From Mohajira to Slendro

This is a tuning that could fit among the "Ethno extras" series
initiated by Margo.
I have always been intrigued by a very classic but singular Mohajira
sequence, that goes :
[5 49 15 147 45 55 135]
or in ascending order :
[147 160 180 196 220 240 270 294]

because of the links it suggests with complementary 7th-limit harmonies.
As I mentionned it to John about his latest scale that contains a
part of it, the secret of this interesting coïncidence between 7th
and 11th limit lies in the 540/539 comma we find between 60/49 and 11/9.
In order to analyse it closer, I developped the "Dudon scale" of
540/539, or the scale formed by the divisors of both 540 and 539 :
D(540/539) = [1 539-135 9 77 5 11 45 3 49 27 7 15]
and also of the 441/440 inside the scale as well :
D(441/440) = [1 9 147 5 21 11 3 49 55-441 7 63]
then the "summing Dudon scale" of both commas :
D(540/539, 441/440) = [1 539-135 9 147 77 5 21 11 45 3
49 27 55-441 7 15 63]

This is how we can complete the original [5 49 15 147 45 55
135] Mohajira with 5 notes (or more of them if you wish), extending
it to many slendros :

! mohajira-to-slendro.scl
!
From Mohajira to Aeolian and Slendros
12
!
21/20
9/8
6/5
49/40
4/3
7/5
3/2
8/5
49/30
9/5
11/6
2/1
! 12 notes selection among two "Dudon scales" :
! D(540/539) = [1 539-135 9 77 5 11 45 3 49 27 7 15]
! D(441/440) = [1 9 147 5 21 11 3 49 55-441 7 63]
! Mohajira scale on white keys : [15 135 147 5 45 49 55]

I have not counted how many slendros you have in here. The black keys
have an ordinary "N" slendro ; the closest to Mohajira is a "M"
slendro with E F# A B C# ;
a Jogyakarta type is E F# G# B C# ; a nice one is of the "SJ" type,
in C# Eb F# G# B that tempers 1029/1024 through the B (=55) of
Mohajira.
There is also a full Aeolian scale in C minor.

Have fun !

Has anyone an idea of what temperaments the tempering of both 540/539
and 441/440 would best refer to ? Miracle ?
- - - - - - -
Jacques

--- In tuning@yahoogroups.com, Jacques Dudon <fotosonix@...> wrote:

> Has anyone an idea of what temperaments the tempering of both 540/539
> and 441/440 would best refer to ? Miracle ?

Just by itself it's jove temperament, but the tuning of harry temperament is basically the same, but it's rank two instead of three. Harry also tempers out 4000/3993. It's also possible to temper out 65625/65536 instead, leading to the 11-limit version of tertiaseptal, or to do both, leading to 202et.

I posted your scale here on the Xenwiki, it seems like quite an interesting 12-note JI scale which some people find to be useful.

> Gene wrote :
> > Has anyone an idea of what temperaments the tempering of both > 540/539
> > and 441/440 would best refer to ? Miracle ?
>
> Just by itself it's jove temperament, but the tuning of harry > temperament is basically the same, but it's rank two instead of > three. Harry also tempers out 4000/3993. It's also possible to > temper out 65625/65536 instead, leading to the 11-limit version of > tertiaseptal, or to do both, leading to 202et.

Sorry, I still haven't catch what was a hobbit temperament, I must say the explanation page is too complex for me... I note that Jove[41] transversal indeed contains all of "Mohajira-to-slendro" ratios, except 8/5 represented by 48/25 instead, 225/224 higher, but I haven't checked possible transpositions (BTW it looks like there is an inversion in this transversal list, between 11/7 and 14/9). So what are Jove, or Harry's generators, or other relevant ones here ?
What about Miracle ? it has a perfect Mohajira and many correct 7th-limit tempered Slendros ; it also tempers 1029/1024, which is relevant in this scale with its [63 : 55 : 48 : 42 : 36] "SJ" type of slendro.

> I posted your scale here on the Xenwiki, it seems like quite an > interesting 12-note JI scale which some people find to be useful.

Thanks ! (I didn't know I had a whole citation in the JI page, agremented with links... it also reminds me I should try to write a few lines about the use of recurrent sequences for eq-b and -c temperaments and what you called "algebraic solutions"... any volunteer to correct my english ?)
- - - - - - -
Jacques

--- In tuning@yahoogroups.com, Jacques Dudon <fotosonix@...> wrote:

> Thanks ! (I didn't know I had a whole citation in the JI page,
> agremented with links... it also reminds me I should try to write a
> few lines about the use of recurrent sequences for eq-b and -c
> temperaments and what you called "algebraic solutions"... any
> volunteer to correct my english ?)

Write the article and I'll correct your English.

--- In tuning@yahoogroups.com, Jacques Dudon <fotosonix@...> wrote:
So
> what are Jove, or Harry's generators, or other relevant ones here ?

Here's Jove:

http://xenharmonic.wikispaces.com/Breed+family

Here's Harry and Tertiaseptal:

http://xenharmonic.wikispaces.com/Breedsmic+temperaments

Miracle I didn't mention because it involves a small reduction in tuning accuracy, but if we allow that other things crowd in as well.