Finding a chromatic pair with the following properties...?

Hi all,

I've been silently absorbing everything posted on the "chromatic
pairs" page. I was wondering if there was a way to find, or if there
exists at all, a chromatic pair with the following properties:

1) The albitonic and chromatic scales are both proper, or the
mega-albitonic and chromatic scales are both proper even if the
albitonic one is not
2) The subgroup involved contains 3, not just 9
3) The subgroup involved contains higher-limit primes, like 7, 11, and
13, even if it doesn't contain 5 (in fact, to hell with 5!)
4) The subgroup involved generally shoots for a "rooted" target chord,
something like 4:6:7:11:13 or 8:11:12:13 - not 10:13:15
5) The subgroup involved doesn't have a generator that's a fifth,
meaning the chromatic pair isn't just the usual diatonic and chromatic
scales

I have yet to find one of these - is there some reason that this
doesn't exist? In fact, I don't even care if it's a subgroup scale or
not. Porcupine is I guess a better example of something I'd be going
for, but for some reason I think it would be artistically cleansing to
get away from 5 for a little while.

-Mike

Try 26-EDO, generator of 3 steps or ~138.46 cents. 4 generators gets you 11/8, 5 gets you 3/2, 6 gets you 13/8, 7 gets you 7/4. MOS's at 9 and 10 notes (albitonic, both proper) and 17 notes (chromatic). There might be a better optimization of it, but this version's pretty good...also found in 17-EDO and 35-EDO, and certainly something higher. Misses 5/4 pretty solidly. Both of your requested triads are present, although 7 has a pretty high complexity...but I have a hard time imagining a scale that gives more 8:11:12:13's for the money.

-Igs

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> Hi all,
>
> I've been silently absorbing everything posted on the "chromatic
> pairs" page. I was wondering if there was a way to find, or if there
> exists at all, a chromatic pair with the following properties:
>
> 1) The albitonic and chromatic scales are both proper, or the
> mega-albitonic and chromatic scales are both proper even if the
> albitonic one is not
> 2) The subgroup involved contains 3, not just 9
> 3) The subgroup involved contains higher-limit primes, like 7, 11, and
> 13, even if it doesn't contain 5 (in fact, to hell with 5!)
> 4) The subgroup involved generally shoots for a "rooted" target chord,
> something like 4:6:7:11:13 or 8:11:12:13 - not 10:13:15
> 5) The subgroup involved doesn't have a generator that's a fifth,
> meaning the chromatic pair isn't just the usual diatonic and chromatic
> scales
>
> I have yet to find one of these - is there some reason that this
> doesn't exist? In fact, I don't even care if it's a subgroup scale or
> not. Porcupine is I guess a better example of something I'd be going
> for, but for some reason I think it would be artistically cleansing to
> get away from 5 for a little while.
>
> -Mike
>

Igs & Mike,

You'll find this scale in my 17-tone paper:
http://xenharmony.wikispaces.com/space/showimage/17puzzle.pdf
Search for "9-tone".

A few years ago I started working on a piece using this scale with 6:7:9:11:13 chords (but haven't had time to finish it):
http://xenharmony.wikispaces.com/space/showimage/17WTjazz.mp3
This is based on the 1978 improvisation mentioned in the paper.

--George

--- In tuning-math@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:
>
> Try 26-EDO, generator of 3 steps or ~138.46 cents. 4 generators gets you 11/8, 5 gets you 3/2, 6 gets you 13/8, 7 gets you 7/4. MOS's at 9 and 10 notes (albitonic, both proper) and 17 notes (chromatic). There might be a better optimization of it, but this version's pretty good...also found in 17-EDO and 35-EDO, and certainly something higher. Misses 5/4 pretty solidly. Both of your requested triads are present, although 7 has a pretty high complexity...but I have a hard time imagining a scale that gives more 8:11:12:13's for the money.
>
> -Igs
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@> wrote:
> >
> > Hi all,
> >
> > I've been silently absorbing everything posted on the "chromatic
> > pairs" page. I was wondering if there was a way to find, or if there
> > exists at all, a chromatic pair with the following properties:
> >
> > 1) The albitonic and chromatic scales are both proper, or the
> > mega-albitonic and chromatic scales are both proper even if the
> > albitonic one is not
> > 2) The subgroup involved contains 3, not just 9
> > 3) The subgroup involved contains higher-limit primes, like 7, 11, and
> > 13, even if it doesn't contain 5 (in fact, to hell with 5!)
> > 4) The subgroup involved generally shoots for a "rooted" target chord,
> > something like 4:6:7:11:13 or 8:11:12:13 - not 10:13:15
> > 5) The subgroup involved doesn't have a generator that's a fifth,
> > meaning the chromatic pair isn't just the usual diatonic and chromatic
> > scales
> >
> > I have yet to find one of these - is there some reason that this
> > doesn't exist? In fact, I don't even care if it's a subgroup scale or
> > not. Porcupine is I guess a better example of something I'd be going
> > for, but for some reason I think it would be artistically cleansing to
> > get away from 5 for a little while.
> >
> > -Mike
> >
>

> MOS's at 9 and 10 notes (albitonic, both proper) and 17 notes (chromatic). There might be a better optimization of it, but this

This should say 8 and 9 notes, right?

Keenan

Hi Igs and Gene and Keenan - thanks for the reply. I guess the MOS's
of 17-equal are an obvious choice for something like this. I
particularly liked this one - I bet the MODMOS's of it are fantastic.
I'll post them once I figure out what I'm doing here :)

-Mike

On Tue, Mar 15, 2011 at 2:13 PM, gdsecor <gdsecor@yahoo.com> wrote:
>
> Igs & Mike,

--- In tuning-math@yahoogroups.com, "Keenan Pepper" <keenanpepper@...> wrote:
> This should say 8 and 9 notes, right?

Yes. Must've counted the root and the octave.

-Igs

--- In tuning-math@yahoogroups.com, "Keenan Pepper" <keenanpepper@...> wrote:
>
> > MOS's at 9 and 10 notes (albitonic, both proper) and 17 notes (chromatic). There might be a better optimization of it, but this
>
> This should say 8 and 9 notes, right?

You wouldn't happen to know the exact definition of Pepper temperament, would you?

On Wed, Mar 16, 2011 at 10:50 PM, genewardsmith
<genewardsmith@sbcglobal.net> wrote:
>
> --- In tuning-math@yahoogroups.com, "Keenan Pepper" <keenanpepper@...> wrote:
> >
> > > MOS's at 9 and 10 notes (albitonic, both proper) and 17 notes (chromatic). There might be a better optimization of it, but this
> >
> > This should say 8 and 9 notes, right?
>
> You wouldn't happen to know the exact definition of Pepper temperament, would you?

Haha! This Pepper temperament is starting to take on mythical
proportions. I imagine it's the most perfect, beautiful temperament
ever discovered, one which renders its composer able to fluidly write
music expressing any color and emotion he wishes, but nobody remembers
what it is.

-Mike

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> > You wouldn't happen to know the exact definition of Pepper temperament, would you?
>
> Haha! This Pepper temperament is starting to take on mythical
> proportions. I imagine it's the most perfect, beautiful temperament
> ever discovered, one which renders its composer able to fluidly write
> music expressing any color and emotion he wishes, but nobody remembers
> what it is.
>
> -Mike

That rumor is fine by me. =)

I do remember something being named after me, but as I recall it was just a specific superpythagorean tuning where the chromatic and diatonic semitones have a ratio of the golden mean. That gives you a nice Fibonacci-like sequence of MOSs - 12, 17, 29, 46, 75... - but other than that it's nothing special.

Keenan

--- In tuning-math@yahoogroups.com, "Keenan Pepper" <keenanpepper@...> wrote:

> I do remember something being named after me, but as I recall it was just a specific superpythagorean tuning where the chromatic and diatonic semitones have a ratio of the golden mean. That gives you a nice Fibonacci-like sequence of MOSs - 12, 17, 29, 46, 75... - but other than that it's nothing special.

Ah, but there's a clue. All I could recall was that it was a no-fives temperament with a fifth about the size of the 46et fifth, but with some precise Margo-like definition.

--- In tuning-math@yahoogroups.com, "Keenan Pepper" <keenanpepper@...> wrote:
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@> wrote:
> > > You wouldn't happen to know the exact definition of Pepper temperament, would you?
> >
> > Haha! This Pepper temperament is starting to take on mythical
> > proportions. I imagine it's the most perfect, beautiful temperament
> > ever discovered, one which renders its composer able to fluidly write
> > music expressing any color and emotion he wishes, but nobody remembers
> > what it is.
> >
> > -Mike
>
> That rumor is fine by me. =)
>
> I do remember something being named after me, but as I recall it was just a specific superpythagorean tuning where the chromatic and diatonic semitones have a ratio of the golden mean. That gives you a nice Fibonacci-like sequence of MOSs - 12, 17, 29, 46, 75... - but other than that it's nothing special.
>
> Keenan

A search for "peppermint" on the main list turned up this message (among numerous others):
/tuning/topicId_91397.html#91408

--George

--- In tuning-math@yahoogroups.com, "gdsecor" <gdsecor@...> wrote:
>
>
>
> --- In tuning-math@yahoogroups.com, "Keenan Pepper" <keenanpepper@> wrote:

> > I do remember something being named after me, but as I recall it was just a specific superpythagorean tuning where the chromatic and diatonic semitones have a ratio of the golden mean. That gives you a nice Fibonacci-like sequence of MOSs - 12, 17, 29, 46, 75... - but other than that it's nothing special.
> >
> > Keenan
>
> A search for "peppermint" on the main list turned up this message (among numerous others):
> /tuning/topicId_91397.html#91408

OK. A less convoluted way of expressing it is that the peppermint fifth is (33+phi)/59 octave, where phi is the golden ratio.