Yahoo Tuning Groups Ultimate Backup tuning Comma and ET relationships revealed by a comma-based lattice

The file I've uploaded here: `MartinGough>Comma lattice
(syntonic, schisma, kleisma)' illustrates a technique that I find
useful for displaying relationships among commas and equal temperaments
in the 5-limit. I think it's also a promising idea for exploring
regular temperaments and higher prime orders.

Here's the gist. In the standard tonal lattice commas are scattered
apparently at random and it's hard to see how they interrelate. But
with a change of basis they can be concentrated near the origin where
their relationships become clear. The dual of this lattice of commas is
a lattice of ETs, which offers further insights.

I'd be a bit surprised if this is a new idea, but since I've
been unable to find any prior mention of it I'll outline the
principle here, starting with some basics (referenced to the 5-limit) to
establish the notation.

A just interval J is the product of a JI tuning vector and a monzo:

J = <v|m>

where

<v| = <2 3 5|

is the JI tuning vector, here expressed in a convenient shorthand in
which an underscore stands for a suitable logarithm function.

This standard coordinate system can be transformed into a rebased system
using a square unimodular matrix [W] and its inverse [N]:

<w| = <v|[W] |n> = [N] |m>

<v| = <w|[N] |m> = [W] |n>

[W] [N] = [N] [W] = [I] |W| = |N| = ±1

where <w| is the rebased tuning vector and |n> is the rebased monzo.

The columns of [W] are the standard monzos for the new basis intervals
and the columns of [N] are rebased monzos for the standard basis
intervals.

Evaluation of an interval in the rebased system follows the usual
pattern:

J = <v|m> = <v|[W][N]|m> = <w|n>

After tempering:

J' = <v'|m> = <v'|[W][N]|m> = <w'|n>

where J' is the number of steps representing interval J in an equal
temperament, and <v'| = <2' 3' 5'| and <w'| =
<w1' w2' w3'| (with integer elements) are the standard and
rebased vals for that temperament.

When changing the basis in this way the unimodular property can be
preserved by proceeding from the identity matrix in a series of steps in
which a multiple of one column (basis interval) is subtracted from
another.

The interesting situation is when all the basis intervals are commas. In
this case the equation

[N] [W] = [I]

can be read as a statement that the rows of [N] are standard vals
representing three ETs which temper the set of commas <w| = <w1 w2 w3|
to <1 0 0|, <0 1 0| and <0 0 1|, respectively. Each of these ETs thus
tempers out two of the basis commas while setting the third to one step
of the temperament.

Every point on the rebased lattice is both a monzo (interval) and a val
(ET). This is of course also true for the standard lattice, but in the
rebased system every lattice point near the origin (as well as more
distant points near the JI zero plane) is the monzo of a comma-sized
interval, and nearly every lattice point is the val for an ET closely
approximating JI.

Expressed in the standard basis the monzos are

|m> = |m2 m3 m5> = [W] |n1 n2 n3>

where n1, n2 and n3 are the coordinates of a general lattice point
representing the coefficients of each basis comma in the monzo, and the
vals are

<v'| = <2' 3' 5'| = <w1' w2' w3'| [ N ]

where w1', w2' and w3' are the coordinates of a general
lattice point representing the sizes of the commas w1, w2 and w3 in
steps of the val's ET.

This lattice of commas/ETs has several interesting properties.

The standard monzos representing the intervals 2, 3 and 5, which in the
standard lattice are orthogonal, fold up under the transformation like
the spokes of a collapsing umbrella until they lie almost in a straight
line. Other simple intervals, being small integer combinations of these
primitive intervals, also lie close to this line. At the same time
smaller intervals, of comma size and above, are pulled in radially and
away from the JI zero plane to populate the space near the origin.
Musically useful commas (those with low complexity and size) cluster
around the origin and spread out more sparsely along the JI zero plane.

Viewed as a val, a rebased lattice point represents a tuning vector for
an ET, and intervals in this ET (for a given monzo lattice point) are
measured by distance in integer steps from a zero plane normal passing
through the origin normal to the tuning vector. Since monzos for simple
intervals now lie in a narrow cone of directions, their sizes in
relation to the octave are approximated well by most vals, even if those
vals have tuning vectors not closely aligned with the (rebased) JI
tuning vector. The exceptions are those vals whose tuning vectors point
nearly perpendicular to the octave monzo, and thus come close to
tempering out the octave and other simple intervals.

With a suitable choice of basis intervals the rebased lattice can
provide a framework for cataloguing both commas and ETs. In the 5-limit
the following basis set (one of many possibilities) proves useful:

<w| = <c sigma k|

where c = syntonic comma, sigma = schisma, k= kleisma. Its
change-of-basis matrices are

[-4 -15 -6] [ 53 84 123]

[W] = [ 4 8 -5] [N] = [-19 -30 -44]

[-1 1 6] [ 12 19 28]

from which it can be seen that the associated basis ETs are 53edo,
19edo and 12edo. (The negative signs in the 19edo val are the
result of the JI tuning vector falling in a different quadrant from the
octave monzo under this transformation.)

The commas can be conveniently displayed in layers of the lattice with
specified values of nk.

The nk = 0 plane contains all the commas tempered out by 12edo,
including, near the origin, schisma, Pythagorean, syntonic, diaschisma,
diesis, major diesis, ripple, misty, passion, and, a little to the
south, close to the JI zero line, the atom of Kirnberger and a 665edo
comma I call schismon (= schisma atom).

The nk = 1 plane contains all the intervals (including commas) tempered
to plus or minus one 12edo step. This group contains more than 30 named
commas, including a vertical sequence of schisma-separated commas which
are all tempered out by 53edo:

semicomma, kleisma, amity, vulture, tricot, monzisma,
counterschisma, mercator

which links up with a diagonal sequence of Pythagorean intervals:

mercator, 41-tone, sublimma, 17-tone, limma, apotome...

Any comma in the nk = 1 plane can substitute for the kleisma as the
third basis comma to form an alternative lattice.

The comma lattice provides a framework for displaying ETs approximating
the 5-limit. In the rebased lattice simple sub-octave intervals are
lattice points lying close to the main diagonal of a rectilinear
`loaf' having the octave (with coordinates |53 -19 12>) at one
corner. Slicing the loaf parallel to its three axes yields 53tet, 19tet
and 12tet, while angled cuts give other ETs.

The zero planes for ETs tempering out a particular comma form sheaves of
planes radiating from that comma's monzo vector. They appear as
lines marking the intersection of their zero planes with the nk = 1
plane, and fall into family groups including:

meantone temperaments: horizontal lines

schismic temperaments: vertical lines

diaschismic temperaments: trailing diagonals

aristoxenean temperaments: leading diagonals

misty temperaments: lines with gradient -2.

Other temperament families (such as kleismic) can be plotted as lines
radiating from the tempered-out comma. Regular temperaments such as
quarter-comma meantone can also be plotted, and the graphic has a number
of other nice features.

Another basis set, |monzisma, raider, atom>, turns up the magnification
to focus on schismina-sized commas and the associated temperaments.

That's it in outline (rather lengthier than I intended). I'm
grateful to anyone who's read this far, and would welcome feedback
on the idea.
Martin Gough

Comma and ET relationships revealed by a comma-based lattice

🔗martingough31 <martingough31@...>

🔗genewardsmith <genewardsmith@...>