Developing a mapping framework for inconsistent temperaments

I just did a search on tuning-math for this, but couldn't find
anything in this vein. So here's an idea about how to work with
inconsistent temperaments that I think is somewhat elegant and fits
into the rest of regular temperament theory.

In the "driftwood" scale, I just posted, I mentioned an alternate
mapping for 9 which enables you to play 4:7:8:9 chords over 5 of the
10 roots of the scale - which is a really cool sound. You can
transpose by fifth any time you want, and there's probably some
interesting possibilities to set up a tonal framework within there
somewhere. The same thing happens in whitewood[14], where you get
consonant 8:9:10:11:12 tetrads over the major roots, but the 8:9 is
different than the mapping you'd get by moving up two 3/2's and down
an octave. However, it's inconsistent, and I don't like that. Every
time I play this alternate 9, I feel guilty and shameful within. 3*3
has equaled 9 for the rest of my life, and it should still equal 9
today too. This is no good.

So here's a rough solution I've worked out:

When you're in meantone, and you're used to having 81/80 tempered out,
it takes a bit of adjusting to move to 5-limit JI. You suddenly have
to deal with comma pumps, and intervals that were equated are no
longer equated. Things don't intuitively work out like you're used to.
In the same vein, someone who's used to dealing with JI is going to
have to deal with new "comma pumps" if they're in an inconsistent
tuning where 9/4 doesn't equal 3/2 * 3/2. Except, these aren't really
comma pumps anymore, but they are, kind of.

So the logical solution to me seems to be, that to deal with
inconsistent temperaments, that you'd treat the inconsistent interval
as its own "prime." So in this case, you'd have the 9 that you'd reach
by two 3's - good for major 9 chords, I imagine, where you really want
to hear 3/2 on top of 3/2 - and then you'd have the 9' that you'd
reach separately, as a direct consonance to the root. This is probably
more what you'd want in driftwood, where you have 4:7:9 chords that
don't involve 3, or in whitewood, where you have a 8:9:10:11:12 chord
that sounds way better if the other mapping for 9 is used, even though
the 9:12 will be different than an octave-inverted 8:12. This is
because the whole smear of notes around 8:9:10 is what's important
perceptualy, not the fact that technically, 8:12 and 9:12 aren't
octave inversions of one another.

So how can we represent this? Well, when musicians talk about the form
of a piece of music, they will often refer to it as something like A A
B A' (where A' is pronounced "A prime"), and the A' is some kind of
variation on A. So perhaps if we're deliberately talking about a
temperament in which we want a different mapping for 9 than just 3*3,
we could call the altered 9 9', pronounced "nine prime."

If we do this, then the chord in driftwood I'm talking about is
actually 4:7:9', and the whitewood chord is 8:9':10:11:12. And
driftwood would be a 2.3.7.9'.13/10 temperament, in which a
distinction is drawn between 9/8 and 9'/8.

In any non-inconsistent temperament, or in JI, composite intervals
like 4'/4, 6'/6, 8'/8, 9'/9 are equated with 1/1. In an inconsistent
temperament, 9'/9 != 1/1. You could also temper things based on this;
say, for instance, you're in a tuning where 9'/8 is flat and 3/2 is
sharp. Thus, 12/9' would be wide. Perhaps you could temper out the
difference between 12/9' and 11/8, and hence eliminate 99'/96.

Or, put briefly: temperament is where a prime becomes composite, and
these inconsistent temperaments are where a composite becomes prime -
so it would be wise to represent them this way.

Some problems with this method - the notation gets pretty messed up if
more than one prime is involved, so you'd need to come up with a
different symbol for each faux-prime you invent. And if you're
multiplying a prime by itself, like if you're tempering out the
difference between two 9'/8s and a 5/4, that should be 81''/80, I
suppose, so that you can distinguish it between 9'/8 * 9'/8 (two 9' s)
and 9'/8 * 9/8 (one 9' and one normal 9).

Thoughts?

-Mike

On 1/18/2011 1:02 AM, Mike Battaglia wrote:

> So the logical solution to me seems to be, that to deal with
> inconsistent temperaments, that you'd treat the inconsistent interval
> as its own "prime." So in this case, you'd have the 9 that you'd reach
> by two 3's - good for major 9 chords, I imagine, where you really want
> to hear 3/2 on top of 3/2 - and then you'd have the 9' that you'd
> reach separately, as a direct consonance to the root.

I've had similar thoughts about inconsistent temperaments, about adding more "primes" to the set. Call them "basic" intervals or something, since they're not actually primes.

Since you don't have a unique factorization for all intervals, you'll need to be explicit about which combination of "basic" intervals you're using. I think probably the simplest way to represent the composite intervals is with an exponent vector (similar to a monzo, but using all of your "basic" intervals as bases for the exponents). So you might have a 2 3 5 7 9' temperament, for instance; a 3/2 would still be [-1 1 0 0 0>, but 9/8 could be either [-3 2 0 0 0> or [-3 0 0 0 1>, and 27/25 could be either [0 3 -2 0 0> or [0 1 -2 0 1>. Even the basic 3/2 could be alternatively represented as [-1 -1 0 0 1> (9'/6).

Then you get into issues of how to optimize the tuning. I imagine something like RMS optimization would still work, but I don't know how reliable the results would be, with more than one way to represent many intervals.

On Tue, Jan 18, 2011 at 10:08 PM, Herman Miller <hmiller@io.com> wrote:
>
> On 1/18/2011 1:02 AM, Mike Battaglia wrote:
>
> > So the logical solution to me seems to be, that to deal with
> > inconsistent temperaments, that you'd treat the inconsistent interval
> > as its own "prime." So in this case, you'd have the 9 that you'd reach
> > by two 3's - good for major 9 chords, I imagine, where you really want
> > to hear 3/2 on top of 3/2 - and then you'd have the 9' that you'd
> > reach separately, as a direct consonance to the root.
>
> I've had similar thoughts about inconsistent temperaments, about adding
> more "primes" to the set. Call them "basic" intervals or something,
> since they're not actually primes.

I like the notion of differentiating between direct-to-root and
indirect-to-root intervals.

I was thinking though, can all inconsistent temperaments be boiled
down to this notion of making a composite interval a faux-prime? Take
the 11-limit hexad in 68-tet for example - there are two mappings for
11. The better one for 11/8 is worse for 11/9, and vice versa. How
does this work? Would you be making 11/9 its own "prime" as well? So
you'd have 2.3.5.7.9.11.(11/9)' or something? I guess that would
extend subgroup temperaments, kind of.

> Since you don't have a unique factorization for all intervals, you'll
> need to be explicit about which combination of "basic" intervals you're
> using. I think probably the simplest way to represent the composite
> intervals is with an exponent vector (similar to a monzo, but using all
> of your "basic" intervals as bases for the exponents). So you might have
> a 2 3 5 7 9' temperament, for instance; a 3/2 would still be [-1 1 0 0
> 0>, but 9/8 could be either [-3 2 0 0 0> or [-3 0 0 0 1>, and 27/25
> could be either [0 3 -2 0 0> or [0 1 -2 0 1>. Even the basic 3/2 could
> be alternatively represented as [-1 -1 0 0 1> (9'/6).

Yeah, I like this. You draw a distinction between 9/8 and 9'/8. The
only problem is, what happens if you have two primes? You need two
different operators, because if you have 9' and 11', and you're
tempering out 99'/98, does 99' equal 9' * 11 or 9 * 11'? You'd have to
call it 9' and 11* and 13^ or something.

> Then you get into issues of how to optimize the tuning. I imagine
> something like RMS optimization would still work, but I don't know how
> reliable the results would be, with more than one way to represent many
> intervals.

Some kind of TOP-based thing would still work, right? If we're saying
that 9' is a different thing than 9, then you just optimize for 2,3,5,
and 9'...

-Mike

On Tue, Jan 18, 2011 at 1:02 AM, Mike Battaglia <battaglia01@gmail.com> wrote:
>
> So how can we represent this? Well, when musicians talk about the form
> of a piece of music, they will often refer to it as something like A A
> B A' (where A' is pronounced "A prime"), and the A' is some kind of
> variation on A. So perhaps if we're deliberately talking about a
> temperament in which we want a different mapping for 9 than just 3*3,
> we could call the altered 9 9', pronounced "nine prime."
>
> If we do this, then the chord in driftwood I'm talking about is
> actually 4:7:9', and the whitewood chord is 8:9':10:11:12. And
> driftwood would be a 2.3.7.9'.13/10 temperament, in which a
> distinction is drawn between 9/8 and 9'/8.

To bring it back to this, it seems like lots of interesting
temperaments have better representations if their inconsistencies are
embraced. For example, mavila has a far better representation of 9/8
than the one you get at by going up two fifths, which is in 23-equal
157 cents. If you go up a mavila fourth and down a mavila 6/5 you end
up in 23-equal at 209 cents, which maps to 10/9. So since 81/80 is
reversed, 10/9 is larger than 9/8.

But if we're in mavila[7], there are decent approximate 8:9:10:12's at
a few points in this scale, although technically they aren't mapped
that way. Using the inconsistent framework mapped out above, we can
say that in mavila, 9'/8 is mapped to that 209 cent interval, whereas
9/8 itself is mapped to the 157 cent interval. This puts us now at
rank 3, since we're adding the 9' as an extra dimension, but we can
temper out the difference between 9'/8 and 10/9, which is 81'/80, and
we're back to rank 2.

Something to think about.

-Mike