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Message: 6675 - Contents - Hide Contents

Date: Mon, 24 Mar 2003 08:01:43

Subject: Re: this T[n] business

From: wallyesterpaulrus

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

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

>> something that *does* work like tenney in this way is the odd 
limit.
> 

> Hmm, this doesn't seem right; the diamond pitches aren't very
> regular at all.  IIRC they get thicker and thicker as they approach
> the identity, but then there's a gap before hitting it.
> 
> -Carl


this is no more true than it is for tenney.

Message: 6676 - Contents - Hide Contents

Date: Mon, 24 Mar 2003 08:35:16

Subject: Re: Diaschismic, Negri and Blackwood 10

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus" 


>> Very true--did I put something up about diaschismic[10] in the 
> 7-limit
>> yet?
> 

> i wrote an entire paper about it :)


I wasn't thinking of pajara, but of the 34&46 system I've been calling
"diaschismic". My endorsement was premature; I knew diaschismic[10]
was not only a 25/24 system, but also a 21/20 system, so I expected to
get asses in place of regular tetrads, and I did. Trouble is, these
are all 5-limit asses, so it's still 5-limit harmony. However, it's in
much better tune 5-limitwise than pajara.

Message: 6677 - Contents - Hide Contents

Date: Mon, 24 Mar 2003 09:53:44

Subject: Re: Diaschismic, Negri and Blackwood 10

From: Graham Breed

Gene Ward Smith wrote:


> I wasn't thinking of pajara, but of the 34&46 system I've been calling
> "diaschismic". My endorsement was premature; I knew diaschismic[10]
> was not only a 25/24 system, but also a 21/20 system, so I expected to
> get asses in place of regular tetrads, and I did. Trouble is, these
> are all 5-limit asses, so it's still 5-limit harmony. However, it's in
> much better tune 5-limitwise than pajara.


Sorry, I've lost the original message.  34-equal isn't consistent in the 
7-limit, so it won't define the system.  Do you mean 46&58?

2  3  5  7
0  1 -2 -8


                  Graham

Message: 6678 - Contents - Hide Contents

Date: Tue, 25 Mar 2003 18:35:18

Subject: Injera[14] and Hexadecimal[9]

From: Gene Ward Smith

Nothing wrong with these, beyond the fact that they are out of tune.

Injera[14]
rms error 11.219

[2, 8, 8, 8, 7, -4]
[1, 9/8, 6/5, 4/3, 7/5, 3/2, 14/9, 5/3, 12/7, 9/5, 27/14, 15/7, 7/3, 18/7]
[[6/5, 9/5, 4/3, 1, 3/2, 9/8, 5/3], [12/7, 18/7, 27/14, 7/5, 15/7,
14/9, 7/3]]

Circle of tetrads generator = 3/2
[1, 5/4, 3/2, 7/4]
[1, 5/4, 3/2, 7/4]
[1, 5/4, 3/2, 7/4]
[1, 6/5, 3/2, 12/7]
[1, 6/5, 3/2, 12/7]
[1, 6/5, 3/2, 12/7]
[1, 6/5, 36/25, 12/7]

Circles of alternative tetrads generator = 3/2
[1, 21/16, 3/2, 15/8] [1, 7/6, 3/2, 5/3] [1, 7/6, 3/2, 7/4]
[1, 21/16, 3/2, 15/8] [1, 7/6, 3/2, 5/3] [1, 7/6, 3/2, 7/4]
[1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 5/3] [1, 7/6, 3/2, 7/4]
[1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 5/3] [1, 7/6, 3/2, 12/7]
[1, 9/7, 3/2, 9/5] [1, 8/7, 3/2, 8/5] [1, 8/7, 3/2, 12/7]
[1, 9/7, 3/2, 9/5] [1, 8/7, 3/2, 8/5] [1, 8/7, 3/2, 12/7]
[1, 9/7, 36/25, 9/5] [1, 8/7, 36/25, 8/5] [1, 8/7, 36/25, 12/7]


Hexadecimal[9]
rms error 18.585

[1, -3, 5, -7, 5, 20]
[1, 9/8, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3, 15/8]
[5/3, 5/4, 15/8, 4/3, 1, 3/2, 9/8, 8/5, 6/5]

Generator = 3/2
[1, 21/16, 3/2, 7/4] [1, 6/5, 3/2, 7/4]
[1, 21/16, 3/2, 7/4] [1, 6/5, 3/2, 7/4]
[1, 21/16, 3/2, 7/4] [1, 6/5, 3/2, 7/4]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 7/4]
[1, 5/4, 3/2, 5/3] [1, 6/5, 3/2, 5/3]
[1, 5/4, 3/2, 5/3] [1, 8/7, 3/2, 5/3]
[1, 5/4, 3/2, 5/3] [1, 8/7, 3/2, 5/3]
[1, 5/4, 3/2, 5/3] [1, 8/7, 3/2, 5/3]
[1, 5/4, 10/7, 5/3] [1, 8/7, 10/7, 5/3]

Message: 6679 - Contents - Hide Contents

Date: Tue, 25 Mar 2003 19:53:41

Subject: Kleismic[8] and Hemifourths[9]

From: Gene Ward Smith

Enough 49/48 for now--on to what I wanted to do, 15/14 as a chroma
from which I expect great things.

Kleismic[8]
rms error 12.274

[6, 5, 3, -6, -12, -7]
[1, 21/20, 7/6, 6/5, 7/5, 10/7, 5/3, 7/4]
[7/6, 7/5, 5/3, 1, 6/5, 10/7, 7/4, 21/20]

6/5 generator tetrad circles
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 7/4] [1, 6/5, 3/2, 9/5] [1, 5/4, 3/2,
9/5]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 7/4] [1, 6/5, 3/2, 5/3] [1, 5/4, 3/2,
5/3]
[1, 5/4, 7/5, 7/4] [1, 6/5, 7/5, 7/4] [1, 6/5, 7/5, 5/3] [1, 5/4, 7/5,
5/3]
[1, 7/6, 7/5, 7/4] [1, 6/5, 7/5, 7/4] [1, 6/5, 7/5, 5/3] [1, 7/6, 7/5,
5/3]
[1, 7/6, 7/5, 7/4] [1, 6/5, 7/5, 7/4] [1, 6/5, 7/5, 5/3] [1, 7/6, 7/5,
5/3]
[1, 7/6, 7/5, 8/5] [1, 6/5, 7/5, 8/5] [1, 6/5, 7/5, 5/3] [1, 7/6, 7/5,
5/3]
[1, 7/6, 7/5, 8/5] [1, 6/5, 7/5, 8/5] [1, 6/5, 7/5, 5/3] [1, 7/6, 7/5,
5/3]
[1, 7/6, 7/5, 8/5] [1, 10/9, 7/5, 8/5] [1, 10/9, 7/5, 5/3] [1, 7/6,
7/5, 5/3]


Hemifourths[9]
rms error 12.690

[2, 8, 1, 8, -4, -20]
[1, 9/8, 7/6, 9/7, 4/3, 3/2, 14/9, 7/4, 9/5]
[9/8, 9/7, 3/2, 7/4, 1, 7/6, 4/3, 14/9, 9/5]

7/6 generator tetrad circles
[1, 7/6, 7/5, 8/5] [1, 6/5, 7/5, 8/5] [1, 6/5, 7/5, 9/5] [1, 7/6, 7/5,
14/9]
[1, 7/6, 7/5, 7/4] [1, 6/5, 7/5, 7/4] [1, 6/5, 7/5, 9/5] [1, 7/6, 7/5,
14/9]
[1, 7/6, 3/2, 7/4] [1, 6/5, 3/2, 7/4] [1, 6/5, 3/2, 9/5] [1, 7/6, 3/2,
14/9]
[1, 7/6, 3/2, 7/4] [1, 9/7, 3/2, 7/4] [1, 9/7, 3/2, 9/5] [1, 7/6, 3/2,
14/9]
[1, 7/6, 3/2, 7/4] [1, 9/7, 3/2, 7/4] [1, 9/7, 3/2, 9/5] [1, 7/6, 3/2,
14/9]
[1, 7/6, 3/2, 7/4] [1, 9/7, 3/2, 7/4] [1, 9/7, 3/2, 27/14] [1, 7/6,
3/2, 14/9]
[1, 7/6, 3/2, 7/4] [1, 9/7, 3/2, 7/4] [1, 9/7, 3/2, 27/14] [1, 7/6,
3/2, 5/3]
[1, 7/6, 3/2, 7/4] [1, 9/7, 3/2, 7/4] [1, 9/7, 3/2, 27/14] [1, 7/6,
3/2, 5/3]
[1, 5/4, 3/2, 7/4] [1, 9/7, 3/2, 7/4] [1, 9/7, 3/2, 27/14] [1, 5/4,
3/2, 5/3]

Message: 6680 - Contents - Hide Contents

Date: Tue, 25 Mar 2003 21:43:19

Subject: Re: Pajara[10]

From: wallyesterpaulrus

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

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

>>>> i don't get the logic of the "so". what implies what, exactly?
>>> 

>>> The major and minor circles are the same.
>> 

>> is this the antecedent or the consequent?
> 

> It's a consequence of 25/24 and 49/48 both being chromas. Didn't we
> get into this once?


not sure . . . the major and minor circles are the same as a 
consequence of 25/24 and 49/48 both being chromas . . . hmm . . . can 
you give the "for dummies" explanation of this? i mean, i know that 
25/24 and 49/48 both being chromas implies that 50/49 is a comma, 
which implies that there will be a half-octave period . . . are you 
saying something beyond that?

Message: 6681 - Contents - Hide Contents

Date: Tue, 25 Mar 2003 03:51:05

Subject: Fifteen notes, 49/48 chroma

From: Gene Ward Smith

Tripletone[15]
rms error 8.101 cents

[3, 0, -6, -7, -18, -14]
[1, 16/15, 10/9, 8/7, 6/5, 5/4, 4/3, 7/5, 10/7, 3/2, 8/5, 5/3, 7/4,
9/5, 15/8]
[[7/5, 16/15, 8/5, 6/5, 9/5], [7/4, 4/3, 1, 3/2, 8/7], [10/9, 5/3,
5/4, 15/8, 10/7]]

Circle of major and minor tetrads generator 3/2
[1, 5/4, 3/2, 12/7] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 12/7] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 7/4]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 7/4]
[1, 5/4, 14/9, 7/4] [1, 49/40, 14/9, 7/4]

Circle of alternative tetrads generator 3/2
[1, 9/7, 3/2, 9/5] [1, 8/7, 3/2, 45/28] [1, 8/7, 3/2, 12/7]
[1, 4/3, 3/2, 9/5] [1, 8/7, 3/2, 5/3] [1, 8/7, 3/2, 12/7]
[1, 4/3, 3/2, 9/5] [1, 8/7, 3/2, 5/3] [1, 8/7, 3/2, 7/4]
[1, 4/3, 3/2, 28/15] [1, 7/6, 3/2, 5/3] [1, 7/6, 3/2, 7/4]
[1, 4/3, 14/9, 28/15] [1, 7/6, 14/9, 5/3] [1, 7/6, 14/9, 7/4]


Porcupine[15]
rms error 6.809 cents

[3, 5, -6, 1, -18, -28]
[1, 25/24, 10/9, 8/7, 6/5, 5/4, 4/3, 25/18, 35/24, 3/2, 8/5, 5/3, 7/4,
9/5, 48/25]
[25/24, 8/7, 5/4, 25/18, 3/2, 5/3, 9/5, 1, 10/9, 6/5, 4/3, 35/24, 8/5,
7/4, 48/25]

Circle of major and minor tetrads generator 10/9
[1, 32/25, 14/9, 7/4] [1, 6/5, 14/9, 7/4]
[1, 32/25, 14/9, 7/4] [1, 6/5, 14/9, 7/4]
[1, 32/25, 14/9, 7/4] [1, 6/5, 14/9, 7/4]
[1, 32/25, 3/2, 7/4] [1, 6/5, 3/2, 7/4]
[1, 32/25, 3/2, 7/4] [1, 6/5, 3/2, 7/4]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 7/4]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 7/4]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 7/4]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 7/4]
[1, 5/4, 3/2, 12/7] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 12/7] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 12/7] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 12/7] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 12/7] [1, 25/21, 3/2, 12/7]
[1, 5/4, 3/2, 12/7] [1, 25/21, 3/2, 12/7]

Circle of alternative tetrads generator 10/9
[1, 4/3, 14/9, 28/15] [1, 7/6, 14/9, 42/25] [1, 7/6, 14/9, 7/4]
[1, 4/3, 14/9, 9/5] [1, 7/6, 14/9, 42/25] [1, 7/6, 14/9, 7/4]
[1, 4/3, 14/9, 9/5] [1, 7/6, 14/9, 5/3] [1, 7/6, 14/9, 7/4]
[1, 4/3, 3/2, 9/5] [1, 7/6, 3/2, 5/3] [1, 7/6, 3/2, 7/4]
[1, 4/3, 3/2, 9/5] [1, 7/6, 3/2, 5/3] [1, 7/6, 3/2, 7/4]
[1, 4/3, 3/2, 9/5] [1, 7/6, 3/2, 5/3] [1, 7/6, 3/2, 7/4]
[1, 4/3, 3/2, 9/5] [1, 8/7, 3/2, 5/3] [1, 8/7, 3/2, 7/4]
[1, 4/3, 3/2, 9/5] [1, 8/7, 3/2, 5/3] [1, 8/7, 3/2, 7/4]
[1, 4/3, 3/2, 9/5] [1, 8/7, 3/2, 5/3] [1, 8/7, 3/2, 7/4]
[1, 4/3, 3/2, 9/5] [1, 8/7, 3/2, 5/3] [1, 8/7, 3/2, 12/7]
[1, 4/3, 3/2, 9/5] [1, 8/7, 3/2, 5/3] [1, 8/7, 3/2, 12/7]
[1, 4/3, 3/2, 9/5] [1, 8/7, 3/2, 5/3] [1, 8/7, 3/2, 12/7]
[1, 9/7, 3/2, 9/5] [1, 8/7, 3/2, 5/3] [1, 8/7, 3/2, 12/7]
[1, 9/7, 3/2, 9/5] [1, 8/7, 3/2, 5/3] [1, 8/7, 3/2, 12/7]
[1, 9/7, 3/2, 9/5] [1, 8/7, 3/2, 5/3] [1, 8/7, 3/2, 12/7]


Superkleismic[15]
rms error 4.0527 cents

[9, 10, -3, -5, -30, -35]
[1, 21/20, 27/25, 8/7, 6/5, 63/50, 21/16, 25/18, 35/24, 32/21, 63/40,
5/3, 7/4, 50/27, 40/21]
[27/25, 21/16, 63/40, 40/21, 8/7, 25/18, 5/3, 1, 6/5, 35/24, 7/4,
21/20, 63/50, 32/21, 50/27]

Circle of major and minor tetrads generator 6/5
[1, 63/50, 32/21, 7/4] [1, 6/5, 32/21, 7/4]
[1, 63/50, 32/21, 7/4] [1, 6/5, 32/21, 7/4]
[1, 63/50, 32/21, 7/4] [1, 6/5, 32/21, 7/4]
[1, 63/50, 32/21, 7/4] [1, 6/5, 32/21, 7/4]
[1, 63/50, 32/21, 7/4] [1, 6/5, 32/21, 7/4]
[1, 63/50, 32/21, 7/4] [1, 6/5, 32/21, 7/4]
[1, 63/50, 32/21, 7/4] [1, 6/5, 32/21, 7/4]
[1, 63/50, 32/21, 7/4] [1, 6/5, 32/21, 7/4]
[1, 63/50, 32/21, 7/4] [1, 6/5, 32/21, 7/4]
[1, 63/50, 3/2, 7/4] [1, 6/5, 3/2, 7/4]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 7/4]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 7/4]
[1, 5/4, 3/2, 12/7] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 12/7] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 12/7] [1, 25/21, 3/2, 12/7]

Circle of alternative tetrads generator 6/5
[1, 4/3, 32/21, 50/27] [1, 7/6, 32/21, 42/25] [1, 7/6, 32/21, 7/4]
[1, 4/3, 32/21, 50/27] [1, 7/6, 32/21, 5/3] [1, 7/6, 32/21, 7/4]
[1, 4/3, 32/21, 50/27] [1, 7/6, 32/21, 5/3] [1, 7/6, 32/21, 7/4]
[1, 4/3, 32/21, 50/27] [1, 8/7, 32/21, 5/3] [1, 8/7, 32/21, 7/4]
[1, 4/3, 32/21, 50/27] [1, 8/7, 32/21, 5/3] [1, 8/7, 32/21, 7/4]
[1, 4/3, 32/21, 50/27] [1, 8/7, 32/21, 5/3] [1, 8/7, 32/21, 7/4]
[1, 21/16, 32/21, 50/27] [1, 8/7, 32/21, 5/3] [1, 8/7, 32/21, 7/4]
[1, 21/16, 32/21, 50/27] [1, 8/7, 32/21, 5/3] [1, 8/7, 32/21, 7/4]
[1, 21/16, 32/21, 9/5] [1, 8/7, 32/21, 5/3] [1, 8/7, 32/21, 7/4]
[1, 21/16, 3/2, 9/5] [1, 8/7, 3/2, 5/3] [1, 8/7, 3/2, 7/4]
[1, 21/16, 3/2, 9/5] [1, 8/7, 3/2, 5/3] [1, 8/7, 3/2, 7/4]
[1, 21/16, 3/2, 9/5] [1, 8/7, 3/2, 5/3] [1, 8/7, 3/2, 7/4]
[1, 21/16, 3/2, 9/5] [1, 8/7, 3/2, 5/3] [1, 8/7, 3/2, 12/7]
[1, 21/16, 3/2, 9/5] [1, 8/7, 3/2, 5/3] [1, 8/7, 3/2, 12/7]
[1, 21/16, 3/2, 9/5] [1, 8/7, 3/2, 5/3] [1, 8/7, 3/2, 12/7]


Quartaminorthirds[15]
rms error 3.066 cents

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

Circle of major and minor tetrads generator 21/20
[1, 5/4, 3/2, 12/7] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 12/7] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 12/7] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 7/4]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 7/4]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 7/4]
[1, 5/4, 32/21, 7/4] [1, 6/5, 32/21, 7/4]
[1, 5/4, 32/21, 7/4] [1, 6/5, 32/21, 7/4]
[1, 5/4, 32/21, 7/4] [1, 6/5, 32/21, 7/4]
[1, 5/4, 32/21, 7/4] [1, 6/5, 32/21, 7/4]
[1, 32/25, 32/21, 7/4] [1, 6/5, 32/21, 7/4]
[1, 32/25, 32/21, 7/4] [1, 49/40, 32/21, 7/4]
[1, 32/25, 32/21, 7/4] [1, 49/40, 32/21, 7/4]
[1, 32/25, 32/21, 7/4] [1, 49/40, 32/21, 7/4]
[1, 32/25, 32/21, 7/4] [1, 49/40, 32/21, 7/4]

Circle of alternative tetrads generator 21/20
[1, 21/16, 3/2, 9/5] [1, 8/7, 3/2, 80/49] [1, 8/7, 3/2, 12/7]
[1, 21/16, 3/2, 9/5] [1, 8/7, 3/2, 80/49] [1, 8/7, 3/2, 12/7]
[1, 21/16, 3/2, 64/35] [1, 8/7, 3/2, 80/49] [1, 8/7, 3/2, 12/7]
[1, 21/16, 3/2, 64/35] [1, 8/7, 3/2, 80/49] [1, 8/7, 3/2, 7/4]
[1, 21/16, 3/2, 64/35] [1, 8/7, 3/2, 5/3] [1, 8/7, 3/2, 7/4]
[1, 21/16, 3/2, 64/35] [1, 8/7, 3/2, 5/3] [1, 8/7, 3/2, 7/4]
[1, 21/16, 32/21, 64/35] [1, 8/7, 32/21, 5/3] [1, 8/7, 32/21, 7/4]
[1, 21/16, 32/21, 64/35] [1, 8/7, 32/21, 5/3] [1, 8/7, 32/21, 7/4]
[1, 21/16, 32/21, 64/35] [1, 8/7, 32/21, 5/3] [1, 8/7, 32/21, 7/4]
[1, 4/3, 32/21, 64/35] [1, 8/7, 32/21, 5/3] [1, 8/7, 32/21, 7/4]
[1, 4/3, 32/21, 64/35] [1, 8/7, 32/21, 5/3] [1, 8/7, 32/21, 7/4]
[1, 4/3, 32/21, 64/35] [1, 8/7, 32/21, 5/3] [1, 8/7, 32/21, 7/4]
[1, 4/3, 32/21, 64/35] [1, 7/6, 32/21, 5/3] [1, 7/6, 32/21, 7/4]
[1, 4/3, 32/21, 64/35] [1, 7/6, 32/21, 5/3] [1, 7/6, 32/21, 7/4]
[1, 4/3, 32/21, 64/35] [1, 7/6, 32/21, 5/3] [1, 7/6, 32/21, 7/4]

Message: 6682 - Contents - Hide Contents

Date: Tue, 25 Mar 2003 16:08:57

Subject: Re: Kleismic[8] and Hemifourths[9]

From: Carl Lumma

>Kleismic[8]
>rms error 12.274


Yeah, it's a good 'un.  But how'd you pick 8?  Once again: where
are you getting the n's????

-Carl

Message: 6683 - Contents - Hide Contents

Date: Tue, 25 Mar 2003 03:52:55

Subject: Nineteen notes, 49/48 chroma

From: Gene Ward Smith

Semisixths[19]
rms error 5.053 cents

[1, 4, -9, 4, -17, -32]
[1, 28/27, 15/14, 10/9, 7/6, 6/5, 5/4, 9/7, 4/3, 7/5, 10/7, 
3/2, 14/9, 8/5, 5/3, 12/7, 9/5, 28/15, 27/14]
[8/5, 28/27, 4/3, 12/7, 10/9, 10/7, 28/15, 6/5, 14/9, 
1, 9/7, 5/3, 15/14, 7/5, 9/5, 7/6, 3/2, 27/14, 5/4]

Circle of major and minor tetrads generator 9/7
[1, 5/4, 3/2, 7/4] [1, 49/40, 3/2, 7/4]
[1, 5/4, 3/2, 7/4] [1, 49/40, 3/2, 7/4]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 7/4]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 7/4]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 7/4]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 7/4]
[1, 5/4, 3/2, 12/7] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 12/7] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 12/7] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 12/7] [1, 6/5, 3/2, 12/7]
[1, 56/45, 3/2, 12/7] [1, 6/5, 3/2, 12/7]
[1, 56/45, 3/2, 12/7] [1, 6/5, 3/2, 12/7]
[1, 56/45, 40/27, 12/7] [1, 6/5, 40/27, 12/7]
[1, 56/45, 40/27, 12/7] [1, 6/5, 40/27, 12/7]
[1, 56/45, 40/27, 12/7] [1, 6/5, 40/27, 12/7]
[1, 56/45, 40/27, 12/7] [1, 6/5, 40/27, 12/7]
[1, 56/45, 40/27, 12/7] [1, 6/5, 40/27, 12/7]
[1, 56/45, 40/27, 12/7] [1, 6/5, 40/27, 12/7]
[1, 56/45, 40/27, 12/7] [1, 6/5, 40/27, 12/7]


Circle of alternative tetrads generator 9/7
[1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 5/3] [1, 7/6, 3/2, 7/4]
[1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 5/3] [1, 7/6, 3/2, 7/4]
[1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 5/3] [1, 7/6, 3/2, 7/4]
[1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 5/3] [1, 7/6, 3/2, 7/4]
[1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 5/3] [1, 7/6, 3/2, 7/4]
[1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 5/3] [1, 7/6, 3/2, 7/4]
[1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 5/3] [1, 7/6, 3/2, 12/7]
[1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 5/3] [1, 7/6, 3/2, 12/7]
[1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 5/3] [1, 7/6, 3/2, 12/7]
[1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 5/3] [1, 7/6, 3/2, 12/7]
[1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 5/3] [1, 7/6, 3/2, 12/7]
[1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 5/3] [1, 7/6, 3/2, 12/7]
[1, 9/7, 40/27, 9/5] [1, 7/6, 40/27, 5/3] [1, 7/6, 40/27, 12/7]
[1, 9/7, 40/27, 9/5] [1, 8/7, 40/27, 5/3] [1, 8/7, 40/27, 12/7]
[1, 9/7, 40/27, 16/9] [1, 8/7, 40/27, 5/3] [1, 8/7, 40/27, 12/7]
[1, 9/7, 40/27, 16/9] [1, 8/7, 40/27, 5/3] [1, 8/7, 40/27, 12/7]
[1, 9/7, 40/27, 16/9] [1, 8/7, 40/27, 5/3] [1, 8/7, 40/27, 12/7]
[1, 9/7, 40/27, 16/9] [1, 8/7, 40/27, 80/49] [1, 8/7, 40/27, 12/7]
[1, 32/25, 40/27, 16/9] [1, 8/7, 40/27, 80/49] [1, 8/7, 40/27, 12/7]


Magic[19]
rms error 4.139 cents

[1, 4, -9, 4, -17, -32]
[1, 25/24, 15/14, 10/9, 7/6, 6/5, 5/4, 9/7, 4/3, 25/18, 35/24, 
3/2, 14/9, 8/5, 5/3, 12/7, 9/5, 15/8, 27/14]
[10/9, 25/18, 12/7, 15/14, 4/3, 5/3, 25/24, 9/7, 8/5, 
1, 5/4, 14/9, 27/14, 6/5, 3/2, 15/8, 7/6, 35/24, 9/5]

Circle of major and minor tetrads generator 5/4
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 7/4]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 7/4]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 7/4]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 7/4]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 7/4]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 7/4]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 7/4]
[1, 5/4, 3/2, 12/7] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 12/7] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 12/7] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 12/7] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 12/7] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 12/7] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 12/7] [1, 6/5, 3/2, 12/7]
[1, 5/4, 40/27, 12/7] [1, 6/5, 40/27, 12/7]
[1, 5/4, 40/27, 12/7] [1, 25/21, 40/27, 12/7]
[1, 5/4, 40/27, 12/7] [1, 25/21, 40/27, 12/7]
[1, 5/4, 40/27, 12/7] [1, 25/21, 40/27, 12/7]
[1, 60/49, 40/27, 12/7] [1, 25/21, 40/27, 12/7]


Circle of alternative tetrads generator 5/4
[1, 21/16, 3/2, 9/5] [1, 7/6, 3/2, 27/16] [1, 7/6, 3/2, 7/4]
[1, 21/16, 3/2, 9/5] [1, 7/6, 3/2, 27/16] [1, 7/6, 3/2, 7/4]
[1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 27/16] [1, 7/6, 3/2, 7/4]
[1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 27/16] [1, 7/6, 3/2, 7/4]
[1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 5/3] [1, 7/6, 3/2, 7/4]
[1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 5/3] [1, 7/6, 3/2, 7/4]
[1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 5/3] [1, 7/6, 3/2, 7/4]
[1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 5/3] [1, 7/6, 3/2, 12/7]
[1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 5/3] [1, 7/6, 3/2, 12/7]
[1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 5/3] [1, 7/6, 3/2, 12/7]
[1, 9/7, 3/2, 16/9] [1, 7/6, 3/2, 5/3] [1, 7/6, 3/2, 12/7]
[1, 9/7, 3/2, 16/9] [1, 7/6, 3/2, 5/3] [1, 7/6, 3/2, 12/7]
[1, 9/7, 3/2, 16/9] [1, 8/7, 3/2, 5/3] [1, 8/7, 3/2, 12/7]
[1, 9/7, 3/2, 16/9] [1, 8/7, 3/2, 5/3] [1, 8/7, 3/2, 12/7]
[1, 9/7, 40/27, 16/9] [1, 8/7, 40/27, 5/3] [1, 8/7, 40/27, 12/7]
[1, 9/7, 40/27, 16/9] [1, 8/7, 40/27, 5/3] [1, 8/7, 40/27, 12/7]
[1, 9/7, 40/27, 16/9] [1, 8/7, 40/27, 5/3] [1, 8/7, 40/27, 12/7]
[1, 9/7, 40/27, 16/9] [1, 8/7, 40/27, 5/3] [1, 8/7, 40/27, 12/7]
[1, 9/7, 40/27, 16/9] [1, 8/7, 40/27, 5/3] [1, 8/7, 40/27, 12/7]


Meantone[19]
rms error 3.665 cents

[1, 4, 10, 4, 13, 12]
[1, 21/20, 15/14, 9/8, 7/6, 6/5, 5/4, 9/7, 4/3, 7/5, 10/7, 3/2, 
14/9, 8/5, 5/3, 12/7, 9/5, 15/8, 27/14]
[12/7, 9/7, 27/14, 10/7, 15/14, 8/5, 6/5, 9/5, 4/3, 
1, 3/2, 9/8, 5/3, 5/4, 15/8, 7/5, 21/20, 14/9, 7/6]

Circle of major and minor tetrads
[1, 5/4, 3/2, 7/4] [1, 49/40, 3/2, 7/4]
[1, 5/4, 3/2, 7/4] [1, 49/40, 3/2, 7/4]
[1, 5/4, 3/2, 7/4] [1, 49/40, 3/2, 7/4]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 7/4]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 7/4]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 7/4]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 7/4]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 7/4]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 7/4]
[1, 5/4, 3/2, 12/7] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 12/7] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 12/7] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 12/7] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 12/7] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 12/7] [1, 6/5, 3/2, 12/7]
[1, 60/49, 3/2, 12/7] [1, 6/5, 3/2, 12/7]
[1, 60/49, 3/2, 12/7] [1, 6/5, 3/2, 12/7]
[1, 60/49, 3/2, 12/7] [1, 6/5, 3/2, 12/7]
[1, 60/49, 72/49, 12/7] [1, 6/5, 72/49, 12/7]


Circle of alternative tetrads
[1, 21/16, 3/2, 49/27] [1, 7/6, 3/2, 5/3] [1, 7/6, 3/2, 7/4]
[1, 21/16, 3/2, 49/27] [1, 7/6, 3/2, 5/3] [1, 7/6, 3/2, 7/4]
[1, 21/16, 3/2, 9/5] [1, 7/6, 3/2, 5/3] [1, 7/6, 3/2, 7/4]
[1, 21/16, 3/2, 9/5] [1, 7/6, 3/2, 5/3] [1, 7/6, 3/2, 7/4]
[1, 21/16, 3/2, 9/5] [1, 7/6, 3/2, 5/3] [1, 7/6, 3/2, 7/4]
[1, 21/16, 3/2, 9/5] [1, 7/6, 3/2, 5/3] [1, 7/6, 3/2, 7/4]
[1, 21/16, 3/2, 9/5] [1, 7/6, 3/2, 5/3] [1, 7/6, 3/2, 7/4]
[1, 21/16, 3/2, 9/5] [1, 7/6, 3/2, 5/3] [1, 7/6, 3/2, 7/4]
[1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 5/3] [1, 7/6, 3/2, 7/4]
[1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 5/3] [1, 7/6, 3/2, 12/7]
[1, 9/7, 3/2, 9/5] [1, 8/7, 3/2, 5/3] [1, 8/7, 3/2, 12/7]
[1, 9/7, 3/2, 9/5] [1, 8/7, 3/2, 5/3] [1, 8/7, 3/2, 12/7]
[1, 9/7, 3/2, 9/5] [1, 8/7, 3/2, 5/3] [1, 8/7, 3/2, 12/7]
[1, 9/7, 3/2, 9/5] [1, 8/7, 3/2, 5/3] [1, 8/7, 3/2, 12/7]
[1, 9/7, 3/2, 9/5] [1, 8/7, 3/2, 5/3] [1, 8/7, 3/2, 12/7]
[1, 9/7, 3/2, 9/5] [1, 8/7, 3/2, 5/3] [1, 8/7, 3/2, 12/7]
[1, 9/7, 3/2, 9/5] [1, 8/7, 3/2, 80/49] [1, 8/7, 3/2, 12/7]
[1, 9/7, 3/2, 9/5] [1, 8/7, 3/2, 80/49] [1, 8/7, 3/2, 12/7]
[1, 9/7, 72/49, 9/5] [1, 8/7, 72/49, 80/49] [1, 8/7, 72/49, 12/7]

Message: 6684 - Contents - Hide Contents

Date: Tue, 25 Mar 2003 03:55:09

Subject: Re: Pajara[10]

From: Gene Ward Smith

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

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

>> This is a 25/24 chroma system, as well as being a 28/27 and 49/48
>> chroma system,
> 

> as shown in "the forms of tonality"


Didn't you tell me you were sending me some stuff, BTW?


>> so I don't give circles for major and minor tetrads,
>> but rather another circle for subminorish stuff. 
> 

> i don't get the logic of the "so". what implies what, exactly?


The major and minor circles are the same.

Message: 6685 - Contents - Hide Contents

Date: Tue, 25 Mar 2003 04:39:31

Subject: Re: Pajara[10]

From: wallyesterpaulrus

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

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

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

>>> This is a 25/24 chroma system, as well as being a 28/27 and 
49/48
>>> chroma system,
>> 

>> as shown in "the forms of tonality"
> 

> Didn't you tell me you were sending me some stuff, BTW?


i'm sending a lot of people a lot of stuff. last call for snail-mail 
addresses -- i'm sending out a paper my a "famous music theorist", 
and if you need a copy of _the forms of tonality_ too, let me know.


>>> so I don't give circles for major and minor tetrads,
>>> but rather another circle for subminorish stuff. 
>> 

>> i don't get the logic of the "so". what implies what, exactly?
> 

> The major and minor circles are the same.


is this the antecedent or the consequent?

Message: 6686 - Contents - Hide Contents

Date: Tue, 25 Mar 2003 07:03:04

Subject: Re: Pajara[10]

From: Gene Ward Smith

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


>>> i don't get the logic of the "so". what implies what, exactly?
>> 

>> The major and minor circles are the same.
> 

> is this the antecedent or the consequent?


It's a consequence of 25/24 and 49/48 both being chromas. Didn't we
get into this once?

Message: 6687 - Contents - Hide Contents

Date: Tue, 25 Mar 2003 07:11:20

Subject: Bleeding edge 25/24 systems

From: Gene Ward Smith

For those who enjoy life there, 4096/3645[13] has a fifth which is a
mere 30.5 cents sharp. If that isn't enough, you can move to
2560/2187[13] which is 41.2 cents sharp. If you prefer flat,
729/640[11] has an rms fifth which is 40 cents flat.

Message: 6688 - Contents - Hide Contents

Date: Wed, 26 Mar 2003 04:22:33

Subject: Designer chromas

From: Gene Ward Smith

It occurred to me that rather than checking known systems, I could try
to specifify what result I wanted and find systems which gave it. I
tried this using the idea that it would be nice to have something where
5/4 changed to 11/8 and 3/2 to 13/8. This means that (11/8)/(5/4) =
11/10 and (13/8)/(3/2) = 13/12 are both chromas, so that
(11/10)/(13/12) = 66/65 is a comma. I started by taking 11-limit
wedgies, adding 66/65 to get a 13-limit wedgie, and selecting the ones
which seemed most interesting. I got supermajor seconds on 20 notes,
and two members of the ever-proliferating meantone family on 14 and 17
notes. In practice these can both be taken as 31-et and identified,
which would mean reducing by [66/65,81/80,99/98,105/104,121/120], the
TM basis for h31. This would slightly change what I have here, in that
15/11 is equated to 11/8, 27/14 to 21/11 and 14/9 to 11/7, etc.

Meantone[14]
rms error 10.260

[66/65, 81/80, 99/98, 105/104]
[1, 4, 10, 18, 15, 4, 13, 25, 20, 12, 28, 20, 16, 5, -15]
[1, 21/20, 14/13, 9/8, 6/5, 5/4, 4/3, 7/5, 10/7, 3/2, 8/5, 5/3, 9/5, 13/7]
[10/7, 14/13, 8/5, 6/5, 9/5, 4/3, 1, 3/2, 9/8, 5/3, 5/4, 13/7, 7/5, 21/20]

Circles of tetrads
[1, 5/4, 3/2, 7/4] [1, 13/10, 3/2, 13/7]
[1, 5/4, 3/2, 7/4] [1, 13/10, 3/2, 13/7]
[1, 5/4, 3/2, 7/4] [1, 13/10, 3/2, 13/7]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 13/7]
[1, 5/4, 3/2, 8/5] [1, 6/5, 3/2, 13/7]
[1, 5/4, 3/2, 8/5] [1, 6/5, 3/2, 13/7]
[1, 5/4, 3/2, 8/5] [1, 6/5, 3/2, 13/7]
[1, 5/4, 3/2, 8/5] [1, 6/5, 3/2, 13/7]
[1, 5/4, 3/2, 8/5] [1, 6/5, 3/2, 13/7]
[1, 5/4, 3/2, 8/5] [1, 6/5, 3/2, 12/7]
[1, 8/7, 3/2, 8/5] [1, 6/5, 3/2, 12/7]
[1, 8/7, 3/2, 8/5] [1, 6/5, 3/2, 12/7]
[1, 8/7, 3/2, 8/5] [1, 6/5, 3/2, 12/7]
[1, 8/7, 15/11, 8/5] [1, 6/5, 15/11, 12/7]

Circles of alternative tetrads
[1, 7/5, 3/2, 35/18] [1, 7/6, 3/2, 5/3] [1, 7/6, 3/2, 7/4]
[1, 7/5, 3/2, 35/18] [1, 7/6, 3/2, 5/3] [1, 7/6, 3/2, 7/4]
[1, 7/5, 3/2, 9/5] [1, 7/6, 3/2, 5/3] [1, 7/6, 3/2, 7/4]
[1, 7/5, 3/2, 9/5] [1, 7/6, 3/2, 5/3] [1, 7/6, 3/2, 7/4]
[1, 7/5, 3/2, 9/5] [1, 7/6, 3/2, 5/3] [1, 7/6, 3/2, 8/5]
[1, 7/5, 3/2, 9/5] [1, 14/13, 3/2, 5/3] [1, 14/13, 3/2, 8/5]
[1, 7/5, 3/2, 9/5] [1, 14/13, 3/2, 5/3] [1, 14/13, 3/2, 8/5]
[1, 7/5, 3/2, 9/5] [1, 14/13, 3/2, 5/3] [1, 14/13, 3/2, 8/5]
[1, 9/7, 3/2, 9/5] [1, 14/13, 3/2, 5/3] [1, 14/13, 3/2, 8/5]
[1, 9/7, 3/2, 9/5] [1, 14/13, 3/2, 5/3] [1, 14/13, 3/2, 8/5]
[1, 9/7, 3/2, 9/5] [1, 14/13, 3/2, 5/3] [1, 14/13, 3/2, 8/5]
[1, 9/7, 3/2, 9/5] [1, 14/13, 3/2, 20/13] [1, 14/13, 3/2, 8/5]
[1, 9/7, 3/2, 9/5] [1, 14/13, 3/2, 20/13] [1, 14/13, 3/2, 8/5]
[1, 9/7, 15/11, 9/5] [1, 14/13, 15/11, 20/13] [1, 14/13, 15/11, 8/5]


Meanplop[17]
rms error 10.082 cents

[66/65, 81/80, 126/125, 385/384]
[1, 4, 10, -13, -16, 4, 13, -24, -29, 12, -44, -52, -71, -82, -7]
[1, 21/20, 15/14, 9/8, 6/5, 5/4, 9/7, 4/3, 7/5, 10/7, 3/2, 14/9, 8/5,
5/3, 9/5, 15/8, 27/14]
[9/7, 27/14, 10/7, 15/14, 8/5, 6/5, 9/5, 4/3, 1, 3/2, 9/8, 5/3, 5/4,
15/8, 7/5, 21/20, 14/9]

Circles of tetrads
[1, 5/4, 3/2, 7/4] [1, 12/11, 3/2, 14/9]
[1, 5/4, 3/2, 7/4] [1, 12/11, 3/2, 14/9]
[1, 5/4, 3/2, 7/4] [1, 12/11, 3/2, 14/9]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 14/9]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 14/9]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 14/9]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 14/9]
[1, 5/4, 3/2, 27/14] [1, 6/5, 3/2, 14/9]
[1, 5/4, 3/2, 27/14] [1, 6/5, 3/2, 14/9]
[1, 5/4, 3/2, 27/14] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 27/14] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 27/14] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 27/14] [1, 6/5, 3/2, 12/7]
[1, 11/8, 3/2, 27/14] [1, 6/5, 3/2, 12/7]
[1, 11/8, 3/2, 27/14] [1, 6/5, 3/2, 12/7]
[1, 11/8, 3/2, 27/14] [1, 6/5, 3/2, 12/7]
[1, 11/8, 13/8, 27/14] [1, 6/5, 13/8, 12/7]

Circles of alternative tetrads
[1, 7/6, 3/2, 18/11] [1, 7/6, 3/2, 5/3] [1, 7/6, 3/2, 7/4]
[1, 7/6, 3/2, 18/11] [1, 7/6, 3/2, 5/3] [1, 7/6, 3/2, 7/4]
[1, 7/6, 3/2, 9/5] [1, 7/6, 3/2, 5/3] [1, 7/6, 3/2, 7/4]
[1, 7/6, 3/2, 9/5] [1, 7/6, 3/2, 5/3] [1, 7/6, 3/2, 7/4]
[1, 7/6, 3/2, 9/5] [1, 7/6, 3/2, 5/3] [1, 7/6, 3/2, 7/4]
[1, 7/6, 3/2, 9/5] [1, 7/6, 3/2, 5/3] [1, 7/6, 3/2, 7/4]
[1, 7/6, 3/2, 9/5] [1, 7/6, 3/2, 5/3] [1, 7/6, 3/2, 7/4]
[1, 7/6, 3/2, 9/5] [1, 7/6, 3/2, 5/3] [1, 7/6, 3/2, 27/14]
[1, 9/7, 3/2, 9/5] [1, 9/7, 3/2, 5/3] [1, 9/7, 3/2, 27/14]
[1, 9/7, 3/2, 9/5] [1, 9/7, 3/2, 5/3] [1, 9/7, 3/2, 27/14]
[1, 9/7, 3/2, 9/5] [1, 9/7, 3/2, 5/3] [1, 9/7, 3/2, 27/14]
[1, 9/7, 3/2, 9/5] [1, 9/7, 3/2, 5/3] [1, 9/7, 3/2, 27/14]
[1, 9/7, 3/2, 9/5] [1, 9/7, 3/2, 5/3] [1, 9/7, 3/2, 27/14]
[1, 9/7, 3/2, 9/5] [1, 9/7, 3/2, 5/3] [1, 9/7, 3/2, 27/14]
[1, 9/7, 3/2, 9/5] [1, 9/7, 3/2, 11/6] [1, 9/7, 3/2, 27/14]
[1, 9/7, 3/2, 9/5] [1, 9/7, 3/2, 11/6] [1, 9/7, 3/2, 27/14]
[1, 9/7, 13/8, 9/5] [1, 9/7, 13/8, 11/6] [1, 9/7, 13/8, 27/14]


Supermajor seconds[20]
rms error 9.326 cents
commas [66/65, 81/80, 99/98, 385/384]

[3, 12, -1, -8, -17, 12, -10, -23, -38, -36, -60, -84, -19, -44, -29]

[1, 33/32, 9/8, 8/7, 7/6, 6/5, 9/7, 21/16, 4/3, 11/8, 16/11, 3/2,
32/21, 11/7, 5/3, 12/7, 7/4, 9/5, 21/11, 35/18]
[6/5, 11/8, 11/7, 9/5, 33/32, 7/6, 4/3, 32/21, 7/4, 1, 8/7, 21/16,
3/2, 12/7, 35/18, 9/8, 9/7, 16/11, 5/3, 21/11]

generator = 8/7

[1, 5/4, 3/2, 21/13] [1, 12/11, 3/2, 12/7]
[1, 5/4, 3/2, 7/4] [1, 12/11, 3/2, 12/7]
[1, 5/4, 3/2, 7/4] [1, 12/11, 3/2, 12/7]
[1, 5/4, 3/2, 7/4] [1, 12/11, 3/2, 12/7]
[1, 5/4, 3/2, 7/4] [1, 12/11, 3/2, 12/7]
[1, 5/4, 3/2, 7/4] [1, 12/11, 3/2, 12/7]
[1, 5/4, 3/2, 7/4] [1, 12/11, 3/2, 12/7]
[1, 5/4, 3/2, 7/4] [1, 12/11, 3/2, 12/7]
[1, 11/8, 3/2, 7/4] [1, 12/11, 3/2, 12/7]
[1, 11/8, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
[1, 11/8, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
[1, 11/8, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
[1, 11/8, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
[1, 11/8, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
[1, 11/8, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
[1, 11/8, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
[1, 11/8, 3/2, 7/4] [1, 6/5, 3/2, 13/7]
[1, 11/8, 13/8, 7/4] [1, 6/5, 13/8, 13/7]
[1, 11/8, 13/8, 7/4] [1, 6/5, 13/8, 13/7]
[1, 11/8, 13/8, 7/4] [1, 6/5, 13/8, 13/7]


[1, 9/7, 3/2, 18/11] [1, 14/13, 3/2, 5/3] [1, 14/13, 3/2, 21/13]
[1, 9/7, 3/2, 18/11] [1, 14/13, 3/2, 5/3] [1, 14/13, 3/2, 7/4]
[1, 9/7, 3/2, 18/11] [1, 14/13, 3/2, 5/3] [1, 14/13, 3/2, 7/4]
[1, 9/7, 3/2, 18/11] [1, 14/13, 3/2, 5/3] [1, 14/13, 3/2, 7/4]
[1, 9/7, 3/2, 18/11] [1, 7/6, 3/2, 5/3] [1, 7/6, 3/2, 7/4]
[1, 9/7, 3/2, 18/11] [1, 7/6, 3/2, 5/3] [1, 7/6, 3/2, 7/4]
[1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 5/3] [1, 7/6, 3/2, 7/4]
[1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 5/3] [1, 7/6, 3/2, 7/4]
[1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 5/3] [1, 7/6, 3/2, 7/4]
[1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 5/3] [1, 7/6, 3/2, 7/4]
[1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 5/3] [1, 7/6, 3/2, 7/4]
[1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 11/6] [1, 7/6, 3/2, 7/4]
[1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 11/6] [1, 7/6, 3/2, 7/4]
[1, 7/5, 3/2, 9/5] [1, 7/6, 3/2, 11/6] [1, 7/6, 3/2, 7/4]
[1, 7/5, 3/2, 9/5] [1, 7/6, 3/2, 11/6] [1, 7/6, 3/2, 7/4]
[1, 7/5, 3/2, 9/5] [1, 7/6, 3/2, 11/6] [1, 7/6, 3/2, 7/4]
[1, 7/5, 3/2, 9/5] [1, 7/6, 3/2, 11/6] [1, 7/6, 3/2, 7/4]
[1, 7/5, 13/8, 9/5] [1, 7/6, 13/8, 11/6] [1, 7/6, 13/8, 7/4]
[1, 7/5, 13/8, 9/5] [1, 7/6, 13/8, 11/6] [1, 7/6, 13/8, 7/4]
[1, 7/5, 13/8, 9/5] [1, 7/6, 13/8, 11/6] [1, 7/6, 13/8, 7/4]

Message: 6689 - Contents - Hide Contents

Date: Wed, 26 Mar 2003 04:42:17

Subject: Re: Kleismic[8] and Hemifourths[9]

From: Gene Ward Smith

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

>> Kleismic[8]
>> rms error 12.274
> 

> Yeah, it's a good 'un.  But how'd you pick 8?  Once again: where
> are you getting the n's????


Now that I've sent you my maple programs, you could find them yourself.

First, read "temper". Now get the wedgie for kleismic. If you know it
comes from the standard vals h19 and h34, you can do

a7v(19, 34) = [6, 5, 3, -6, -12, -7]

If you happen to know it has 49/48 and 126/125 as a TM basis, you can
get it as

a7i(49/48, 126/125) = [6, 5, 3, -6, -12, -7]

Or maybe you just know the kleisma is a comma, and think 49/48 is
also. You can try

a7i(49/48, 15625/15552) = [-12, -10, -6, 12, 24, 14]. You can feed
this into the reduction to standard wedgie function, and get

kleisma:=wedgie7(a7i(49/48, 15625/15552));

       [6, 5, 3, -6, -12, -7]


>


Now you can check and see what kleisma does with various commas

a7d(kleisma, 25/24);

  [4, 6, 9, 11]

Four notes--not much of a scale

a7d(kleisma, 36/35);

  [4, 6, 9, 11]

Same four notes. Oh well. 

a7d(kleisma, 49/48);

  [0, 0, 0, 0]

You didn't forget 49/48 was a comma, did you? :)

a7d(kleisma, 21/20);

  [4, 6, 9, 11]

Darn! This is getting annoying!

a7d(kleisma, 15/14);

  [8, 12, 18, 22]

Success at last! This is the 8-note scale you asked about. The
torsional aspect is benign.

a7d(kleisma, 28/27);

  [-15, -24, -35, -42]

Nice number of notes, but the equivalences from 28/27 are a little goofy.

a7d(kleisma, 16/15);

  [-11, -18, -26, -31]

Same comment, but probably worth exploring.

Message: 6690 - Contents - Hide Contents

Date: Wed, 26 Mar 2003 08:45:33

Subject: More nifty 15/14 chroma scales

From: Gene Ward Smith

Nonkleismic[12]
rms error 3.320 cents

[126/125, 1728/1715]
[10, 9, 7, -9, -17, -9]
[1, 36/35, 7/6, 6/5, 49/40, 7/5, 10/7, 35/24, 49/30, 5/3, 12/7, 35/18]
[49/30, 35/18, 7/6, 7/5, 5/3, 1, 6/5, 10/7, 12/7, 36/35, 49/40, 35/24]

Circle of 6/5
[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, 7/5, 7/4] [1, 6/5, 7/5, 12/7]
[1, 7/6, 7/5, 7/4] [1, 6/5, 7/5, 12/7]
[1, 7/6, 7/5, 7/4] [1, 6/5, 7/5, 12/7]
[1, 7/6, 7/5, 49/30] [1, 6/5, 7/5, 12/7]
[1, 7/6, 7/5, 49/30] [1, 6/5, 7/5, 12/7]
[1, 7/6, 7/5, 49/30] [1, 6/5, 7/5, 12/7]
[1, 7/6, 7/5, 49/30] [1, 6/5, 7/5, 12/7]
[1, 7/6, 7/5, 49/30] [1, 6/5, 7/5, 8/5]
[1, 7/6, 7/5, 49/30] [1, 6/5, 7/5, 8/5]
[1, 7/6, 7/5, 49/30] [1, 10/9, 7/5, 8/5]

Alternative circle of 6/5
[1, 6/5, 3/2, 9/5] [1, 5/4, 3/2, 9/5]
[1, 6/5, 3/2, 5/3] [1, 5/4, 3/2, 5/3]
[1, 6/5, 7/5, 5/3] [1, 5/4, 7/5, 5/3]
[1, 6/5, 7/5, 5/3] [1, 7/6, 7/5, 5/3]
[1, 6/5, 7/5, 5/3] [1, 7/6, 7/5, 5/3]
[1, 6/5, 7/5, 5/3] [1, 7/6, 7/5, 5/3]
[1, 6/5, 7/5, 5/3] [1, 7/6, 7/5, 5/3]
[1, 6/5, 7/5, 5/3] [1, 7/6, 7/5, 5/3]
[1, 6/5, 7/5, 5/3] [1, 7/6, 7/5, 5/3]
[1, 6/5, 7/5, 5/3] [1, 7/6, 7/5, 5/3]
[1, 6/5, 7/5, 5/3] [1, 7/6, 7/5, 5/3]
[1, 10/9, 7/5, 5/3] [1, 7/6, 7/5, 5/3]


Superpythagorean[12]
rms error 6.410 cents

[64/63, 245/243]
[1, 4, -2, 4, -6, -16]
[1, 28/27, 8/7, 7/6, 9/7, 4/3, 35/24, 3/2, 14/9, 12/7, 7/4, 27/14]
[28/27, 14/9, 7/6, 7/4, 4/3, 1, 3/2, 8/7, 12/7, 9/7, 27/14, 35/24]

Circles of fifths
[1, 5/4, 3/2, 15/8] [1, 9/7, 3/2, 12/7]
[1, 5/4, 3/2, 15/8] [1, 9/7, 3/2, 12/7]
[1, 5/4, 3/2, 7/4] [1, 9/7, 3/2, 12/7]
[1, 7/6, 3/2, 7/4] [1, 9/7, 3/2, 12/7]
[1, 7/6, 3/2, 7/4] [1, 9/7, 3/2, 12/7]
[1, 7/6, 3/2, 7/4] [1, 9/7, 3/2, 12/7]
[1, 7/6, 3/2, 7/4] [1, 9/7, 3/2, 12/7]
[1, 7/6, 3/2, 7/4] [1, 9/7, 3/2, 12/7]
[1, 7/6, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
[1, 7/6, 3/2, 7/4] [1, 6/5, 3/2, 8/5]
[1, 7/6, 3/2, 7/4] [1, 6/5, 3/2, 8/5]
[1, 7/6, 7/5, 7/4] [1, 6/5, 7/5, 8/5]

Alternative circles of fifths
[1, 9/7, 3/2, 27/14] [1, 5/4, 3/2, 5/3] [1, 5/4, 3/2, 15/8]
[1, 9/7, 3/2, 27/14] [1, 5/4, 3/2, 5/3] [1, 5/4, 3/2, 15/8]
[1, 9/7, 3/2, 27/14] [1, 5/4, 3/2, 5/3] [1, 5/4, 3/2, 7/4]
[1, 9/7, 3/2, 27/14] [1, 7/6, 3/2, 5/3] [1, 7/6, 3/2, 7/4]
[1, 9/7, 3/2, 27/14] [1, 7/6, 3/2, 14/9] [1, 7/6, 3/2, 7/4]
[1, 9/7, 3/2, 27/14] [1, 7/6, 3/2, 14/9] [1, 7/6, 3/2, 7/4]
[1, 9/7, 3/2, 27/14] [1, 7/6, 3/2, 14/9] [1, 7/6, 3/2, 7/4]
[1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 14/9] [1, 7/6, 3/2, 7/4]
[1, 6/5, 3/2, 9/5] [1, 7/6, 3/2, 14/9] [1, 7/6, 3/2, 7/4]
[1, 6/5, 3/2, 9/5] [1, 7/6, 3/2, 14/9] [1, 7/6, 3/2, 7/4]
[1, 6/5, 3/2, 9/5] [1, 7/6, 3/2, 14/9] [1, 7/6, 3/2, 7/4]
[1, 6/5, 7/5, 9/5] [1, 7/6, 7/5, 14/9] [1, 7/6, 7/5, 7/4]


Dominant sevenths[7]
rms error 20.163 cents

[36/35, 81/80]
[1, 4, -2, 4, -6, -16]
[1, 8/7, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3, 7/4]
[8/5, 6/5, 7/4, 4/3, 1, 3/2, 8/7, 5/3, 5/4]

Circle of fifths
[1, 5/4, 3/2, 15/8] [1, 5/4, 3/2, 5/3]
[1, 5/4, 3/2, 15/8] [1, 5/4, 3/2, 5/3]
[1, 5/4, 3/2, 7/4] [1, 5/4, 3/2, 5/3]
[1, 6/5, 3/2, 7/4] [1, 6/5, 3/2, 5/3]
[1, 6/5, 3/2, 7/4] [1, 6/5, 3/2, 8/5]
[1, 6/5, 3/2, 7/4] [1, 6/5, 3/2, 8/5]
[1, 6/5, 7/5, 7/4] [1, 6/5, 7/5, 8/5]

Message: 6691 - Contents - Hide Contents

Date: Wed, 26 Mar 2003 00:54:51

Subject: Re: T[7] with 25/24 chroma

From: Carl Lumma

>part of "glassic" is in porcupine. you can hear the consecutive 
>triads -- major, major, minor, minor -- as the root descends stepwise 
>from "8ve" to "5th". thus part of you wants to believe it's 
>mixolydian, but melodically it's singing quite a different tune.


Oh yes, I hear that!  Great work!

-Carl

Message: 6692 - Contents - Hide Contents

Date: Wed, 26 Mar 2003 01:03:37

Subject: Re: Kleismic[8] and Hemifourths[9]

From: Carl Lumma

>> >eah, it's a good 'un.  But how'd you pick 8?  Once again: where
>> are you getting the n's????
>

>Now that I've sent you my maple programs, you could find them yourself.


Thanks for the tutorial.  I glanced at your stuff, but probably won't
get a chance to look at it until next week.


>Same comment, but probably worth exploring.


So, you're doing it by hand, was all I was looking for.  I wonder if
there's any way to predict the good stopping points.  I suppose they
occur when the chromatic uv is simple? .  .  .


>Pelogic[7]
>
>135/128
>[1, 9/8, 5/4, 4/3, 3/2, 8/5, 15/8]
>[5/4, 15/8, 4/3, 1, 3/2, 9/8, 8/5]
//
>Dominant sevenths[7]
>rms error 20.163 cents
//
>Hemifourths[9]
>rms error 12.690
//
>Tertiathirds[9]
>rms error 12.189 cents
//
>Hexadecimal[9]
>rms error 18.585


Ahh -- it's all coming too fast!

-Carl

Message: 6693 - Contents - Hide Contents

Date: Wed, 26 Mar 2003 09:28:33

Subject: Re: Kleismic[8] and Hemifourths[9]

From: Gene Ward Smith

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


> So, you're doing it by hand, was all I was looking for.  I wonder if
> there's any way to predict the good stopping points.  I suppose they
> occur when the chromatic uv is simple? .  .  .


It's not just when it is simple, it's when it moves chord elements
around in a way which is useful. This is why I favor 15/14, 21/20,
25/24 and 36/35.


> Ahh -- it's all coming too fast!


I could look at ennealimmal[27] and ennealimmal[36] instead--that
would put you back to sleep!

Message: 6694 - Contents - Hide Contents

Date: Thu, 27 Mar 2003 22:01:15

Subject: Muggles[13] and Porcupine[14]

From: Gene Ward Smith

Muggles[13]
rms error 8.727 cents

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

Circles of major thirds
[1, 5/4, 3/2, 15/8] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 15/8] [1, 6/5, 3/2, 8/5]
[1, 5/4, 3/2, 15/8] [1, 6/5, 3/2, 8/5]
[1, 5/4, 3/2, 15/8] [1, 6/5, 3/2, 8/5]
[1, 5/4, 3/2, 15/8] [1, 6/5, 3/2, 8/5]
[1, 5/4, 3/2, 15/8] [1, 6/5, 3/2, 8/5]
[1, 5/4, 3/2, 15/8] [1, 6/5, 3/2, 8/5]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 8/5]
[1, 5/4, 7/5, 7/4] [1, 6/5, 7/5, 8/5]
[1, 5/4, 7/5, 7/4] [1, 10/9, 7/5, 8/5]
[1, 5/4, 7/5, 7/4] [1, 10/9, 7/5, 8/5]
[1, 5/4, 7/5, 7/4] [1, 10/9, 7/5, 8/5]
[1, 7/6, 7/5, 7/4] [1, 10/9, 7/5, 8/5]

Alternative circles of major thirds
[1, 6/5, 3/2, 9/5] [1, 5/4, 3/2, 9/5]
[1, 6/5, 3/2, 9/5] [1, 5/4, 3/2, 9/5]
[1, 6/5, 3/2, 9/5] [1, 5/4, 3/2, 9/5]
[1, 6/5, 3/2, 9/5] [1, 5/4, 3/2, 9/5]
[1, 6/5, 3/2, 5/3] [1, 5/4, 3/2, 5/3]
[1, 6/5, 3/2, 5/3] [1, 5/4, 3/2, 5/3]
[1, 6/5, 3/2, 5/3] [1, 5/4, 3/2, 5/3]
[1, 6/5, 3/2, 5/3] [1, 5/4, 3/2, 5/3]
[1, 6/5, 7/5, 5/3] [1, 5/4, 7/5, 5/3]
[1, 10/9, 7/5, 5/3] [1, 5/4, 7/5, 5/3]
[1, 10/9, 7/5, 5/3] [1, 5/4, 7/5, 5/3]
[1, 10/9, 7/5, 5/3] [1, 5/4, 7/5, 5/3]
[1, 10/9, 7/5, 5/3] [1, 7/6, 7/5, 5/3]

Porcupine[14]
rms error 6.809 cents

[64/63, 250/243]
[3, 5, -6, 1, -18, -28]
[1, 10/9, 8/7, 6/5, 5/4, 4/3, 25/18, 35/24, 3/2, 8/5, 5/3, 7/4, 9/5,
25/12]
[25/12, 8/7, 5/4, 25/18, 3/2, 5/3, 9/5, 1, 10/9, 6/5, 4/3, 35/24, 8/5,
7/4]

Circles of tetrads generator = 10/9
[1, 7/6, 7/5, 7/4] [1, 6/5, 7/5, 8/5]
[1, 7/6, 7/5, 7/4] [1, 6/5, 7/5, 8/5]
[1, 7/6, 7/5, 7/4] [1, 6/5, 7/5, 8/5]
[1, 7/6, 3/2, 7/4] [1, 6/5, 3/2, 8/5]
[1, 7/6, 3/2, 7/4] [1, 6/5, 3/2, 8/5]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 8/5]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 8/5]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 8/5]
[1, 5/4, 3/2, 15/8] [1, 6/5, 3/2, 8/5]
[1, 5/4, 3/2, 15/8] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 15/8] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 15/8] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 15/8] [1, 9/7, 3/2, 12/7]
[1, 5/4, 3/2, 15/8] [1, 9/7, 3/2, 12/7]

Circles of alternative tetrads 
[1, 6/5, 7/5, 42/25] [1, 7/6, 7/5, 14/9]
[1, 6/5, 7/5, 9/5] [1, 7/6, 7/5, 14/9]
[1, 6/5, 7/5, 9/5] [1, 7/6, 7/5, 14/9]
[1, 6/5, 3/2, 9/5] [1, 7/6, 7/5, 14/9]
[1, 6/5, 3/2, 9/5] [1, 7/6, 7/5, 14/9]
[1, 6/5, 3/2, 9/5] [1, 7/6, 7/5, 14/9]
[1, 6/5, 3/2, 9/5] [1, 7/6, 7/5, 14/9]
[1, 6/5, 3/2, 9/5] [1, 7/6, 7/5, 14/9]
[1, 6/5, 3/2, 9/5] [1, 7/6, 7/5, 14/9]
[1, 6/5, 3/2, 9/5] [1, 7/6, 7/5, 14/9]
[1, 6/5, 3/2, 9/5] [1, 7/6, 7/5, 14/9]
[1, 6/5, 3/2, 9/5] [1, 7/6, 7/5, 14/9]
[1, 9/7, 3/2, 9/5] [1, 7/6, 7/5, 14/9]
[1, 9/7, 3/2, 9/5] [1, 7/6, 7/5, 14/9]

Message: 6695 - Contents - Hide Contents

Date: Fri, 28 Mar 2003 20:19:56

Subject: Still more 15/14 systems

From: Gene Ward Smith

Hemikleismic[31]
rms error 1.897 cents


[4000/3969, 6144/6125]
[12, 10, -9, -12, -48, -49]
[1, 64/63, 25/24, 21/20, 35/32, 10/9, 8/7, 125/108, 6/5, 128/105, 5/4,
63/50, 21/16, 4/3, 48/35, 25/18, 36/25, 35/24, 3/2, 32/21, 63/40, 8/5,
105/64, 5/3, 125/72, 7/4, 9/5, 64/35, 40/21, 48/25, 63/32]
[64/63, 10/9, 128/105, 4/3, 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, 3/2, 105/64, 9/5, 63/32]

Generator = 35/32

[1, 5/4, 3/2, 15/8] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 15/8] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 15/8] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 15/8] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 15/8] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 15/8] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 15/8] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 15/8] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 15/8] [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, 8/5]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 8/5]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 8/5]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 8/5]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 8/5]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 8/5]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 8/5]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 8/5]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 8/5]
[1, 5/4, 7/5, 7/4] [1, 6/5, 7/5, 8/5]
[1, 5/4, 7/5, 7/4] [1, 6/5, 7/5, 8/5]
[1, 7/6, 7/5, 7/4] [1, 6/5, 7/5, 8/5]
[1, 7/6, 7/5, 7/4] [1, 6/5, 7/5, 8/5]
[1, 7/6, 7/5, 7/4] [1, 6/5, 7/5, 8/5]
[1, 7/6, 7/5, 7/4] [1, 6/5, 7/5, 8/5]
[1, 7/6, 7/5, 7/4] [1, 6/5, 7/5, 8/5]
[1, 7/6, 7/5, 7/4] [1, 6/5, 7/5, 8/5]
[1, 7/6, 7/5, 7/4] [1, 6/5, 7/5, 8/5]
[1, 7/6, 7/5, 7/4] [1, 6/5, 7/5, 8/5]
[1, 7/6, 7/5, 7/4] [1, 28/25, 7/5, 8/5]
[1, 7/6, 7/5, 7/4] [1, 28/25, 7/5, 8/5]


[1, 6/5, 3/2, 9/5] [1, 5/4, 3/2, 25/14]
[1, 6/5, 3/2, 9/5] [1, 5/4, 3/2, 25/14]
[1, 6/5, 3/2, 9/5] [1, 5/4, 3/2, 5/3]
[1, 6/5, 3/2, 9/5] [1, 5/4, 3/2, 5/3]
[1, 6/5, 3/2, 9/5] [1, 5/4, 3/2, 5/3]
[1, 6/5, 3/2, 9/5] [1, 5/4, 3/2, 5/3]
[1, 6/5, 3/2, 9/5] [1, 5/4, 3/2, 5/3]
[1, 6/5, 3/2, 9/5] [1, 5/4, 3/2, 5/3]
[1, 6/5, 3/2, 9/5] [1, 5/4, 3/2, 5/3]
[1, 6/5, 3/2, 9/5] [1, 5/4, 3/2, 5/3]
[1, 6/5, 3/2, 9/5] [1, 5/4, 3/2, 5/3]
[1, 6/5, 3/2, 9/5] [1, 5/4, 3/2, 5/3]
[1, 6/5, 3/2, 9/5] [1, 5/4, 3/2, 5/3]
[1, 6/5, 3/2, 9/5] [1, 5/4, 3/2, 5/3]
[1, 6/5, 3/2, 9/5] [1, 5/4, 3/2, 5/3]
[1, 6/5, 3/2, 9/5] [1, 5/4, 3/2, 5/3]
[1, 6/5, 3/2, 9/5] [1, 5/4, 3/2, 5/3]
[1, 6/5, 3/2, 42/25] [1, 5/4, 3/2, 5/3]
[1, 6/5, 3/2, 42/25] [1, 5/4, 3/2, 5/3]
[1, 6/5, 7/5, 42/25] [1, 5/4, 7/5, 5/3]
[1, 6/5, 7/5, 42/25] [1, 5/4, 7/5, 5/3]
[1, 6/5, 7/5, 42/25] [1, 7/6, 7/5, 5/3]
[1, 6/5, 7/5, 42/25] [1, 7/6, 7/5, 5/3]
[1, 6/5, 7/5, 42/25] [1, 7/6, 7/5, 5/3]
[1, 6/5, 7/5, 42/25] [1, 7/6, 7/5, 5/3]
[1, 6/5, 7/5, 42/25] [1, 7/6, 7/5, 5/3]
[1, 6/5, 7/5, 42/25] [1, 7/6, 7/5, 5/3]
[1, 6/5, 7/5, 42/25] [1, 7/6, 7/5, 5/3]
[1, 6/5, 7/5, 42/25] [1, 7/6, 7/5, 5/3]
[1, 28/25, 7/5, 42/25] [1, 7/6, 7/5, 5/3]
[1, 28/25, 7/5, 42/25] [1, 7/6, 7/5, 5/3]


Amity[35]
rms error 0.846 cents

[4375/4374, 5120/5103]
[5, 13, -17, 9, -41, -76]
[35/36, 63/64, 1, 27/25, 35/32, 10/9, 9/8, 8/7, 32/27, 6/5, 128/105,
100/81, 5/4, 21/16, 4/3, 27/20, 48/35, 25/18, 36/25, 35/24, 40/27,
3/2, 32/21, 8/5, 81/50, 105/64, 5/3, 27/16, 7/4, 16/9, 9/5, 64/35,
50/27, 81/40, 72/35]
[7/4, 36/25, 32/27, 35/36, 8/5, 21/16, 27/25, 16/9, 35/24, 6/5, 63/64,
81/50, 4/3, 35/32, 9/5, 40/27, 128/105, 1, 105/64, 27/20, 10/9, 64/35,
3/2, 100/81, 81/40, 5/3, 48/35, 9/8, 50/27, 32/21, 5/4, 72/35, 27/16,
25/18, 8/7]

Generator = 35/32

[1, 5/4, 3/2, 15/8] [1, 9/7, 3/2, 12/7]
[1, 5/4, 3/2, 15/8] [1, 9/7, 3/2, 12/7]
[1, 5/4, 3/2, 15/8] [1, 9/7, 3/2, 12/7]
[1, 5/4, 3/2, 15/8] [1, 9/7, 3/2, 12/7]
[1, 5/4, 3/2, 15/8] [1, 9/7, 3/2, 12/7]
[1, 5/4, 3/2, 15/8] [1, 9/7, 3/2, 12/7]
[1, 5/4, 3/2, 15/8] [1, 9/7, 3/2, 12/7]
[1, 5/4, 3/2, 15/8] [1, 9/7, 3/2, 12/7]
[1, 5/4, 3/2, 15/8] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 15/8] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 15/8] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 15/8] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 15/8] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 15/8] [1, 6/5, 3/2, 8/5]
[1, 5/4, 3/2, 15/8] [1, 6/5, 3/2, 8/5]
[1, 5/4, 3/2, 15/8] [1, 6/5, 3/2, 8/5]
[1, 5/4, 3/2, 15/8] [1, 6/5, 3/2, 8/5]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 8/5]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 8/5]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 8/5]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 8/5]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 8/5]
[1, 7/6, 3/2, 7/4] [1, 6/5, 3/2, 8/5]
[1, 7/6, 3/2, 7/4] [1, 6/5, 3/2, 8/5]
[1, 7/6, 3/2, 7/4] [1, 6/5, 3/2, 8/5]
[1, 7/6, 3/2, 7/4] [1, 6/5, 3/2, 8/5]
[1, 7/6, 3/2, 7/4] [1, 6/5, 3/2, 8/5]
[1, 7/6, 3/2, 7/4] [1, 6/5, 3/2, 8/5]
[1, 7/6, 3/2, 7/4] [1, 6/5, 3/2, 8/5]
[1, 7/6, 3/2, 7/4] [1, 6/5, 3/2, 8/5]
[1, 7/6, 7/5, 7/4] [1, 6/5, 7/5, 8/5]
[1, 7/6, 7/5, 7/4] [1, 6/5, 7/5, 8/5]
[1, 7/6, 7/5, 7/4] [1, 6/5, 7/5, 8/5]
[1, 7/6, 7/5, 7/4] [1, 6/5, 7/5, 8/5]
[1, 7/6, 7/5, 7/4] [1, 6/5, 7/5, 8/5]


[1, 9/7, 3/2, 27/14] [1, 5/4, 3/2, 5/3]
[1, 9/7, 3/2, 27/14] [1, 5/4, 3/2, 5/3]
[1, 9/7, 3/2, 27/14] [1, 5/4, 3/2, 5/3]
[1, 9/7, 3/2, 9/5] [1, 5/4, 3/2, 5/3]
[1, 9/7, 3/2, 9/5] [1, 5/4, 3/2, 5/3]
[1, 9/7, 3/2, 9/5] [1, 5/4, 3/2, 5/3]
[1, 9/7, 3/2, 9/5] [1, 5/4, 3/2, 5/3]
[1, 9/7, 3/2, 9/5] [1, 5/4, 3/2, 5/3]
[1, 6/5, 3/2, 9/5] [1, 5/4, 3/2, 5/3]
[1, 6/5, 3/2, 9/5] [1, 5/4, 3/2, 5/3]
[1, 6/5, 3/2, 9/5] [1, 5/4, 3/2, 5/3]
[1, 6/5, 3/2, 9/5] [1, 5/4, 3/2, 5/3]
[1, 6/5, 3/2, 9/5] [1, 5/4, 3/2, 5/3]
[1, 6/5, 3/2, 9/5] [1, 5/4, 3/2, 5/3]
[1, 6/5, 3/2, 9/5] [1, 5/4, 3/2, 5/3]
[1, 6/5, 3/2, 9/5] [1, 5/4, 3/2, 5/3]
[1, 6/5, 3/2, 9/5] [1, 5/4, 3/2, 5/3]
[1, 6/5, 3/2, 9/5] [1, 5/4, 3/2, 5/3]
[1, 6/5, 3/2, 9/5] [1, 5/4, 3/2, 5/3]
[1, 6/5, 3/2, 9/5] [1, 5/4, 3/2, 5/3]
[1, 6/5, 3/2, 9/5] [1, 5/4, 3/2, 5/3]
[1, 6/5, 3/2, 9/5] [1, 5/4, 3/2, 5/3]
[1, 6/5, 3/2, 9/5] [1, 7/6, 3/2, 5/3]
[1, 6/5, 3/2, 9/5] [1, 7/6, 3/2, 5/3]
[1, 6/5, 3/2, 9/5] [1, 7/6, 3/2, 5/3]
[1, 6/5, 3/2, 9/5] [1, 7/6, 3/2, 5/3]
[1, 6/5, 3/2, 9/5] [1, 7/6, 3/2, 5/3]
[1, 6/5, 3/2, 9/5] [1, 7/6, 3/2, 14/9]
[1, 6/5, 3/2, 9/5] [1, 7/6, 3/2, 14/9]
[1, 6/5, 3/2, 9/5] [1, 7/6, 3/2, 14/9]
[1, 6/5, 7/5, 9/5] [1, 7/6, 7/5, 14/9]
[1, 6/5, 7/5, 9/5] [1, 7/6, 7/5, 14/9]
[1, 6/5, 7/5, 9/5] [1, 7/6, 7/5, 14/9]
[1, 6/5, 7/5, 9/5] [1, 7/6, 7/5, 14/9]
[1, 6/5, 7/5, 9/5] [1, 7/6, 7/5, 14/9]

Message: 6696 - Contents - Hide Contents

Date: Fri, 28 Mar 2003 07:06:47

Subject: More 15/14 chroma systems

From: Gene Ward Smith

Quartaminorthirds[17]
rms error 3.066 cents

[126/125, 1029/1024]
[9, 5, -3, -13, -30, -21]
[1, 21/20, 35/32, 8/7, 6/5, 5/4, 21/16, 48/35, 7/5, 
10/7, 35/24, 32/21, 8/5, 5/3, 7/4, 64/35, 40/21]
[7/5, 35/24, 32/21, 8/5, 5/3, 7/4, 64/35, 40/21, 
1, 21/20, 35/32, 8/7, 6/5, 5/4, 21/16, 48/35, 10/7]

Generator = 8/7

Tetrad circles

[1, 5/4, 3/2, 15/8] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 15/8] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 15/8] [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, 8/5]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 8/5]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 8/5]
[1, 5/4, 7/5, 7/4] [1, 6/5, 7/5, 8/5]
[1, 5/4, 7/5, 7/4] [1, 6/5, 7/5, 8/5]
[1, 5/4, 7/5, 7/4] [1, 6/5, 7/5, 8/5]
[1, 5/4, 7/5, 7/4] [1, 6/5, 7/5, 8/5]
[1, 7/6, 7/5, 7/4] [1, 6/5, 7/5, 8/5]
[1, 7/6, 7/5, 7/4] [1, 10/9, 7/5, 8/5]
[1, 7/6, 7/5, 7/4] [1, 10/9, 7/5, 8/5]
[1, 7/6, 7/5, 7/4] [1, 10/9, 7/5, 8/5]
[1, 7/6, 7/5, 7/4] [1, 10/9, 7/5, 8/5]


[1, 6/5, 3/2, 9/5] [1, 5/4, 3/2, 9/5]
[1, 6/5, 3/2, 9/5] [1, 5/4, 3/2, 9/5]
[1, 6/5, 3/2, 9/5] [1, 5/4, 3/2, 9/5]
[1, 6/5, 3/2, 9/5] [1, 5/4, 3/2, 9/5]
[1, 6/5, 3/2, 5/3] [1, 5/4, 3/2, 5/3]
[1, 6/5, 3/2, 5/3] [1, 5/4, 3/2, 5/3]
[1, 6/5, 3/2, 5/3] [1, 5/4, 3/2, 5/3]
[1, 6/5, 3/2, 5/3] [1, 5/4, 3/2, 5/3]
[1, 6/5, 7/5, 5/3] [1, 5/4, 7/5, 5/3]
[1, 6/5, 7/5, 5/3] [1, 5/4, 7/5, 5/3]
[1, 6/5, 7/5, 5/3] [1, 5/4, 7/5, 5/3]
[1, 6/5, 7/5, 5/3] [1, 5/4, 7/5, 5/3]
[1, 6/5, 7/5, 5/3] [1, 7/6, 7/5, 5/3]
[1, 10/9, 7/5, 5/3] [1, 7/6, 7/5, 5/3]
[1, 10/9, 7/5, 5/3] [1, 7/6, 7/5, 5/3]
[1, 10/9, 7/5, 5/3] [1, 7/6, 7/5, 5/3]
[1, 10/9, 7/5, 5/3] [1, 7/6, 7/5, 5/3]


Semififths[21]
rms error 3.732 cents

[81/80, 6144/6125]
[2, 8, -11, 8, -23, -48]
[1, 36/35, 16/15, 35/32, 9/8, 6/5, 128/105, 5/4, 21/16, 4/3, 48/35,
35/24, 3/2, 32/21, 8/5, 105/64, 5/3, 9/5, 64/35, 15/8, 35/18]
[16/15, 21/16, 8/5, 35/18, 6/5, 35/24, 9/5, 35/32, 4/3, 105/64, 1,
128/105, 3/2, 64/35, 9/8, 48/35, 5/3, 36/35, 5/4, 32/21, 15/8]

Generator = 128/105

Circles of tetrads

[1, 5/4, 3/2, 15/8] [1, 9/7, 3/2, 12/7]
[1, 5/4, 3/2, 15/8] [1, 9/7, 3/2, 12/7]
[1, 5/4, 3/2, 15/8] [1, 9/7, 3/2, 12/7]
[1, 5/4, 3/2, 15/8] [1, 9/7, 3/2, 12/7]
[1, 5/4, 3/2, 15/8] [1, 9/7, 3/2, 12/7]
[1, 5/4, 3/2, 15/8] [1, 9/7, 3/2, 12/7]
[1, 5/4, 3/2, 15/8] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 15/8] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 15/8] [1, 6/5, 3/2, 8/5]
[1, 5/4, 3/2, 15/8] [1, 6/5, 3/2, 8/5]
[1, 5/4, 3/2, 15/8] [1, 6/5, 3/2, 8/5]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 8/5]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 8/5]
[1, 7/6, 3/2, 7/4] [1, 6/5, 3/2, 8/5]
[1, 7/6, 3/2, 7/4] [1, 6/5, 3/2, 8/5]
[1, 7/6, 3/2, 7/4] [1, 6/5, 3/2, 8/5]
[1, 7/6, 3/2, 7/4] [1, 6/5, 3/2, 8/5]
[1, 7/6, 3/2, 7/4] [1, 6/5, 3/2, 8/5]
[1, 7/6, 3/2, 7/4] [1, 6/5, 3/2, 8/5]
[1, 7/6, 7/5, 7/4] [1, 6/5, 7/5, 8/5]
[1, 7/6, 7/5, 7/4] [1, 6/5, 7/5, 8/5]


[1, 9/7, 3/2, 27/14] [1, 5/4, 3/2, 5/3]
[1, 9/7, 3/2, 27/14] [1, 5/4, 3/2, 5/3]
[1, 9/7, 3/2, 27/14] [1, 5/4, 3/2, 5/3]
[1, 9/7, 3/2, 27/14] [1, 5/4, 3/2, 5/3]
[1, 9/7, 3/2, 9/5] [1, 5/4, 3/2, 5/3]
[1, 9/7, 3/2, 9/5] [1, 5/4, 3/2, 5/3]
[1, 6/5, 3/2, 9/5] [1, 5/4, 3/2, 5/3]
[1, 6/5, 3/2, 9/5] [1, 5/4, 3/2, 5/3]
[1, 6/5, 3/2, 9/5] [1, 5/4, 3/2, 5/3]
[1, 6/5, 3/2, 9/5] [1, 5/4, 3/2, 5/3]
[1, 6/5, 3/2, 9/5] [1, 5/4, 3/2, 5/3]
[1, 6/5, 3/2, 9/5] [1, 5/4, 3/2, 5/3]
[1, 6/5, 3/2, 9/5] [1, 5/4, 3/2, 5/3]
[1, 6/5, 3/2, 9/5] [1, 7/6, 3/2, 5/3]
[1, 6/5, 3/2, 9/5] [1, 7/6, 3/2, 5/3]
[1, 6/5, 3/2, 9/5] [1, 7/6, 3/2, 14/9]
[1, 6/5, 3/2, 9/5] [1, 7/6, 3/2, 14/9]
[1, 6/5, 3/2, 9/5] [1, 7/6, 3/2, 14/9]
[1, 6/5, 3/2, 9/5] [1, 7/6, 3/2, 14/9]
[1, 6/5, 7/5, 9/5] [1, 7/6, 7/5, 14/9]
[1, 6/5, 7/5, 9/5] [1, 7/6, 7/5, 14/9]


Superkleismic[22]
rms error 4.053 cents

[875/864, 1029/1024]
[9, 10, -3, -5, -30, -35]
[1, 21/20, 27/25, 10/9, 8/7, 6/5, 5/4, 63/50, 21/16, 4/3, 25/18, 35/24, 
3/2, 32/21, 63/40, 8/5, 5/3, 7/4, 9/5, 50/27, 40/21, 35/18]
[5/4, 3/2, 9/5, 27/25, 21/16, 63/40, 40/21, 8/7, 25/18, 5/3, 
1, 6/5, 35/24, 7/4, 21/20, 63/50, 32/21, 50/27, 10/9, 4/3, 8/5, 35/18]

Generator = 6/5

Circles of tetrads 

[1, 7/6, 7/5, 7/4] [1, 6/5, 7/5, 8/5]
[1, 7/6, 7/5, 7/4] [1, 6/5, 7/5, 8/5]
[1, 7/6, 7/5, 7/4] [1, 6/5, 7/5, 8/5]
[1, 7/6, 7/5, 7/4] [1, 6/5, 7/5, 8/5]
[1, 7/6, 7/5, 7/4] [1, 6/5, 7/5, 8/5]
[1, 7/6, 7/5, 7/4] [1, 6/5, 7/5, 8/5]
[1, 7/6, 7/5, 7/4] [1, 6/5, 7/5, 8/5]
[1, 7/6, 7/5, 7/4] [1, 6/5, 7/5, 8/5]
[1, 7/6, 7/5, 7/4] [1, 6/5, 7/5, 8/5]
[1, 7/6, 3/2, 7/4] [1, 6/5, 3/2, 8/5]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 8/5]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 8/5]
[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, 15/8] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 15/8] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 15/8] [1, 9/7, 3/2, 12/7]


[1, 6/5, 7/5, 42/25] [1, 7/6, 7/5, 14/9]
[1, 6/5, 7/5, 42/25] [1, 7/6, 7/5, 5/3]
[1, 6/5, 7/5, 42/25] [1, 7/6, 7/5, 5/3]
[1, 6/5, 7/5, 42/25] [1, 7/6, 7/5, 5/3]
[1, 6/5, 7/5, 42/25] [1, 7/6, 7/5, 5/3]
[1, 6/5, 7/5, 42/25] [1, 7/6, 7/5, 5/3]
[1, 6/5, 7/5, 42/25] [1, 7/6, 7/5, 5/3]
[1, 6/5, 7/5, 42/25] [1, 7/6, 7/5, 5/3]
[1, 6/5, 7/5, 9/5] [1, 7/6, 7/5, 5/3]
[1, 6/5, 3/2, 9/5] [1, 7/6, 3/2, 5/3]
[1, 6/5, 3/2, 9/5] [1, 5/4, 3/2, 5/3]
[1, 6/5, 3/2, 9/5] [1, 5/4, 3/2, 5/3]
[1, 6/5, 3/2, 9/5] [1, 5/4, 3/2, 5/3]
[1, 6/5, 3/2, 9/5] [1, 5/4, 3/2, 5/3]
[1, 6/5, 3/2, 9/5] [1, 5/4, 3/2, 5/3]
[1, 6/5, 3/2, 9/5] [1, 5/4, 3/2, 5/3]
[1, 6/5, 3/2, 9/5] [1, 5/4, 3/2, 5/3]
[1, 6/5, 3/2, 9/5] [1, 5/4, 3/2, 5/3]
[1, 6/5, 3/2, 9/5] [1, 5/4, 3/2, 5/3]
[1, 6/5, 3/2, 9/5] [1, 5/4, 3/2, 5/3]
[1, 6/5, 3/2, 9/5] [1, 5/4, 3/2, 5/3]
[1, 9/7, 3/2, 9/5] [1, 5/4, 3/2, 5/3]

Message: 6697 - Contents - Hide Contents

Date: Fri, 28 Mar 2003 07:14:50

Subject: Goldston-Yildrim theorem

From: Gene Ward Smith

Since there seems to be some degree of interest on this list for this
sort of thing, I will mention that my buddy Danny Goldston together
with Cem Yildirim of Turky have proven the prime gap conjecture, which
now becomes the Goldston-Yildrim theorem. This tells us that if G(p) is
the gap between the prime p and the next prime above it, then

lim inf G(p)/log(p) = 0

This is quite a ways yet from the twin primes conjecture, which Danny
and just about everyone else thinks is true, but it is a major step
forward.

Message: 6698 - Contents - Hide Contents

Date: Sun, 30 Mar 2003 23:40:16

Subject: Re: 225/225 & 21/20 scales

From: Carl Lumma

>[4, -3, 2, -14, -8, 13] [[1, 2, 2, 3], [0, -4, 3, -2]]
>
>bad 2644.480844 comp 14.72969740 rms 12.18857055
>graham 7 scale size 9 ratio 1.285714

What's this?

-Carl

Message: 6699 - Contents - Hide Contents

Date: Mon, 31 Mar 2003 07:04:08

Subject: 225/225 & 21/20 scales

From: Gene Ward Smith

By requiring that 225/224 be a comma and 21/20 a chroma, I got a number
of equations and inequalities which made it possible to directly search
the wedgies. This involved searching for 225/224 wedgies, of the form

[a, b, 2*b+2*a, d, 2*d+5*a, -2*d+5*b]

and such that |b| <= a. The form insures that 225/224 is a comma, and
the inequality insures that the size of scale with 21/20 as a chroma is
at least as large as the Graham complexity. I searched for a up to 31,
and filtered the result by requiring that the rms error be less than
15, and the geometric badness less than 20000. At the top of the list
is Orwell[18].

It should be noted this does not always work so easily; for instance
225/224 and 15/14 or 441/440 and 21/20 lead to problems. We are also
not guaranteed that everything on this list will actually work out
in practice; I haven't discussed the obstructions much but they do arise.


[7, -3, 8, -21, -7, 27] [[1, 0, 3, 1], [0, 7, -3, 8]]

bad 1673.187049 comp 25.42062964 rms 2.589237496
graham 11 scale size 18 ratio 1.636364


[6, 5, 22, -6, 18, 37] [[1, 0, 1, -3], [0, 6, 5, 22]]

bad 1891.474472 comp 34.26986563 rms 1.610555448
graham 22 scale size 23 ratio 1.045455


[5, 1, 12, -10, 5, 25] [[1, 0, 2, -1], [0, 5, 1, 12]]

bad 1935.541443 comp 21.62473825 rms 4.139050792
graham 12 scale size 16 ratio 1.333333


[4, -3, 2, -14, -8, 13] [[1, 2, 2, 3], [0, -4, 3, -2]]

bad 2644.480844 comp 14.72969740 rms 12.18857055
graham 7 scale size 9 ratio 1.285714


[12, -2, 20, -31, -2, 52] [[2, 1, 5, 2], [0, 6, -1, 10]]

bad 3656.269932 comp 45.66691577 rms 1.753213831
graham 22 scale size 34 ratio 1.545455


[13, -10, 6, -46, -27, 42] [[1, 2, 2, 3], [0, -13, 10, -6]]

bad 3872.384714 comp 48.03151022 rms 1.678518039
graham 23 scale size 29 ratio 1.260870


[8, 1, 18, -17, 6, 39] [[1, -1, 2, -3], [0, 8, 1, 18]]

bad 3960.691733 comp 33.57393404 rms 3.513715352
graham 18 scale size 25 ratio 1.388889


[13, 2, 30, -27, 11, 64] [[1, 6, 3, 13], [0, -13, -2, -30]]

bad 4674.708063 comp 55.18330983 rms 1.535108242
graham 30 scale size 41 ratio 1.366667


[11, -6, 10, -35, -15, 40] [[1, 4, 1, 5], [0, -11, 6, -10]]

bad 5333.482178 comp 39.77267622 rms 3.371640164
graham 17 scale size 27 ratio 1.588235


[4, 4, 16, -3, 14, 26] [[4, 6, 9, 10], [0, 1, 1, 4]]

bad 5517.328547 comp 24.55359179 rms 9.151636960
graham 16 scale size 16 ratio 1.000000


[14, -6, 16, -42, -14, 54] [[2, 0, 6, 2], [0, 14, -6, 16]]

bad 6692.748195 comp 50.84125928 rms 2.589237496
graham 22 scale size 36 ratio 1.636364


[11, 6, 34, -16, 23, 62] [[1, 1, 2, 1], [0, 11, 6, 34]]

bad 7170.265679 comp 55.28147479 rms 2.346259298
graham 34 scale size 39 ratio 1.147059


[9, -7, 4, -32, -19, 29] [[1, 2, 2, 3], [0, -9, 7, -4]]

bad 7369.743470 comp 33.30471422 rms 6.644173248
graham 16 scale size 20 ratio 1.250000


[19, -5, 28, -52, -9, 79] [[1, -1, 3, -1], [0, 19, -5, 28]]

bad 7433.465748 comp 70.68042097 rms 1.487966281
graham 33 scale size 52 ratio 1.575758


[12, 10, 44, -12, 36, 74] [[2, 0, 2, -6], [0, 12, 10, 44]]

bad 7565.897894 comp 68.53973127 rms 1.610555448
graham 44 scale size 46 ratio 1.045455


[18, -9, 18, -56, -22, 67] [[9, 14, 21, 25], [0, 2, -1, 2]]

bad 7569.695667 comp 65.10901923 rms 1.785649073
graham 27 scale size 45 ratio 1.666667


[9, 5, 28, -13, 19, 51] [[1, 1, 2, 1], [0, 9, 5, 28]]

bad 7646.641388 comp 45.43499477 rms 3.704160176
graham 28 scale size 32 ratio 1.142857


[10, 2, 24, -20, 10, 50] [[2, 0, 4, -2], [0, 10, 2, 24]]

bad 7742.165768 comp 43.24947649 rms 4.139050792
graham 24 scale size 32 ratio 1.333333


[10, -10, 0, -39, -28, 28] [[10, 16, 23, 28], [0, -1, 1, 0]]

bad 7826.501965 comp 39.17133416 rms 5.100713978
graham 20 scale size 20 ratio 1.000000


[19, -17, 4, -71, -47, 57] [[1, -7, 10, 1], [0, 19, -17, 4]]

bad 7855.035658 comp 72.18391247 rms 1.507534728
graham 36 scale size 40 ratio 1.111111


[14, 6, 40, -23, 24, 76] [[2, 4, 5, 8], [0, -7, -3, -20]]

bad 8407.532757 comp 66.79114641 rms 1.884650276
graham 40 scale size 48 ratio 1.200000


[9, -2, 14, -24, -3, 38] [[1, 3, 2, 5], [0, -9, 2, -14]]

bad 8811.637847 comp 33.78049801 rms 7.721906590
graham 16 scale size 25 ratio 1.562500


[18, 3, 42, -37, 16, 89] [[3, 5, 7, 9], [0, -6, -1, -14]]

bad 9988.496510 comp 76.80133416 rms 1.693411845
graham 42 scale size 57 ratio 1.357143


[10, -3, 14, -28, -6, 41] [[1, 6, 1, 9], [0, -10, 3, -14]]

bad 10168.63993 comp 36.93793274 rms 7.452769630
graham 17 scale size 27 ratio 1.588235


[20, -13, 14, -67, -34, 69] [[1, -1, 4, 1], [0, 20, -13, 14]]

bad 10378.94397 comp 72.69891283 rms 1.963800357
graham 33 scale size 47 ratio 1.424242


[17, -13, 8, -60, -35, 55] [[1, 2, 2, 3], [0, -17, 13, -8]]

bad 10577.11642 comp 62.75966800 rms 2.685381606
graham 30 scale size 38 ratio 1.266667


[8, -6, 4, -28, -16, 26] [[2, 4, 4, 6], [0, -8, 6, -4]]

bad 10577.92338 comp 29.45939480 rms 12.18857055
graham 14 scale size 18 ratio 1.285714


[15, -2, 26, -38, -1, 66] [[1, 4, 2, 7], [0, -15, 2, -26]]

bad 10687.83946 comp 57.60877001 rms 3.220421490
graham 28 scale size 43 ratio 1.535714


[20, -1, 38, -48, 4, 91] [[1, 8, 2, 15], [0, -20, 1, -38]]

bad 10785.36676 comp 78.79332776 rms 1.737224847
graham 39 scale size 59 ratio 1.512821


[17, -1, 32, -41, 3, 77] [[1, 7, 2, 13], [0, -17, 1, -32]]

bad 10962.29742 comp 66.78311603 rms 2.457922714
graham 33 scale size 50 ratio 1.515152


[7, 2, 18, -13, 9, 36] [[1, 4, 3, 9], [0, -7, -2, -18]]

bad 11045.68014 comp 31.38144021 rms 11.21622520
graham 18 scale size 23 ratio 1.277778


[19, 7, 52, -33, 29, 101] [[1, -2, 1, -7], [0, 19, 7, 52]]

bad 11567.12301 comp 88.27816455 rms 1.484290010
graham 52 scale size 64 ratio 1.230769


[25, -12, 26, -77, -29, 94] [[1, -4, 5, -3], [0, 25, -12, 26]]

bad 12185.51922 comp 90.49265062 rms 1.488049644
graham 38 scale size 63 ratio 1.657895


[15, -14, 2, -57, -39, 44] [[1, 3, 1, 3], [0, -15, 14, -2]]

bad 12223.03079 comp 57.58995263 rms 3.685407058
graham 29 scale size 31 ratio 1.068966


[8, 6, 28, -9, 22, 48] [[2, 5, 6, 12], [0, -4, -3, -14]]

bad 12236.41269 comp 44.04524617 rms 6.307482186
graham 28 scale size 30 ratio 1.071429


[10, 9, 38, -9, 32, 63] [[1, -1, 0, -7], [0, 10, 9, 38]]

bad 12949.78760 comp 58.80133944 rms 3.745313768
graham 38 scale size 39 ratio 1.026316


[16, -5, 22, -45, -10, 65] [[1, 9, 0, 13], [0, -16, 5, -22]]

bad 13459.64211 comp 58.97163144 rms 3.870323144
graham 27 scale size 43 ratio 1.592593


[24, -16, 16, -81, -42, 82] [[8, 12, 19, 22], [0, 3, -2, 2]]

bad 13518.53011 comp 87.38177875 rms 1.770466488
graham 40 scale size 56 ratio 1.400000


[25, -24, 2, -96, -67, 72] [[1, 4, 0, 3], [0, -25, 24, -2]]

bad 13886.41196 comp 96.72994445 rms 1.484117098
graham 49 scale size 51 ratio 1.040816


[15, 10, 50, -19, 37, 88] [[5, 7, 11, 11], [0, 3, 2, 10]]

bad 13983.81149 comp 79.58043642 rms 2.208070460
graham 50 scale size 55 ratio 1.100000


[11, 1, 24, -24, 7, 53] [[1, -2, 2, -5], [0, 11, 1, 24]]

bad 14084.32133 comp 45.56036466 rms 6.785182636
graham 24 scale size 34 ratio 1.416667


[24, -4, 40, -62, -4, 104] [[4, 2, 10, 4], [0, 12, -2, 20]]

bad 14625.07972 comp 91.33383152 rms 1.753213831
graham 44 scale size 68 ratio 1.545455


[7, 4, 22, -10, 15, 40] [[1, 1, 2, 1], [0, 7, 4, 22]]

bad 14634.49544 comp 35.59004417 rms 11.55368971
graham 22 scale size 25 ratio 1.136364


[26, -8, 36, -73, -16, 106] [[2, 2, 5, 4], [0, 13, -4, 18]]

bad 14810.48305 comp 95.90825535 rms 1.610116282
graham 44 scale size 70 ratio 1.590909


[25, 0, 50, -58, 9, 116] [[25, 40, 58, 71], [0, -1, 0, -2]]

bad 14998.42416 comp 100.1661880 rms 1.494869699
graham 50 scale size 75 ratio 1.500000


[21, -9, 24, -63, -21, 81] [[3, 0, 9, 3], [0, 21, -9, 24]]

bad 15058.68343 comp 76.26188891 rms 2.589237496
graham 33 scale size 54 ratio 1.636364


[26, -20, 12, -92, -54, 84] [[2, 4, 4, 6], [0, -26, 20, -12]]

bad 15489.53885 comp 96.06302042 rms 1.678518039
graham 46 scale size 58 ratio 1.260870


[16, 2, 36, -34, 12, 78] [[2, -2, 4, -6], [0, 16, 2, 36]]

bad 15842.76693 comp 67.14786808 rms 3.513715352
graham 36 scale size 50 ratio 1.388889


[11, -11, 0, -43, -31, 31] [[11, 17, 26, 31], [0, 1, -1, 0]]

bad 15984.88332 comp 43.08846759 rms 8.609687186
graham 22 scale size 22 ratio 1.000000


[20, 11, 62, -29, 42, 113] [[1, 1, 2, 1], [0, 20, 11, 62]]

bad 16004.08424 comp 100.7161993 rms 1.577728140
graham 62 scale size 71 ratio 1.145161


[17, 11, 56, -22, 41, 99] [[1, -2, 0, -9], [0, 17, 11, 56]]

bad 16052.65711 comp 89.40526263 rms 2.008263812
graham 56 scale size 62 ratio 1.107143


[21, 3, 48, -44, 17, 103] [[3, 5, 7, 9], [0, -7, -1, -16]]

bad 16311.24569 comp 88.75350222 rms 2.070694980
graham 48 scale size 66 ratio 1.375000


[16, 7, 46, -26, 28, 87] [[1, -6, -1, -19], [0, 16, 7, 46]]

bad 16435.37138 comp 76.63496479 rms 2.798501846
graham 46 scale size 55 ratio 1.195652


[16, -10, 12, -53, -26, 56] [[2, 1, 6, 4], [0, 8, -5, 6]]

bad 16488.92261 comp 58.03978434 rms 4.894864788
graham 26 scale size 38 ratio 1.461538


[15, -9, 12, -49, -23, 53] [[3, 3, 8, 7], [0, 5, -3, 4]]

bad 16785.97003 comp 54.32725934 rms 5.687361716
graham 24 scale size 36 ratio 1.500000


[18, 15, 66, -18, 54, 111] [[3, 0, 3, -9], [0, 18, 15, 66]]

bad 17023.27025 comp 102.8095969 rms 1.610555448
graham 66 scale size 69 ratio 1.045455


[23, -8, 30, -66, -17, 92] [[1, -9, 6, -11], [0, 23, -8, 30]]

bad 17130.95974 comp 84.30066077 rms 2.410569942
graham 38 scale size 61 ratio 1.605263


[15, 3, 36, -30, 15, 75] [[3, 0, 6, -3], [0, 15, 3, 36]]

bad 17419.87298 comp 64.87421474 rms 4.139050792
graham 36 scale size 48 ratio 1.333333


[21, -21, 0, -82, -59, 59] [[21, 33, 49, 59], [0, 1, -1, 0]]

bad 17502.75864 comp 82.25980175 rms 2.586611398
graham 42 scale size 42 ratio 1.000000


[23, -20, 6, -85, -55, 70] [[1, 10, -5, 5], [0, -23, 20, -6]]

bad 17672.69851 comp 86.82372813 rms 2.344369208
graham 43 scale size 49 ratio 1.139535


[26, 4, 60, -54, 22, 128] [[2, 12, 6, 26], [0, -26, -4, -60]]

bad 18698.83223 comp 110.3666196 rms 1.535108242
graham 60 scale size 82 ratio 1.366667


[14, -1, 26, -34, 2, 63] [[1, 6, 2, 11], [0, -14, 1, -26]]

bad 18845.84375 comp 54.77717330 rms 6.280820060
graham 27 scale size 41 ratio 1.518519


[12, -7, 10, -39, -18, 43] [[1, -3, 5, -1], [0, 12, -7, 10]]

bad 19574.18705 comp 43.42843741 rms 10.37851763
graham 19 scale size 29 ratio 1.526316


[12, 5, 34, -20, 20, 65] [[1, -4, 0, -13], [0, 12, 5, 34]]

bad 19840.40051 comp 56.94920411 rms 6.117516042
graham 34 scale size 41 ratio 1.205882

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