Domes and wakalixes and common practice harmony

The Zarlino major scale is a MODMOS of porcupine; it requires two
alterations to get to it. Since the Zarlino major scale is porcupine
MODMOS, it thus follows that all of the domes of the 81/80 # 25/24
Fokker block are also MODMOS's of porcupine, except for the dome which
is also a meantone-dicot-porcupine wakalix - this isn't a MODMOS, but
an MOS.

Another way to think of this is that we're obtaining the Zarlino major
scale in porcupine as follows: start with porcupine[7]. Then, view
this as a tempered version of a 5-limit Fokker block in which the
unison vectors are 250/243 and 25/24. Note that every dome of this
block will trivially be the same porcupine[7] MOS, up to modal
rotation, so you can shift around the domes without anything changing.
Pick the dome of this block that's the meantone-dicot-porcupine
wakalix.

Now, since this is a wakalix, reframe this block by changing the
250/243 to 81/80, keeping the pitches the same. Now start exploring
the domes of this new block. These will no longer be porcupine[7]
MOS's, but will instead be MODMOS's of porcupine[7]. These domes
should look familiar to anyone who's tried to detemper meantone[7] in
different ways, but this time under a new, porcupinized perspective.
Not only are many of these rather easy to obtain via a small number of
alterations to the base MOS, but in porcupine, 81/80 is equated with
25/24, thus giving a new meaning to these supposedly "comma-shifted"
equivalents of meantone[7]. It's particularly notable that the Zarlino
major scale is one of these domes, and that it exists as a 2-note
alteration of porcupine[7], making it a particularly "close" MODMOS of
porcupine[7].

One can continue this process of shifting domes and reframing
wakalixes forever, treating the wakalixes as "grand central stations"
to reframe the block and get a new set of domes, and in so doing,
you'll continually arrive at different sets of MODMOS's to use. Some
of these will require more alterations from the base MOS than others;
however, all of them will be epimorphic under the same val.

-Mike

On Thu, Mar 29, 2012 at 6:58 PM, Mike Battaglia <battaglia01@gmail.com> wrote:
>
> One can continue this process of shifting domes and reframing
> wakalixes forever, treating the wakalixes as "grand central stations"
> to reframe the block and get a new set of domes, and in so doing,
> you'll continually arrive at different sets of MODMOS's to use. Some
> of these will require more alterations from the base MOS than others;
> however, all of them will be epimorphic under the same val.

What I find most interesting about this is that this leads to a
different perspective to take on temperaments. In the rank-2, 5-limit
case, it ends up leading to a constant network of domes which have
beautiful harmonic properties, same as any Fokker block, and which
continually transmute into one another. However, since the lattice is
actually tempered, this means that certain domes will be either fully
or partially tempered down into a much smoother structure which has
beautiful melodic properties as well.

Mathematically, this is a generalization of how we already currently
treat rank-1 temperaments: they're viewed as entities which support
rank-2 temperaments, but for which some MOS's contain "ambiguous"
intervals (e.g. dim4 and maj3 in 12-EDO). Similarly, we can view
rank-2 temperaments as entities which support rank-3 temperaments, and
freely think in terms of Fokker blocks and domes, but note that these
temperaments also contain "ambiguous" intervals - except this time the
so-called ambiguities are points where the domes partly or fully turn
into MOS's.

Probably the most notable thing of all, however, is that some of the
MODMOS's of meantone also yield to the same analysis. For instance,
the harmonic minor scale can be treated as 1/1 9/8 6/5 4/3 3/2 8/5
15/8 2/1, which is a Fokker block with unison vectors 135/128 and
25/24. The harmonic major scale is a dome of the same Fokker block!
This means that those scales are MOS's in mavila and dicot, and for
those who can handle mavila, they're absolutely beautiful.

This shouldn't be too surprising, because these scales were chosen
specifically to have lots of nicely compact 5-limit harmonic
properties while remaining epimorphic under <7 11 16|. This is the
exact mathematical descriptor for what happens when one splits the
general concept of "minor" up into a set of different 7-note scales,
all which are shifted around by a chromatic semitone of one another.
This is exactly what this operation is, in regular temperament terms -
splitting up a (tempered) region of the 5-limit lattice into scales
which are epimorphic under <7 11 16| that shift slightly in their
5-limit harmonic properties. Of course!

I think that this sort of analysis might yield many fruitful insights
into "common practice" harmony, hopefully including ways to generalize
it to other rank-2 tuning systems, such as porcupine, to other vals
besides 7p, and and hopefully to other limits as well. If it proves
useful there, then I have a strong hunch that the 7-limit extension of
this concept will also prove useful for analyzing jazz, which also
takes incredibly complex chords and finds 7-note epimorphic scales
that support them, but which incorporates greater domal motion in
order to handle the increased complexity of the various 7-limit
temperaments that 12-EDO supports.

One thing, though: I'm not sure if the melodic minor scale makes any
sense as a Fokker block. I'm not sure that it needs to for the basic
idea above to still be useful (MODBLOCKs, anyone?), but for
posterity's sake, can anyone figure out how to work it out? These
pitches look interesting, but I'm not sure what the unison vectors
are: {1/1 9/8 6/5 4/3 3/2 27/16 15/8 2/1}

-Mike

On Thu, Mar 29, 2012 at 7:24 PM, Mike Battaglia <battaglia01@gmail.com> wrote:
>
> One thing, though: I'm not sure if the melodic minor scale makes any
> sense as a Fokker block. I'm not sure that it needs to for the basic
> idea above to still be useful (MODBLOCKs, anyone?), but for
> posterity's sake, can anyone figure out how to work it out? These
> pitches look interesting, but I'm not sure what the unison vectors
> are: {1/1 9/8 6/5 4/3 3/2 27/16 15/8 2/1}

The above isn't a Fokker block, but I found a Fokkerized melodic minor
scale in Gene's Fokker block archives here:

/tuning-math/files/gene/fokker/

The blocks that work are "porchrome4" and "porchrome5." Here's what we get:

! porchrome4.scl
Fourth 25/24&250/243 scale = inverse porchrome5
7
!
10/9
5/4
25/18
3/2
5/3
9/5
2

! porchrome5.scl
Fifth 25/24&250/243 scale = inverse porchrome4
7
!
9/8
5/4
25/18
3/2
5/3
9/5
2

These both temper down to an MOS two ways (porcupine, dicot) and down
to a 1-alteration symmetrical MODMOS at least one way (meantone).

So there you go. All four of the usual western parent scales for
harmony are themselves domes of Fokker blocks, all of which hang
around the general meantone/mavila/dicot/porcupine region of the
spectrum. This shouldn't be terribly surprising, since they were
probably invented specifically for their compact 5-limit harmonic
properties and epimorphicity under <7 11 16|. Melodic minor, on the
other hand, was supposedly invented for its melodic properties, but
since the Fokker lattice unites melody and harmony so nicely, this
turns out to be the same thing.

I don't expect it'll always work out so neatly, but it did this time.
I think though, that in general, it'll be prudent to come to a better
understanding of the mathematical relationship between wakalixes and
MODMOS's.

-Mike

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

> One can continue this process of shifting domes and reframing
> wakalixes forever, treating the wakalixes as "grand central stations"
> to reframe the block and get a new set of domes, and in so doing,
> you'll continually arrive at different sets of MODMOS's to use. Some
> of these will require more alterations from the base MOS than others;
> however, all of them will be epimorphic under the same val.

One kind of near-wakalix structure which shows up at times and might be relevant to this process is a MOS with r-2 temperaments, and a MODMOS with another, where r is the rank of the scale.