41 Diatonic

So your scale is really a different tuning of Balzano's 11 from 30?

It could be stated that way, but that's not how it came about. Really, I
just 'discovered' when looking at all the scales I found when manually
enumerating the 2d grids some year ago. The abstract bit about scales came
later, and not all scales that can be formed from a/b sequences can be
found as segments of a cyclic generator of a Cn.

Anyway, the point I was trying to make was that, justas the 7-diatonic has
various 'mean-tone' interpretations in 19 and 31 equal, so this particular
11-tone diatonic also has analogous 'mena-tone' interpretations.

This made me ask an important question about scales:

Given the keyboard layout for ol' 12-equal, the black keys are enharmonic
equivalents caused by tempering, and the same is true for 19 and 31; then
for other diatonics, these represent 'tempered' diatonics of just intoned
forms. Hence stuff in my art. about ratios and background shapes. In which
case, looking at the 19-tone 11 tone diatonic, we can assume that the 41
equal 11-tone diatonic must be also a tempered scale.

In this connection, it is interesting to note that the 20 equal 11-tone
diatonic, as sugg. Zweifel, and discussed by me, has the pattern

aaaabaaaaab (9a+2b)

if a=2 and b=1 chromatic = 20 equal
if a=3 and b=2 chromatic = 31 equal
if a=4 and b=3 chromatic = 42 equal
if a=5 and b=3 chromatic = 51 equal
etc..

Strange how these two diatonics sort of 'hang about' each other

19 : 8+3
20 : 9+2
30 : 8+3
31 : 9+2
41 : 8+3
42 : 9+2
(if it were a sequence, I'd guess at:
52 : 8+3
53 : 9+2 but it sort of isn't, past a=5)

Mark

In-Reply-To: <4843.194.203.13.66.1019121786.squirrel@email.argonet.co.uk>
Me:
> So your scale is really a different tuning of Balzano's 11 from 30?

Mark:
> It could be stated that way, but that's not how it came about. Really, I
> just 'discovered' when looking at all the scales I found when manually
> enumerating the 2d grids some year ago. The abstract bit about scales
> came
> later, and not all scales that can be formed from a/b sequences can be
> found as segments of a cyclic generator of a Cn.

An a/b sequence that's a segment of a cyclic generator is "MOS" or "Well
Formed".

> Anyway, the point I was trying to make was that, justas the 7-diatonic
> has
> various 'mean-tone' interpretations in 19 and 31 equal, so this
> particular
> 11-tone diatonic also has analogous 'mena-tone' interpretations.

Yes, that's what I started to work out when you originally posted it to
the big list. But you didn't follow up on my comments, so I didn't give
the reasoning. In fact, I was using this temperament:

11/30, 443.8 cent generator

basis:
(1.0, 0.3697965324659695)

mapping by period and generator:
[(1, 0), (-1, 7), (-1, 9), (-2, 13)]

mapping by steps:
[(19, 11), (30, 17), (44, 25), (53, 30)]

highest interval width: 13
complexity measure: 13 (19 for smallest MOS)
highest error: 0.006241 (7.489 cents)
unique

It can be derived using my cross-platform Python library by

>>> h19 = temper.PrimeET(19, temper.primes[:3])
>>> g11 = temper.PrimeET(11, temper.primes[:3])
>>> g11.basis[2]=25
>>> g11.basis[3]=30
>>> g11&h19

You'd given the scale in terms of 19-equal then, so this temperament is
consistent with 19-equal. You don't get all 7-limit intervals in the 11
note scale, but most of them are there. Only 7:4 and 8:7 are missing, I
think. Also 9:8 is missing from the 9-limit. There's also a tuning with
a 442.9 cent generator that gets the 5-limit to within 1.5 cents. We're
looking at this segment of the scale (or Farey or Stern Brocot) tree

19 11
30
49 41

It's tempting to think the temperament will work with 41-equal, but in
fact it doesn't. The temperament which does unify 41 and 19 is what I
call "magic" and it's this bit of the scale tree:

19 22
41
60 63

22=11*2 and 60=30*2, so the first temperament only has half the number of
notes for some magic scales. That explains my description of it as "half
magic" which nobody seemed to notice.

The 41 diatonic under discussion now is very half magic, because it has to
work with both 41 and 19. That comes form its derivation as alternating
intervals which have to be 7:9 and 9:8. The 9:8 is two steps of the 11
note diatonic. There's no way this can work with a general 9-limit
temperament. An octave plus the 9:8 has to be two fifths. So two fifths
are 11+4=15 diatonic steps. Which means one of the fifths has to be 7 and
the other 8 steps.

So what's going on? Well, it looks like the 11 note diatonic is every
other note from the 22 note magic MOS. Like this

2 1 2 2 2 2 2 2 1 2 2 2 2 2 2 2 1 2 2 2 2 2
3 4 4 4 3 4 4 4 3 4 4

You can't describe the whole 9-limit (or even the whole 5-limit) by
modulating between 11 note diatonics in this temperament. That's
certainly a defect, but not one covered by the current scoring system.

> In this connection, it is interesting to note that the 20 equal 11-tone
> diatonic, as sugg. Zweifel, and discussed by me, has the pattern
>
> aaaabaaaaab (9a+2b)
>
> if a=2 and b=1 chromatic = 20 equal
> if a=3 and b=2 chromatic = 31 equal
> if a=4 and b=3 chromatic = 42 equal
> if a=5 and b=3 chromatic = 51 equal
> etc..
>
> Strange how these two diatonics sort of 'hang about' each other
>
> 19 : 8+3
> 20 : 9+2
> 30 : 8+3
> 31 : 9+2
> 41 : 8+3
> 42 : 9+2
> (if it were a sequence, I'd guess at:
> 52 : 8+3
> 53 : 9+2 but it sort of isn't, past a=5)

It looks like you're fumbling for a scale tree. 9a+2b is

9 2
11
20 13
29 31 44 15
38 49 51 42 55 57 28 17

and the ones you're looking at are between 20 and 11. The similarity
between the two diatonics is presumably that the both have 11 notes, and
are built on chromatics that differ by 10 notes. To get them on the same
tree, you need to go to

2 3
5
7 8
9 12 13 11
11 16 19 17 18 21 19 14

Note that meantone is on there as well. It covers all linear temperaments
with an octave period and a generator between a third and a half of an
octave. 7 and 9 give pelogic, there will be more of interest.

Graham

In-Reply-To: <4843.194.203.13.66.1019121786.squirrel@email.argonet.co.uk>
Mark Gould wrote:

> In this connection, it is interesting to note that the 20 equal 11-tone
> diatonic, as sugg. Zweifel, and discussed by me, has the pattern
>
> aaaabaaaaab (9a+2b)
>
> if a=2 and b=1 chromatic = 20 equal
> if a=3 and b=2 chromatic = 31 equal
> if a=4 and b=3 chromatic = 42 equal
> if a=5 and b=3 chromatic = 51 equal
> etc..

Sorry to give the impression I'm talking to myself, but there are two
entries in Carl's list that fit this pattern.

09- David Rothenberg's generalized diatonic in 31-tet
[0 5 8 11 14 17 22 25 28 31]
3. efficiency 0.74
4. strictly proper
5. no, but 9th is 15:8 in 7 of 9 modes
6. yes; 8th is 5:3 or 7:4 in 9 of 9 modes

09- Balzano's generalized diatonic in 20-tet
[0 2 5 7 9 11 14 16 18 20]
3. efficiency 0.74
4. strictly proper
5. no.
6. yes; 8th is 5:3 or 7:4 in 9 of 9 modes
9th is 9:5 or 15:8 in 9 of 9 modes

They should be unified. The "9th" works as 9:5 in 31-equal as well. 29
belongs to the pattern too, as 3 4 3 3 3 4 3 3 3. I don't know if it fits
the implied temperament. The efficiencies being identical should have
been a clue.

Graham

In-Reply-To: <4843.194.203.13.66.1019121786.squirrel@email.argonet.co.uk>
Mark Gould wrote:

> In this connection, it is interesting to note that the 20 equal 11-tone
> diatonic, as sugg. Zweifel, and discussed by me, has the pattern
>
> aaaabaaaaab (9a+2b)
>
> if a=2 and b=1 chromatic = 20 equal
> if a=3 and b=2 chromatic = 31 equal
> if a=4 and b=3 chromatic = 42 equal
> if a=5 and b=3 chromatic = 51 equal
> etc..

In which case it's related to this entry from Carl's list:

09- David Rothenberg's generalized diatonic in 31-tet
[0 5 8 11 14 17 22 25 28 31]
3. efficiency 0.74
4. strictly proper
5. no, but 9th is 15:8 in 7 of 9 modes
6. yes; 8th is 5:3 or 7:4 in 9 of 9 modes

5 3 3 3 3 5 3 3 3 /31
3 2 2 2 2 3 2 2 2 /20
2 1 1 1 1 2 1 1 1 /11

Graham