Classifying Lumma scales

A lumma can be defined as a 12-note scale which is the image under
marvel (225/224-planar) tempering of a 7-limit epimorphic scale with
<12 19 28 34| as its val. It seems to me that a good way of analyzing
and classifying these is to give the chain of fifths in both meantone
and superpyth.

I picked on superpyth because it has a fifth as generator, like
meantone, and has no comma in common with meantone. The alternatives
would be father, which is much too coarse, and pontiac, which would
involve looking at a 53-et chain and not a 22-et chain, and seems like
it would be less useful for scales of the size we are looking at.

Because the generators are both fifths, we can standardize by moving
both chains up and down by fifths; hence we can start the meantone
chain at 0 and start the superpyth chain at whereever it arrives after
moving the meantone chain to 0. The meantone chain is a modmos--it
gives a complete set of representatives modulo 12. The first level of
the classification is the meantone modmos, and any two lumma scales
the same in this department bear a clear family likeness; we can then
investigate more fully via the superpyth chain.

One way to look at all of this is that meantone gives you the
information from 12 and 19 et, and superpyth from 22 and 5 et. The 5
et info is the 225/224 info; 5 sends 225/224 to 1 and 64/63, 81/80 and
245/243 all go to 0. We can express everything in the 7-limit as a
product of 64/63, 81/80, 245/243 and 225/224, and so dropping 225/224
all we need is the first three--which means the info from 12, 19 and
22. If we are ignoring octaves (which we can do since we are looking
at scales) we are just left with the meantone and superpyth generator
vals.

Anyway, here are some examples, where the su suffix means the
superpyth chain. Centaur, of course, has 0-11 for a modmos, since I
got meande and meandin by detempering Meantone[12]. A question one
might ask about this is to what extent the meantone and superpyth
chains are fungible, so that we can get a decent lumma by simply
picking any one of the first and putting it together with any one of
the second; this would reduce the problem to one of finding good 12
note meantone modmos and good 12 note syperpyth scales.

Class

classmodmos := [0, 3, 4, 6, 7, 8, 9, 10, 11, 13, 14, 17]
classsu := [-16, -8, -7, -5, 1, 2, 3, 4, 10, 12, 13, 21]

Prism

prismmodmos := [0, 1, 4, 5, 6, 7, 8, 9, 10, 11, 14, 15]
prismsu := [-16, -15, -7, -6, -5, -4, 2, 3, 4, 5, 13, 14]

Centaur

centmodmos := [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]
centsu := [-10, -9, -3, -2, -1, 0, 1, 2, 3, 9, 10, 11]

>A lumma can be defined as a 12-note scale which is the image under
>marvel (225/224-planar) tempering of a 7-limit epimorphic scale with
><12 19 28 34| as its val. It seems to me that a good way of analyzing
>and classifying these is to give the chain of fifths in both meantone
>and superpyth.
>
>I picked on superpyth because it has a fifth as generator, like
>meantone, and has no comma in common with meantone. The alternatives
>would be father, which is much too coarse, and pontiac, which would
>involve looking at a 53-et chain and not a 22-et chain, and seems like
>it would be less useful for scales of the size we are looking at.
>
>Because the generators are both fifths, we can standardize by moving
>both chains up and down by fifths; hence we can start the meantone
>chain at 0 and start the superpyth chain at whereever it arrives after
>moving the meantone chain to 0. The meantone chain is a modmos--it
>gives a complete set of representatives modulo 12. The first level of
>the classification is the meantone modmos, and any two lumma scales
>the same in this department bear a clear family likeness; we can then
>investigate more fully via the superpyth chain.
>
>One way to look at all of this is that meantone gives you the
>information from 12 and 19 et, and superpyth from 22 and 5 et. The 5
>et info is the 225/224 info; 5 sends 225/224 to 1 and 64/63, 81/80 and
>245/243 all go to 0. We can express everything in the 7-limit as a
>product of 64/63, 81/80, 245/243 and 225/224, and so dropping 225/224
>all we need is the first three--which means the info from 12, 19 and
>22. If we are ignoring octaves (which we can do since we are looking
>at scales) we are just left with the meantone and superpyth generator
>vals.
>
>Anyway, here are some examples, where the su suffix means the
>superpyth chain. Centaur, of course, has 0-11 for a modmos, since I
>got meande and meandin by detempering Meantone[12]. A question one
>might ask about this is to what extent the meantone and superpyth
>chains are fungible, so that we can get a decent lumma by simply
>picking any one of the first and putting it together with any one of
>the second; this would reduce the problem to one of finding good 12
>note meantone modmos and good 12 note syperpyth scales.
>
>Class
>
>classmodmos := [0, 3, 4, 6, 7, 8, 9, 10, 11, 13, 14, 17]
>classsu := [-16, -8, -7, -5, 1, 2, 3, 4, 10, 12, 13, 21]
>
>Prism
>
>prismmodmos := [0, 1, 4, 5, 6, 7, 8, 9, 10, 11, 14, 15]
>prismsu := [-16, -15, -7, -6, -5, -4, 2, 3, 4, 5, 13, 14]
>
>Centaur
>
>centmodmos := [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]
>centsu := [-10, -9, -3, -2, -1, 0, 1, 2, 3, 9, 10, 11]

whoa.

-C.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

> A question one
> might ask about this is to what extent the meantone and superpyth
> chains are fungible, so that we can get a decent lumma by simply
> picking any one of the first and putting it together with any one of
> the second; this would reduce the problem to one of finding good 12
> note meantone modmos and good 12 note syperpyth scales.

The answer to this question is "no", since we want the resulting scale
to be epimorphic. When two scale steps of the meantone chain differ by
something congruent to 1 mod 12, we want the corresponding difference
in the superpyth chain to give us an interval in the range 650 to 750
cents. We can multiply the fifth by something which is a comma of
12-et and which is in the range -50 to 50 cents to accomplish this;
the meantone difference will then be 1 mod 12, and the superpyth
distance can take care of itself, though the ones which keep the chain
of superpyth fifths in a restricted range are the most interesting.

Here's a list of "interesting" 5-limit commas of 12-et, meaning the
size is under 50 cents and the superpyth distance less than 23;
showing the comma, the superpyth distance, and the size in cents:

32805/32768 17 1.953720
2048/2025 -22 19.552569
81/80 -5 21.506290
531441/524288 12 23.460011
43046721/41943040 7 44.966301

The method for lifting a meantone-superpyth pair to a lumma is
interesting, in that it involves a use of temperaments which is
completely new. If we take the pair of monzos z1 = |46 -40 5 2> and
z2 = |66 -57 7 3> we find that z1 is a vum/provo/comma for 12 and 19,
and equivalent to 3/4 for 22 and 5. Similarly z2 is a comma for 22 and
5, and is a 3/2 for 12 and 19. This means z1 is mapped to 0 by
meantone and 1 by superpyth, and z2 is mapped to 1 by meantone and 0
by superpyth. If superpyth maps to a and meantone maps to b, an
interval which gives this mapping is z1^a * z2^b.

This interval is going to be large and ugly, but we can reduce it via
a curious sort of formal temperament--meaning something our machinery
can treat as a temperament. If we take the vals for the generator
mapping of meantone and superpyth, namely <0 1 4 10| and <0 1 9 -2|,
we can wedge them together and get a wedgie for a "temperament" which
is utterly useless for tempering anything, but whose "commas" are what
we want for reducing the above product. The wedgie is <0 0 0 5 -12
-98|, and the "commas" are 2 and |0 98 -5 -12>. To lift to a lumma we
can reduce with respect to 225/224 and the above interval, transpose
one of the intervals to 1/1, and reduce to the octave.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

> we want for reducing the above product. The wedgie is <0 0 0 5 -12
> -98|, and the "commas" are 2 and |0 98 -5 -12>.

Should be |0 98 -12 -5>.