Re: more on composing in X tuning (was Good 5-limit scale generators)

I wrote,
>>> But in fact I'm suggesting that the
>>> _only_
>>> 7-tone scales that have _more_ complete 5-limit triads than this
>one
>>> (with
>>> errors not worse than 12-tET) are diatonic. Do you know of any
>>> counter-example?

Graham Breed wrote:
>> More than 4 5-limit triads in a 7 note scale? I most certainly do!
>>
>> A-----E
>> / \ / \
>> / \ / \
>> F-----C-----G
>> \ / \ /
>> \ / \ /
>> Ab----Eb

Paul Erlich wrote:
>Dave probably meant scales with a single
>generator, and specifically MOS scales.

Thanks Paul, I wish I had. But in fact I was just plain wrong.
Thanks Graham, You have pointed out a bug in my triple-chain spreadsheet.
The spreadsheet should have thrown this up as three chains of fifths a
third of an octave apart. In fact you can get 7 triads with a different
(but still non-diatonic) choice of 7 notes from this tuning.

G#Ab-D#Eb
/ \ /
/ \ /
E-----B
/ \ / \
/ \ / \
C-----G-----D
/ \ /
/ \ /
G#Ab-D#Eb

The smallest proper MOS it embeds in has 9 notes (with 12 triads) (e.g. in
12-tET). Is it correct to use the term MOS for something with n>1 chains of
the generator spaced 1/n octave apart?

I also forgot you can get 5 triads with 7 notes using 4 chains 1/4 octave
apart. The smallest MOS it embeds in is 8 notes (with 8 triads).

Graham Breed:
>> I worked all this stuff out a while back, and wrote it up at:
>>
>> <http://x31eq.com/genera.htm>

Oh yeah! Thanks.

So it seems that the only 7 note scale with 4 or more triads that isn't in
12-tET or 19-tET (or meantone in general) is that one with a single chain
of 163 cent generators (3/22 octave) and it's very nice that it is a proper
MOS.

>> I notice a misteak there,

Please fix it up. There's an indefinite amount of future for it to be wrong
in.

-- Dave Keenan
http://dkeenan.com

I wrote:
>>I also forgot you can get 5 triads with 7 notes using 4 chains 1/4 octave
>>apart. The smallest MOS it embeds in is 8 notes (with 8 triads).

Carl Lumma wrote:
>What scale is this?

I was mistaken in calling it a MOS, but it _is_ proper. Note that the
generator here can be either a fifth or a major third. They end up as the
same 8 note scale. In 12-tET it is 12121212. Outside of 12-tET the minor
thirds must stay at 300 cents but you are free to narrow the fifths to
improve the major thirds. The 8 triads are

D#Eb--A#Bb
\ / \
\ / \
F#----C#
\ / \
\ / \
A-----E
\ / \
\ / \
C-----G
\ / \
\ / \
D#Eb--A#Bb

Delete any note and you lose 3 triads. Of course once you've deleted a
note, you might as well lose the lone third as well and tune the whole
thing Just.

Here's the 9 note proper scale with 12 triads, based on 3 chains of fifths
(or minor thirds) 1/3 octave apart. In 12-tET it's 121121121. The major
thirds are stuck at 400 cents but you can widen the fifths to improve the
minor thirds. If you lose an outside note you lose 3 triads. Lose a second
note on the same side and you only lose another 2 triads, leaving 7 triads
in 7 notes.

C#Db--G#Ab--D#Eb
/ \ / \ /
/ \ / \ /
A-----E-----B
/ \ / \ /
/ \ / \ /
F-----C-----G
/ \ / \ /
/ \ / \ /
C#Db--G#Ab--D#Eb

It would be interesting to see if I come up with all Graham Breed's 5-limit
scales (plus the extra one in 22-tET) by a completely different method.
I'll list the good two-chain generators (with a half-octave) soon.

Do these symmetrical 8 and 9 note scales have standard names?

By the way, why is this stuff in a thread called "more on composing in X
tuning"? It doesn't seem to have anything to do with composing per se.
"Good 5-limit scale generators" seemed like a perfectly good title to me.

I'm only reading the list via the website now, so if you particularly want
me to read something it should have "Dave Keenan" in it somewhere. Usually
this will occur naturally as "Dave Keenan wrote:".

Regards,
-- Dave Keenan
http://dkeenan.com