Simple proposed nomenclature for identifying MODMOS's

Major = meantone[7]
Melodic minor = meantone[7]#1 = meantone[7]b7
Harmonic minor = meantone[7]#2 = meantone[7]b3b7
Harmonic major = meantone[7]b3 = meantone[7]#2#5

So if you haven't picked up on the pattern, "meantone[7]" means lydian
mode, aka all generators going up - which for the purposes of this I
propose just be the "standard mode" for whatever temperament we're
talking about. The numbers being sharpened or flattened indicate the
scale degree in the given MOS that's changing. So for meantone[7]#2,
that means lydian #2, which is indeed a mode of harmonic minor.
Remember again that the # and b accidentals generally mean alteration
by L=c-s, although you are always free to denote whatever accidentals
you want for specific MOS's.

There are a few other ways to do this as well, like letting the #'s
and b's denote which position in the generator chain is altered, or
doing everything in relation to a different scale than all generators
going up vs down. However, having the #'s and b's be this way is
consistent with existing terminology, and although you're free to call
Lssssss something like "porcupine major" and talk about "porcupine
major b4" and so on, since there are an extraordinary amount of
temperaments this can be a quick and standard way to identify the
MODMOS's for all of them. We could also pick a different reference
mode, e.g. all generators going down or split it evenly, but this
seems like the simplest way to do things.

Thoughts?

-Mike

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> Thoughts?

You've not standardized the generator, and I'm not clear on what to do if the period isn't an octave.

On Tue, May 17, 2011 at 11:58 AM, genewardsmith
<genewardsmith@sbcglobal.net> wrote:
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > Thoughts?
>
> You've not standardized the generator, and I'm not clear on what to do if the period isn't an octave.

Good point. Should we just make the generator the one that's less than
600 cents, as is the convention right now? If so, then everything I
said up there is inverted, and the "base mode" for meantone[7] would
be Locrian rather than Lydian.

If the period isn't an octave, why would that pose a problem?
Pajara[10]#5 just sharps the fifth of the scale with all generators
going up. When you say if the period isn't an octave, do you mean if
the equivalence interval isn't an octave?

-Mike

On 5/17/2011 2:25 PM, Mike Battaglia wrote:
> On Tue, May 17, 2011 at 11:58 AM, genewardsmith
> <genewardsmith@sbcglobal.net> wrote:
>>
>> --- In tuning-math@yahoogroups.com, Mike Battaglia<battaglia01@...> wrote:
>>
>>> Thoughts?
>>
>> You've not standardized the generator, and I'm not clear on what to do if the period isn't an octave.
>
> Good point. Should we just make the generator the one that's less than
> 600 cents, as is the convention right now? If so, then everything I
> said up there is inverted, and the "base mode" for meantone[7] would
> be Locrian rather than Lydian.

In most cases having a generator less than half the period will work, although there is at least one case where you could have ambiguity depending on how you optimize the tuning (7&14c, jamesbond). The two mappings for jamesbond are:

[<7 11 16 19|, <0 0 0 1|]
[<7 11 16 20|, <0 0 0 -1|]

In this case we'll need to pick one arbitrarily, and if we look at 11-limit extensions we'll find 7&14c as "septimal".

[<7 11 16 20 24|, <0 0 0 -1 0|]

So I'll suggest [<7 11 16 20|, <0 0 0 -1|] as a "standard" mapping for 7-limit jamesbond. If any other cases come up, we can deal with them individually, but this is the only one I'm familiar with.

> If the period isn't an octave, why would that pose a problem?
> Pajara[10]#5 just sharps the fifth of the scale with all generators
> going up. When you say if the period isn't an octave, do you mean if
> the equivalence interval isn't an octave?
>
> -Mike

Well, if pajara[10] is identified with Paul's "symmetrical" modes, then you could use pajara[10]#5 to denote the "pentachordal" modes. But pajara[10]#10 would be equivalent. Now consider pajara[10]#4#7, aka pajara[10]#2#9. Which one of those labels should you use?