Scheme.
> Using all ratios with Tenney height less than 3000, I counted
> how many approximated to each degree of 23-, 50-, and 100-et.
> I didn't bother to plot the results, but a roughly equal
> number of ratios seem to fall in all bins in each case,
> whether I enforced octave equivalence or not.  So it seems my
> assertion is wrong; simple ratios don't tend to be bigger.


that's the great thing about tenney complexity (as opposed to farey, 
mann, etc.)!

Message: 6646 - Contents - Hide Contents

Date: Sat, 22 Mar 2003 20:13:16

Subject: Re: this T[n] business

From: Carl Lumma

>> >. The last version of this thread thread (see msg. 6017)
>> left off with how to identify important commas.  Gene's
>> mentioned square and triangular numbers as being better,
>> though I'm not sure why...
>
>don't know


I suppose squareness and triangularity are types of
compositeness.


>> though I imagine a high level of
>> non-primeness in general would be good, since it increases
>> chances of turning a simple interval into another simple
>> interval.  For the same reason, complexity (Tenney or
>> heuristic) might be good.
>

>complexity would be bad (inverse complexity would be good),


Yep, that's what I meant.  Which is why n*d was in the den.
of my suggested measures.


>> I got fed up with Excel and tested this assertion in Scheme.
>> Using all ratios with Tenney height less than 3000, I counted
>> how many approximated to each degree of 23-, 50-, and 100-et.
>> I didn't bother to plot the results, but a roughly equal
>> number of ratios seem to fall in all bins in each case,
>> whether I enforced octave equivalence or not.  So it seems my
>> assertion is wrong; simple ratios don't tend to be bigger.
>

>that's the great thing about tenney complexity (as opposed to
>farey, mann, etc.)!


Ah, you've said that before, I think!

-Carl

Message: 6647 - Contents - Hide Contents

Date: Sun, 23 Mar 2003 20:19:16

Subject: Diaschismic, Negri and Blackwood 10

From: Gene Ward Smith

Diaschismic[10] looks like an attractive alternative to meantone[7].

Diaschismic[10]
rms error 2.61 cents

2048/2025
[1, 16/15, 9/8, 5/4, 4/3, 45/32, 3/2, 8/5, 16/9, 15/8]
[[1, 3/2, 9/8, 16/9, 4/3], [45/32, 16/15, 8/5, 5/4, 15/8]]

Two identical circles of fifths
[1, 6/5, 3/2]
[1, 6/5, 3/2]
[1, 5/4, 3/2]
[1, 5/4, 3/2]
[1, 5/4, 25/16]

Negri[10]
rms error 5.94 cents

16875/16384
[1, 16/15, 75/64, 5/4, 4/3, 64/45, 3/2, 8/5, 128/75, 15/8]
[3/2, 8/5, 128/75, 15/8, 1, 16/15, 75/64, 5/4, 4/3, 64/45]

Generator = 16/15
[1, 5/4, 25/16]
[1, 5/4, 25/16]
[1, 5/4, 25/16]
[1, 5/4, 25/16]
[1, 5/4, 3/2]
[1, 5/4, 3/2]
[1, 5/4, 3/2]
[1, 6/5, 3/2]
[1, 6/5, 3/2]
[1, 6/5, 3/2]

Blackwood[10]
rms error 12.76 cents

256/243
[1, 10/9, 9/8, 5/4, 4/3, 40/27, 3/2, 5/3, 16/9, 15/8]
[[16/9, 4/3, 1, 3/2, 9/8], [15/8, 40/27, 10/9, 5/3, 5/4]]

Circles of fifths
[1, 5/4, 3/2] [1, 6/5, 3/2]
[1, 5/4, 3/2] [1, 6/5, 3/2]
[1, 5/4, 3/2] [1, 6/5, 3/2]
[1, 5/4, 3/2] [1, 6/5, 3/2]
[1, 5/4, 3/2] [1, 6/5, 3/2]

Message: 6648 - Contents - Hide Contents

Date: Sun, 23 Mar 2003 21:22:55

Subject: Schismic[17] and Miracle[20]

From: Gene Ward Smith

We can make the same comment about these as I made for Orwell[13];
that the augmented triads, from the perspective of the corresponding
septimal temperament, are [1,5/4,14/9] 225/224-tempered asses. Note
however that 5-limit schismic is a microtemperament, and 7-limit
schismic is not nearly as well in tune.

Looking at this, I find myself completely unconvinced by George's
17-revolution alternative history. Going from meantone to 12-equal is
one thing, but I don't think you could ever get people used to
schismic[17] used for 5-limit harmony to move to 17-equal. What do the
17-revolutionaries think of schismic[17]?

Schismic[17]
rms error 0.162 cents

32805/32768
[1, 135/128, 16/15, 9/8, 32/27, 5/4, 81/64, 4/3, 45/32, 64/45, 3/2,
128/81, 8/5, 27/16, 16/9, 15/8, 243/128]
[5/4, 15/8, 45/32, 135/128, 128/81, 32/27, 16/9, 4/3, 1, 3/2, 9/8, 27/
16, 81/64, 243/128, 64/45, 16/15, 8/5]

Circle of fifths
[1, 6/5, 3/2]
[1, 6/5, 3/2]
[1, 6/5, 3/2]
[1, 6/5, 3/2]
[1, 6/5, 3/2]
[1, 6/5, 3/2]
[1, 6/5, 3/2]
[1, 6/5, 3/2]
[1, 5/4, 3/2]
[1, 5/4, 3/2]
[1, 5/4, 3/2]
[1, 5/4, 3/2]
[1, 5/4, 3/2]
[1, 5/4, 3/2]
[1, 5/4, 3/2]
[1, 5/4, 3/2]
[1, 5/4, 25/16]

Miracle[20]
rms error = 1.981 cents

34171875/33554432
[1, 16/15, 1125/1024, 256/225, 75/64, 4096/3375, 5/4, 675/512, 4/3,
45/32, 64/45, 3/2, 1024/675, 8/5, 3375/2048, 128/75, 225/128,
2048/1125, 15/8, 2025/1024]
[1125/1024, 75/64, 5/4, 4/3, 64/45, 1024/675, 3375/2048, 225/128,
15/8, 1, 16/15, 256/225, 4096/3375, 675/512, 45/32, 3/2, 8/5, 128/75,
2048/1125, 2025/1024]

Generator = 16/15
[1, 6/5, 3/2]
[1, 6/5, 3/2]
[1, 6/5, 3/2]
[1, 6/5, 3/2]
[1, 6/5, 3/2]
[1, 6/5, 3/2]
[1, 6/5, 3/2]
[1, 5/4, 3/2]
[1, 5/4, 3/2]
[1, 5/4, 3/2]
[1, 5/4, 3/2]
[1, 5/4, 3/2]
[1, 5/4, 3/2]
[1, 5/4, 3/2]
[1, 5/4, 25/16]
[1, 5/4, 25/16]
[1, 5/4, 25/16]
[1, 5/4, 25/16]
[1, 5/4, 25/16]
[1, 5/4, 25/16]

Message: 6649 - Contents - Hide Contents

Date: Sun, 23 Mar 2003 07:11:55

Subject: Orwell[13] and Quadrafifhs[14]

From: Gene Ward Smith

It should be noted that the augmented triads in orwell, considered in
terms of septimal orwell, can be thought of as [1, 5/4, 14/9] as well,
a tempered ass triad, or whatever it should be called.

Orwell[13]

2109375/2097152
[1, 16/15, 1125/1024, 75/64, 5/4, 32/25, 512/375, 375/256, 25/16, 8/5,
128/75, 1875/1024, 15/8]
[25/16, 1875/1024, 16/15, 5/4, 375/256, 128/75, 1, 75/64, 512/375,
8/5, 15/8, 1125/1024, 32/25]

Generator = 75/64
[1, 6/5, 3/2]
[1, 6/5, 3/2]
[1, 6/5, 3/2]
[1, 5/4, 3/2]
[1, 5/4, 3/2]
[1, 5/4, 3/2]
[1, 5/4, 25/16]
[1, 5/4, 25/16]
[1, 5/4, 25/16]
[1, 5/4, 25/16]
[1, 5/4, 25/16]
[1, 5/4, 25/16]
[1, 5/4, 25/16]


Quadrafifths[14]

20000/19683
[1, 81/80, 27/25, 10/9, 6/5, 100/81, 4/3, 27/20, 40/27, 3/2, 81/50,
5/3, 9/5, 50/27]
[27/25, 6/5, 4/3, 40/27, 81/50, 9/5, 1, 10/9, 100/81, 27/20, 3/2, 5/3,
50/27, 81/80]

Generator = 10/9
[1, 5/4, 3/2]
[1, 5/4, 3/2]
[1, 5/4, 3/2]
[1, 5/4, 3/2]
[1, 5/4, 3/2]
[1, 6/5, 3/2]
[1, 6/5, 3/2]
[1, 6/5, 3/2]
[1, 6/5, 3/2]
[1, 6/5, 3/2]
[1, 6/5, 36/25]
[1, 6/5, 36/25]
[1, 6/5, 36/25]
[1, 6/5, 36/25]

Circles of fifths
[1, 5/4, 3/2] [1, 5/4, 3/2]
[1, 6/5, 3/2] [1, 6/5, 3/2]
[1, 6/5, 36/25] [1, 6/5, 36/25]
[1, 5/4, 3/2] [1, 5/4, 3/2]
[1, 6/5, 3/2] [1, 6/5, 3/2]
[1, 6/5, 3/2] [1, 6/5, 36/25]
[1, 6/5, 36/25] [1, 5/4, 3/2]

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