Re: [tuning-math] Digest Number 1254 diminished scales in edo 88

Why Monz would choose alternating steps of 3 and 5 19 edo intervals to attempt to generate a diminished scale escapes me.
(except that he is playing a 19edo instrument) - been there tried that;-)

As I see it musically, what you are attempting to do is produce many equal ascending intervals of flat thirds (bIII): e.g.C-Eb-Gb-Bbb-Dbb-Fbb-Abbb etc.
i.e. Intervals of One Large + one small intervals

In 19 edo - the Large interval is 3 steps and the small is 2 = 5 steps

http://www.lucytune.com/tuning/equal_temp.html

If you consider it in the tradition of 12edo
The diminished splits the octave into four equal intervals hence 3 semitones.
The augmented splits the octave into three equal intervals hence 4 semitones.

For all meantone-type systems with the fifth of less than 700 cents, you will find that four diminished intervals (L+s) will be greater than an octave, 4(L+s) = 8L+4s.
and three augmented intervals (2L) will be less than an octave.3(2L) = 6L.

http://www.lucytune.com/tuning/mean_tone.html

How about thinking it out in 88 edo if you want to play with an edo tuning?

The Large in 88edo is 14 and 9 = 23 steps
Diminished =23 steps * 4 = 92
Augmented =28 steps * 3 = 84

You can them play with all sorts of interval differences between numbers of Dim & Aug steps and octaves.
Although it looks to be that using 88edo arrives four steps sharp or flat of an octave.

Which BTW is not necessarily what happens with a spiral meantone system not derived from an edo; but that's another kettle of screaming fish......

Message: 12
Date: Sat, 09 Apr 2005 02:40:07 -0000
From: "monz" <monz@tonalsoft.com>
Subject: 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