Mazzola notation

On page 126 of Topos, we find a 5-limit scale presented in the form of
3D lattice diagram. The scale is attributed to Martin Vogel, but it
turns out to be familiar--the Malcolm Monochord yet again. On page
507, he gives a matrix transforming the {2,3,5} basis to one more
"transparent", inverting it, we find that the new basis elements are
9/8, 3/4, and 5/4. This is intersesting--the octave is not a basis
element, and instead we have the fifth and the tone. Since 9/8 plays a
harmonic role as a sort of flat 8/7 in the dominant seventh chord as
well as being a harmonic tone in the ninth chord, it makes more sense
to me to move it to the third position, changing at the same time 3/4
to 3/2, and get [3/2, 5/4, 9/8] as a 5-limit basis. Mazzola draws the
Malcolm scale via a square 3D lattice under this basis change, which
"make its symmetries evident". We might call the
[3/2, 5/4, 9/8] basis the Mazzola notation, and the square lattices he
draws Mazzola lattices, I suppose.

On the next page he gives a 5-limit transformation in terms of his
notation; if we translate it back into {2,3,5} it becomes
2 |-> 1/2, 3 |-> 3, 5 |-> 3/5, which seems a little warped and which
doesn't seem to do what he wants it to do. If you instead simply use
x |-> 2/x, you convert Malcolm into the New Albion scale, which
appears to be the intent. In the 3D lattice in Mazzola's basis, you
can see the near-symmetry clearly enough, but not better than on the
usual planar lattice 5-limit classes, i think.

Anyway, it's a different perspective on the 5-limit, and one that
brings potential 7 and 9 limit reinterpretations to the fore. One
could simply take a 5-limit scale, and change all of the 9/8
generators into 8/7 generators; since [2,3/2,5/4,8/7] is a notation
for the 7-limit, this makes a certain warped kind of sense. For
example, if we do this to Malcolm from 1 <= scale element < 2 we get
1, 21/20, 8/7, 6/5, 5/4, 21/16, 10/7, 3/2, 63/40, 105/64, 441/256,
15/8, 2; which could use some 1029/1024 tempering but is fine for
people who like triads and 1-5/4-7/4 chords.

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

> On page 126 of Topos, we find a 5-limit scale presented
> in the form of 3D lattice diagram. The scale is attributed
> to Martin Vogel, but it turns out to be familiar--the
> Malcolm Monochord yet again. On page 507, he gives a matrix
> transforming the {2,3,5} basis to one more "transparent",
> inverting it, we find that the new basis elements are
> 9/8, 3/4, and 5/4. This is intersesting--the octave is
> not a basis element, and instead we have the fifth and the
> tone. Since 9/8 plays a harmonic role as a sort of flat 8/7
> in the dominant seventh chord as well as being a harmonic
> tone in the ninth chord, it makes more sense to me to move
> it to the third position, changing at the same time 3/4
> to 3/2, and get [3/2, 5/4, 9/8] as a 5-limit basis. Mazzola
> draws the Malcolm scale via a square 3D lattice under this
> basis change, which "make its symmetries evident". We might
> call the [3/2, 5/4, 9/8] basis the Mazzola notation, and
> the square lattices he draws Mazzola lattices, I suppose.

this sounds a lot like what Regener did, which i document here:

http://tonalsoft.com/enc/transform.htm

-monz

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:

> this sounds a lot like what Regener did, which i document here:
>
> http://tonalsoft.com/enc/transform.htm

Mazzola is expressing everything in the 5-limit in terms of three
numbers, whereas Regener is not; there is no implicit tempering
involved in what Mazzola does. In Regener's system, four diatones and
a quint give us a 3/2, and seven diatones an octave. However there
little point in a system which uses two generators for the 3-limit,
and then fails to get it in tune, so it is tempting to suppose that it
must be intended to represent the 5-limit.

If a fifth is (4,1) and an octave is (7,0), then a tone is (1,2), and
a meantone major third (2,4), and so a minor third will be (2,-3), and
a chromatic semitone (0,7). Monz gives a chromatic semitone above a
fifth as a (4,8), so it seems we can conclude meantone is what is
intended. You can get good meantone from 98 or 105 such that a
chromatic semitone is 7 steps; in 98 (43+55=98, so Monz ought to like
this tuning well enough) a diatone would be 14 steps, and a quint 1,
whereas in 105 (31+74=105) we would get a diatone of 15 steps and a
quint of 1.