latest generalized diatonic review

At Graham's suggestion I've tried to make my gd rules more simple
and objective. Resulting in the following spec:

http://lumma.org/spec.txt

I applied it to the usual suspects, and the results are shown in
this excel spreadsheet:

http://lumma.org/results.xls

The scala files I used, a text-file version of the results with
notes:

http://lumma.org/gd.zip

There are 28 scales in all. There are only two scales which are
in the top 14 of all four areas: the usual diatonic in 12-tet and
Balzano's nonatonic in 20-tet. There are 7 scales common to the
top-14 of the 3rd and 4th areas:

octatonic
blackwood
symmaj
diatonic
balzano
hahn-433
pentmaj

-Carl

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

> http://lumma.org/spec.txt

This spec does not pinpoint the features of a scale which make it a good one, IMHO. Part of the problem is that you assume a sort of rough linearity ("higher values are better.") Are they really? For
number of scale steps, you restrict to 5-10, which is reasonable, but claim that in that range lower is better; I don't agree. The numbers given by "modal transposition" reflect scale regularity to some extent, but seem to suggest an equal division is melodically perfect--in fact, like number of scale steps, this is highly nonlinear, and you want to do a Goldilocks and come out somewhere in the sweet spot. The numbers for the Miracle-10 MOS (0.75) and Porcupine-7 (0.73) reflect the bland and pudding-like quality which makes them less interesting than they might be, but the Orwell-9 (0.71), despite its comparitive regularity, is melodically wonderful, like Meantone-7 (ie, diatonic) at 0.61. You need a number to reflect the difference, which is crucial to the sound of these scales!

>> http://lumma.org/spec.txt
>
>This spec does not pinpoint the features of a scale which make it a good
>one, IMHO.

It isn't supposed to do that -- just the features that make scales like
the diatonic scale, in its application in Western music.

>Part of the problem is that you assume a sort of rough linearity
>("higher values are better.") Are they really?

Sure.

>For number of scale steps, you restrict to 5-10, which is reasonable,
>but claim that in that range lower is better; I don't agree.

Me either. Lower values make pitch tracking easier. I didn't say this
desirable. I do cut it off at 5, which is enough to prevent the scale
from sounding like a chord in most cases.

>The numbers given by "modal transposition" reflect scale regularity to
>some extent, but seem to suggest an equal division is melodically
>perfect--in fact, like number of scale steps, this is highly nonlinear,
>and you want to do a Goldilocks and come out somewhere in the sweet spot.

An equal division is supposedly perfect with respect to modal
transposition, but will be poor with respect to mode autonomy.

>The numbers for the Miracle-10 MOS (0.75) and Porcupine-7 (0.73) reflect
>the bland and pudding-like quality which makes them less interesting than >they might be, but the Orwell-9 (0.71), despite its comparitive
>regularity, is melodically wonderful, like Meantone-7 (ie, diatonic) at
>0.61. You need a number to reflect the difference, which is crucial to
>the sound of these scales!

So what do you think is going on here?

-Carl

Carl, maybe there's an error in 08-star.scl?
When you take the 46-tET version of the scale
Gene posted you get 5 7 3 9 3 7 5 7 which is
different and has a higher stability.

07-graham.scl is a mode of harmonic major in 31-tET.

Manuel

>Carl, maybe there's an error in 08-star.scl?
>When you take the 46-tET version of the scale
>Gene posted you get 5 7 3 9 3 7 5 7 which is
>different and has a higher stability.

Gene?

0 3 12 15 22 27 34 [37] 46

>07-graham.scl is a mode of harmonic major in 31-tET.

Yeah, I saw that. Graham asked me to put it in.

-Carl

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

> >Carl, maybe there's an error in 08-star.scl?
> >When you take the 46-tET version of the scale
> >Gene posted you get 5 7 3 9 3 7 5 7 which is
> >different and has a higher stability.
>
> Gene?

I should have written [1,25/24,6/5,5/4,36/25,3/2,5/3,9/5], the 46-et
version of which is [0, 3, 12, 15, 24, 27, 34, 39]. However, the
alternative with second degree being approximately 27/25 is very much
worthy of notice also--like star, it is a 126/125-tempered version of
a Fokker block, consisting of two parallel chains of minor thirds,
with a lot of nice harmonic properties. Being a new star, maybe
it's "nova" :)

>I should have written [1,25/24,6/5,5/4,36/25,3/2,5/3,9/5], the 46-et
>version of which is [0, 3, 12, 15, 24, 27, 34, 39]. However, the
>alternative with second degree being approximately 27/25 is very much
>worthy of notice also--like star, it is a 126/125-tempered version of
>a Fokker block, consisting of two parallel chains of minor thirds,
>with a lot of nice harmonic properties. Being a new star, maybe
>it's "nova" :)

Just for my sanity, fill in the blanks:

star = 0 _ _ _ _ _ _ _ 46
nova = 0 _ _ _ _ _ _ _ 46

Thanks!

-Carl

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

> Just for my sanity, fill in the blanks:
>
> star = 0 3 12 15 24 27 34 39 46
> nova = 0 5 12 15 24 27 34 39 46

>> star = 0 3 12 15 24 27 34 39 46
>> nova = 0 5 12 15 24 27 34 39 46

Thanks, Gene. So what I've been calling star is indeed star.
I've added nova to the search. It comes out just above star
because its stability is higher while all its other values are
the same. It also has one more 3:2 in it (vs. star's extra 8:5).

I'll update the search on my site as warranted by the number of new
scales.

-Carl

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

> I'll update the search on my site as warranted by the number of new
> scales.

Did you see my recent posting to the main list? I'd be interested in your assessment of Qm(2) and Qm(3).

>>I'll update the search on my site as warranted by the number of new
>>scales.
>
>Did you see my recent posting to the main list? I'd be interested in your
>assessment of Qm(2) and Qm(3).

I get the main list and Columbia in digest format, so it may be another
day until I see your post.

-Carl

>>I'll update the search on my site as warranted by the number of new
>>scales.
>
>Did you see my recent posting to the main list? I'd be interested in
>your assessment of Qm(2) and Qm(3).

Oh, I guess Qm(2) is possible. I'll have to make it up. It'd be
great if you could post Scala files for these.

Qm(3) is a mode of this scale:

>10-tone Fokker-Lumma, e=27 c=5, in 72-tET
>(0 5 12 19 28 35 42 49 58 65) -> ((32 $ 39 % rms) (20 $ 24 % mad))

Which brings up an important point: who's keeping track of these
discoveries? Perhaps we should begin a database, with keys for
both interval and rank-order matrices. For example,

Qm(3) interval matrix:

((7 14 21 30 37 44 49 56 63 72)
(7 14 23 30 37 42 49 56 65 72)
(7 16 23 30 35 42 49 58 65 72)
(9 16 23 28 35 42 51 58 65 72)
(7 14 19 26 33 42 49 56 63 72)
(7 12 19 26 35 42 49 56 65 72)
(5 12 19 28 35 42 49 58 65 72)
(7 14 23 30 37 44 53 60 67 72)
(7 16 23 30 37 46 53 60 65 72)
(9 16 23 30 39 46 53 58 65 72))

Qm(3) rank-order matrix:

((2 5 8 12 15 18 20 23 26 29)
(2 5 9 12 15 17 20 23 27 29)
(2 6 9 12 14 17 20 24 27 29)
(3 6 9 11 14 17 21 24 27 29)
(2 5 7 10 13 17 20 23 26 29)
(2 4 7 10 14 17 20 23 27 29)
(1 4 7 11 14 17 20 24 27 29)
(2 5 9 12 15 18 22 25 28 29)
(2 6 9 12 15 19 22 25 27 29)
(3 6 9 12 16 19 22 24 27 29))

Manuel,

() What's the best way to get Scala to represent scales as degrees
of an et?

() As far as inputting scales as et subsets, I do "equal n" and then
"select". Is that the Official Way?

() Think we could get View -> rank-order matrix? (Yes, I'm using
2.05 now).

-Carl

--- In tuning-math@y..., Carl Lumma <carl@l...> wrote:
> >>I'll update the search on my site as warranted by the number of new
> >>scales.
> >
> >Did you see my recent posting to the main list? I'd be interested in
> >your assessment of Qm(2) and Qm(3).
>
> Oh, I guess Qm(2) is possible. I'll have to make it up. It'd be
> great if you could post Scala files for these.

Here are my own personal Scala files for them:

! qm2.scl
! [0, 7, 23, 30, 42, 49, 65]
Qm(2) 7-note quasi-miracle scale
7
!
116.6666667
383.3333333
500.
700.
816.6666667
1083.333333
2/1

! qm3a.scl
! [0, 7, 16, 23, 30, 35, 42, 49, 58, 65]
Qm(3) 10-note quasi-miracle scale, mode A
10
!
116.6666667
266.6666667
383.3333333
500.
583.3333333
700.
816.6666667
966.6666667
1083.333333
2/1

! qm3b.scl
! [0, 7, 14, 23, 30, 37, 42, 49, 56, 65]
Qm(3) 10-note quasi-miracle scale, mode B
10
!
116.6666667
233.3333333
383.3333333
500.
616.6666667
700.
816.6666667
933.3333333
1083.333333
2/1

> Qm(3) is a mode of this scale:
>
> >10-tone Fokker-Lumma, e=27 c=5, in 72-tET
> >(0 5 12 19 28 35 42 49 58 65) -> ((32 $ 39 % rms) (20 $ 24 % mad))
>
> Which brings up an important point: who's keeping track of these
> discoveries?

Nobody, I think. Where is the Fokker-Lumma 2-parameter family described?

>Here are my own personal Scala files for them:

Thanks.

>> Qm(3) is a mode of this scale:
>>
>> >10-tone Fokker-Lumma, e=27 c=5, in 72-tET
>> >(0 5 12 19 28 35 42 49 58 65) -> ((32 $ 39 % rms) (20 $ 24 % mad))
>>
>> Which brings up an important point: who's keeping track of these
>> discoveries?
>
>Nobody, I think. Where is the Fokker-Lumma 2-parameter family described?

Don't know what you mean by 2-parameter. AFAIK Fokker-Lumma refers
to any scale with a 225:224 in the map.

-Carl