Yahoo Tuning Groups Ultimate Backup tuning-math Circons (DE circulating consonances for MOS) for the three Ms

We probably don't want more jargon, but if anyone but me finds these
interesting and thinks we do, apparently they don't have a name so
I'll call them "circons". Below are "circons" for MOS of the classic
MMM temperaments--meantone, magic, and miracle. I mostly stayed within
the 9-limit for these. The 16 note MOS of magic, with a circle of
10/7s and 3/2s, the 11 note MOS of miracle with the same, and Canasta,
which can be obtained as a circle of major and minor whole tones, may
be worth noting in particular.

I haven't gotten any feedback but it seems to me it is cool and
potentially useful to know such as the fact that Canasta can be
composed out of a cycle of whole tones. It might be a good way to
produce mutant blendings of meantone and miracle.

==Meantone==

Meantone[5]

3 steps
3/2 : {0,1,3,4}
8/5 : {2}

Meantone[7]

2 steps
6/5 : {2,3,5,6}
5/4 : {0,1,4}

4 steps
10/7 : {3}
3/2 : {0,1,2,4,5,6}

Meantone[12]

3 steps
7/6 : {0,2,7}
6/5 : {1,3,4,5,6,8,9,10,11}

6 steps
7/5 : {0,2,4,7,9,11}
10/7 : {1,3,5,6,8,10}

Meantone[19]

4 steps
8/7 : {2,5,7,8,10,13,15,16,18}
7/6 : {0,1,3,4,6,9,11,12,14,17}

==Magic==

Magic[7]

3 steps
9/7 : {1,3,4,5,6}
3/2 : {0,2}

Magic[10]

5 steps
4/3 : {1,3,5,7,8}
3/2 : {0,2,3,6,9}

Magic[13]

4 steps
8/7 : {9}
5/4 : {0,1,2,3,4,5,6,7,8,10,11,12}

Magic[16]

4 steps
8/7 : {1,6,11,12}
6/5 : {0,2,3,4,5,7,8,9,10,13,14,15,16}

9 steps
10/7 : {1,6,7,11,12}
3/2 : {0,2,3,4,5,8,9,10,13,14,15,16}

Magic[19]

4 steps
8/7 : {1,2,7,8,13,14,15}
7/6 : {0,3,4,5,6,9,10,11,12,16,17,18}

Magic[22]

4 steps
9/8 : {0,4,5,6,7,11,12,13,14,19,20,21}
8/7 : {1,2,3,8,9,10,15,16,17,18}

6 steps
6/5 : {0,2,3,4,5,6,7,9,10,11,12,13,14,17,18,19,20,21}
11/9 := {1,8,15,16}

11 steps
7/5: {0,4,5,6,7,12,13,14,19,20,21}
10/7 : {1,2,3,8,9,10,11,15,16,17,18}

==Miracle==

Miracle[10]

2 steps
8/7 : {0,2,3,4,5,6,7}
7/6 : {8,9}

Miracle[11]

6 steps
10/7 : {5,6,7,8,9,10}
3/2 : {0,1,2,3,4}

Miracle[21] (Blackjack)

5 steps
7/6 : {1,3,5,7,9,11,13,15,16,17,18,19,20}
6/5 : {0,2,4,6,8,10,12,14}

Miracle[31] (Canasta)

5 steps
10/9 : {1,4,7,10,13,16,19,22,25,26,28,29}
9/8 : {0,2,3,5,6,8,9,11,12,14,15,17,18,20,21,23,24,27,30}

Miracle[41] (Studloco)

6 steps
11/10 : {0,3,4,7,8,11,12,15,16,19,20,23,24,27,28,31,32,36,40}
10/9 : {1,2,5,6,9,10,13,14,17,18,21,22,25,26,29,30,33,34,35,37,38,39}

🔗Jacob <jbarton@rice.edu>

🔗Paul Erlich <perlich@aya.yale.edu>

--- In tuning-math@yahoogroups.com, "Jacob" <jbarton@r...> wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...> wrote:
> > I haven't gotten any feedback but it seems to me it is cool and
> > potentially useful to know such as the fact that Canasta can be
> > composed out of a cycle of whole tones. It might be a good way to
> > produce mutant blendings of meantone and miracle.
>
> I'm very interested in finding mutant blendings of scales. I don't
understand this list,
> though; could you make an example of explaining one of them?
> >
> > ==Meantone==
> >
> >
> > Meantone[5]
> >
> > 3 steps
> > 3/2 : {0,1,3,4}
> > 8/5 : {2}
>
> what do you do with the numbers inside the brackets?

I think it just means step #s 0, 1, 3, and 4 are 3/2s, and step #2 is
8/5; each of these intervals subtending 3 steps of the resulting
pentatonic scale.

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> --- In tuning-math@yahoogroups.com, "Jacob" <jbarton@r...> wrote:

> > what do you do with the numbers inside the brackets?
>
> I think it just means step #s 0, 1, 3, and 4 are 3/2s, and step #2 is
> 8/5; each of these intervals subtending 3 steps of the resulting
> pentatonic scale.

By Meantone[5] I meant the 5-note MOS of meantone, but if you meant the
number inside the braces, yes. Above scale steps 0,1,3 and 4 we have a
3/2, and above scale step 2 an 8/5. If we have a MOS of n notes, and go
along regularly taking m steps out of the n notes, we get such an
interval pattern. I'm interested in when the intervals in question are
both consonant, leading to a circle of consonances defining the scale.
Since we started from a MOS, we of course define a MOS in this way, but
circles of consonances can be used to construct other scales. When the
scales become large these often become absurdly irregular even when the
intervals in the circle are DE, but not always--for a counterexample,
Ennealimmal[45]. For smaller scales, we can get nice "modmos" variants
of a MOS.

Anyway, the question does not seem to be much explored, despite the
interest which attaches to the circle of thirds for the diatonic scale.
It seems to me these things count as basic structural features of a
scale. Paul, did you happen to notice that your favorite Pajara
[10]/Diaschismic[10] scale has such a circle of thirds?

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Paul Erlich <perlich@aya.yale.edu>

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

> Anyway, the question does not seem to be much explored, despite the
> interest which attaches to the circle of thirds for the diatonic
scale.
> It seems to me these things count as basic structural features of a
> scale. Paul, did you happen to notice that your favorite Pajara
> [10]/Diaschismic[10] scale has such a circle of thirds?

Yes, and this only because both of the things you call 'thirds'
are "fourths" in the 10-note scales. The circle-of-thirds progression
was a starting point for what became "Decatonic Swing".

The fact that all the thirds/"fourths" are consonant is familiar
from "country" applications of this 10-note scale, commonly licks in
parallel 6ths ("8ths") or 3rds ("4ths"). To that you add the fact that
the numbers 3 and 10 have no common divisors, and the resulting
implication is your circle.