two (7,11,13) temperaments

I found an interesting pair of temperaments that I wonder if Gene or anyone else has come across before. They are linear (7,11,13) temperaments--that is, 13-limit with no threes and no fives. They might come in handy if you like ratios of 7 11 and 13 and you don't care about 5ths or 3rds.

The (7,11,13)-map for the first one is [8,7,2] and the generator (optimized for minimax error for the six intervals shown) is 421.383 cents, a little bigger than 14:11. It works like this (generators that would provide pure target intervals are shown on the right; all intervals have been octave-reduced):

8 generators represents 7:4 => g~421.103c
7 generators represents 11:8 => g~421.617c
2 generators represents 13:8 => g~420.264c

1 generator represents 14:11 => g~417.508c
6 generators represents 14:13 => g~421.383c
5 generators represents 22:13 => g~422.158c

The optimal generator 421.383c will make a MOS when extended to chains of 5,8,11,14,17,20,23,43,63,83...

The other temperament has a generator just a tiny bit narrower, and the map is [-12,-13,2]. So the complexity of two of the three primary target intervals is higher than in the first temperament, but the representations are much more accurate:

12 generators represents 8:7 => g~419.265c
13 generators represents 16:11 => g~419.129c
2 generators represents 13:8 => g~420.264c

1 generator represents 14:11 => g~417.508c
14 generators represents 13:7 => g~419.407c
15 generators represents 13:11 => g~419.281c

The minimax-optimized generator for this one is 419.280 cents--all six target intervals are then represented with an accuracy of less than 2 cents. It will make a MOS if extended to chains of 5,8,11,14,17,20,37,57,94...

If no one has come across these before, I'd suggest "nigerian" for the second one (because of the 419-c generator) and then "cameroon" for its neighbour.

The tipping point between them (like 700c separating meantone/schismic) is 420c or 7deg20. Twenty scale-steps feels right for this pair; 20 nigerian generators yields step sizes of 57.840c and 72.240c in the pattern ssssssLssssssLsssssL; cameroon gets 64.149c and 36.489c in the pattern LLLLLLsLLLLLLsLLLLLs.

One other very accurate generator I found for a (7,11,13) temperament is optimized at 130.107 cents. The map is [-11,-5,-12] and it works like this:
1 generator represents 14:13
5 generators represents 16:11
6 generators represents 11:7
7 generators represents 22:13
11 generators represents 8:7
12 generators represents 16:13

Again, all six target intervals can be represented within 2 cents. MOS scales appear at 5,6,7,8,9,10,19,28,37,46,83...

My search resulted in a dozen or so other (7,11,13) temperaments, but the three above look most promising.

>If no one has come across these before, I'd suggest "nigerian" for the
>second one (because of the 419-c generator)

What's the connection?

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>
> >I'd suggest "nigerian" for the
> >second one (because of the 419-c generator)
>
> What's the connection?

see http://www.google.com/search?q=nigerian+419

or http://www.419eater.com/html/419faq.htm

>> >I'd suggest "nigerian" for the
>> >second one (because of the 419-c generator)
>>
>> What's the connection?
>
>see http://www.google.com/search?q=nigerian+419
>
>or http://www.419eater.com/html/419faq.htm

Oh, riight. I was thinking some ethnomusicology thing.
Silly me. :)

-Carl

--- In tuning-math@yahoogroups.com, Jon Wild <wild@...> wrote:

> 8 generators represents 7:4 => g~421.103c

c1 = 7/(14/11)^8 = 214358881/210827008

> 7 generators represents 11:8 => g~421.617c

(11/2)/(14/11)^7 = c1

> 2 generators represents 13:8 => g~420.264c

c2 = (13/8)/(14/11)^2 = 1573/1568

> 1 generator represents 14:11 => g~417.508c
> 6 generators represents 14:13 => g~421.383c

c3 = (56/13)/(14/11)^6 = 1771561/1747928

> 5 generators represents 22:13 => g~422.158c

(44/13)/(14/11)^5 = c3

There are two independent commas, and we can take the TM basis for
them, which is {1573/1568, 15488/15379}. One way to extend it to the
complete 13-limit is as the 17&37 temperament, which we might call
wild temperament:

<<4 19 -8 -7 -2 21 -24 -25 -18 -72 -82 -75 8 24 19||

> 12 generators represents 8:7 => g~419.265c

r1 = (128/7)/(14/11)^12 = 3138428376721/3100448333024

> 13 generators represents 16:11 => g~419.129c

(256/11)/(14/11)^13 = r1

> 2 generators represents 13:8 => g~420.264c

r2 = 1573/1568

> 1 generator represents 14:11 => g~417.508c
> 14 generators represents 13:7 => g~419.407c

r3 = (208/7)/(14/11)^14

> 15 generators represents 13:11 => g~419.281c

(416/11)/(14/11)^15 = r3

The TM basis is now {1573/1568, 33787663/33554432}. 166, which I was
touting for all of its nice commas, is a good tuning for this.