spiraling 19-edo diminished scale

i've been playing around on a 19-edo keyboard
at the Sonic Arts Gallery, with a version of
the diminished scale in 19. it alternates steps of
3 degrees of 19 and 2 degrees of 19.

it is similar to the 12-edo version of diminished,
which is octatonic, except that the 8th note of the
scale is 1 degree of 19 above the octave. thus, the
scale keeps introducing new pitches until it has
gone thru 38 steps, at which point it cycles back
to the beginning of the pattern.

here's the whole scale. the first "19edo" column
(the 2nd from left) gives the total number of
19edo degrees from the starting point. the 3rd column
shows the number of octaves needed to be subtracted
to "normalize" the note into the reference octave,
and the second "19edo" column (4th from left) gives
the octave-reduced number of 19edo degrees. the final
column shows the cents equivalent. i'd love to see
some mathematical analyses of this ... in particular,
i can't find a generator (number of 19edo degrees)
which will generate it.

scale . 19edo . . . . 19edo
step . degree . 8ve . degree . ~cents

.. 1 .... 0 .... 0 .... 0 ........ 0
.. 2 .... 3 .... 0 .... 3 ...... 189
.. 3 .... 5 .... 0 .... 5 ...... 316
.. 4 .... 8 .... 0 .... 8 ...... 505
.. 5 ... 10 .... 0 ... 10 ...... 632
.. 6 ... 13 .... 0 ... 13 ...... 821
.. 7 ... 15 .... 0 ... 15 ...... 947
.. 8 ... 18 .... 0 ... 18 ..... 1137
.. 9 ... 20 .... 1 .... 1 ....... 63
. 10 ... 23 .... 1 .... 4 ...... 253
. 11 ... 25 .... 1 .... 6 ...... 379
. 12 ... 28 .... 1 .... 9 ...... 568
. 13 ... 30 .... 1 ... 11 ...... 695
. 14 ... 33 .... 1 ... 14 ...... 884
. 15 ... 35 .... 1 ... 16 ..... 1011
. 16 ... 38 .... 2 .... 0 ........ 0
. 17 ... 40 .... 2 .... 2 ...... 126
. 18 ... 43 .... 2 .... 5 ...... 316
. 19 ... 45 .... 2 .... 7 ...... 442
. 20 ... 48 .... 2 ... 10 ...... 632
. 21 ... 50 .... 2 ... 12 ...... 758
. 22 ... 53 .... 2 ... 15 ...... 947
. 23 ... 55 .... 2 ... 17 ..... 1074
. 24 ... 58 .... 3 .... 1 ....... 63
. 25 ... 60 .... 3 .... 3 ...... 189
. 26 ... 63 .... 3 .... 6 ...... 379
. 27 ... 65 .... 3 .... 8 ...... 505
. 28 ... 68 .... 3 ... 11 ...... 695
. 29 ... 70 .... 3 ... 13 ...... 821
. 30 ... 73 .... 3 ... 16 ..... 1011
. 31 ... 75 .... 3 ... 18 ..... 1137
. 32 ... 78 .... 4 .... 2 ...... 126
. 33 ... 80 .... 4 .... 4 ...... 253
. 34 ... 83 .... 4 .... 7 ...... 442
. 35 ... 85 .... 4 .... 9 ...... 568
. 36 ... 88 .... 4 ... 12 ...... 758
. 37 ... 90 .... 4 ... 14 ...... 884
. 38 ... 93 .... 4 ... 17 ..... 1074
. 39 ... 95 .... 5 .... 0 ........ 0

-monz

--- In tuning-math@yahoogroups.com, "monz" <monz@t...> wrote:
>
> i've been playing around on a 19-edo keyboard
> at the Sonic Arts Gallery, with a version of
> the diminished scale in 19. it alternates steps of
> 3 degrees of 19 and 2 degrees of 19.

I guess you could consider it a MOS with period of two octaves and
generator of five steps of 19. In other words take the 15-note MOS of
38-equal with 5/38 as a generator, and then stretch everything by a
factor of two. Since five steps of 19 gives an excellent 6/5, this has
got a lot of minor thirds going for it, but since you can't represent
an octave in this setup, it is a subgroup system--it represents a
subgroup of the 5-limit, not the whole thing. Since (6/5)^5 ~ 5/2
modulo 15625/15552, we get 5 times odd powers of two in this system,
but not times even powers. On the other hand, from (6/5)^6 ~ 3, we
discover that we get 3 times even powers, but not odd powers. It is a
{3, 4, 10} subgroup system.

hi Gene,

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

> --- In tuning-math@yahoogroups.com, "monz" <monz@t...> wrote:
> >
> > i've been playing around on a 19-edo keyboard
> > at the Sonic Arts Gallery, with a version of
> > the diminished scale in 19. it alternates steps of
> > 3 degrees of 19 and 2 degrees of 19.
>
> I guess you could consider it a MOS with period of
> two octaves and generator of five steps of 19.

i'm not seeing that.
i see it repeating after 5 octaves, not 2.

-monz

--- In tuning-math@yahoogroups.com, "monz" <monz@t...> wrote:

> i'm not seeing that.
> i see it repeating after 5 octaves, not 2.

Somehow I got the idea you said it repeated after 2. The point about
five octaves is that (6/5)^19 ~ 2^5, and these are the same in 19-et.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> --- In tuning-math@yahoogroups.com, "monz" <monz@t...> wrote:
>
> > i'm not seeing that.
> > i see it repeating after 5 octaves, not 2.
>
> Somehow I got the idea you said it repeated after 2.
> The point about five octaves is that (6/5)^19 ~ 2^5,
> and these are the same in 19-et.

right, that makes sense. but here's what i find
intriguing about this scale:

because 19 is a prime number, one may use any of
its intervals as a generator and it will generate
the entire scale, unlike the case with 12-edo, where
certain intervals only generate subset scales.
so using 5 degrees as a generator will generate the
entire 19-edo after stepping thru 19 generators.

cycling thru this diminished scale for 2 octaves
generates 15 pitches of 19-edo and does bring the
pitch back to the starting pitch, but the
diminished scale isn't finished cycling because
the next step is 2 degrees in size rather than 3
as at the beginning.

stepping thru the next octave of pitches reuses
some of the notes that have already been generated,
and also generates the remaining 4. so by the time
the 23rd note is reached, all degrees of 19-edo have
been generated. but still, the diminished scale
doesn't stop there. it keeps going until the end
of the 5th octave, which encompasses 38 notes.
so every degree of 19-edo has to be used twice
to generate the entire repeating diminished scale
pattern.

here's an ASCII schematic pitch-height graph of
the whole scale, showing the octave reductions:

.......18.....................18
......................17.............17
..............16.............16
......15.............15
.............14.....................14
.....13.....................13
....................12.............12
............11.............11
....10.............10
...........9......................9
...8......................8
..................7..............7
..........6..............6
..5..............5
.........4......................4
.3......................3
................2..............2
........1..............1
0..............0......................0

-monz
http://tonalsoft.com