Comma-based lattices

When plotted on the standard tonal lattice (in which the basis intervals have prime number frequency ratios up to some prime limit p) commas form a widely scattered cloud in which no obvious structure is discernible. But rebasing to a lattice in which the basis intervals are themselves of comma size has the effect of drawing a set of similar-sized commas into the region near the origin, where their interrelationships become apparent. The dual of such a comma-based lattice is a lattice of equal temperaments (ETs), which provides a means of visualising the relationships between ETs and commas.

The theory behind this technique is set out below, illustrated for the 5-limit but extending in a straightforward way to any prime limit. An example of its application in the 5-limit can be viewed in this spreadsheet and this image.

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

[math]\qquad J = \langle v \vert m \rangle[/math]

where

[math]\qquad \langle v \vert = \langle \underline{2} \; \underline{3} \; \underline{5} \vert[/math]

is the JI tuning vector, here expressed in a convenient shorthand in which an underscore denotes a suitable logarithm function. (The choice of logarithmic base is arbitrary and sets the unit for interval measurement.)

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

[math]\qquad \langle w \vert = \langle v \vert W \qquad \qquad \vert n \rangle = N \vert m \rangle \\ \qquad \langle v \vert = \langle w \vert N \qquad \qquad \, \vert m \rangle = W \vert n \rangle \\ \qquad WN = NW = I \qquad \vert W \vert = \vert N \vert = \pm 1[/math]

where

[math]\qquad \langle w \vert = \langle w_1 \; w_2 \; w_3 \vert[/math]

is the rebased tuning vector (the elements of which are the sizes of the basis intervals) and

[math]\qquad \vert n \rangle = \vert n_1 \; n_2 \; n_3 \rangle[/math]

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 procedure:

[math]\qquad J = \langle v \vert m \rangle = \langle w \vert N \, W \vert n \rangle = \langle w \vert n \rangle[/math]

After tempering (denoted by a dash):

[math]\qquad J' = \langle v' \vert m \rangle = \langle w' \vert n \rangle[/math]

where J’ is the number of steps representing interval J in an equal temperament, and

[math]\qquad \langle v’ \vert = \langle \underline{2}’ \; \underline{3}’ \; \underline{5}’ \vert \\ \qquad \langle w’ \vert = \langle w_1’ \; w_2’ \; w_3’ \vert[/math]

(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

[math]\qquad NW = I[/math]

can be read as a statement that the rows of N are standard vals representing a set of ETs which temper the basis commas (whose monzos are the columns of W) to <1 0 0|, <0 1 0| and <0 0 1|, respectively. Each of these ETs sets one of the basis commas to a single step of the temperament while tempering out all the others.

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 every lattice point which is not too close to a certain plane is the val for an ET approximating JI.

Expressed in the standard basis the monzos are

[math]\qquad \vert m \rangle = \vert m \negthinspace \underline{_2} \; m \negthinspace \underline{_3} \; m \negthinspace \underline{_5} \rangle = W \vert n_1 \; n_2 \; n_3 \rangle[/math]

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

[math]\qquad \langle v’ \vert = \langle \underline{2}’ \; \underline{3}’ \; \underline{5}’ \vert = \langle w_1’ \; w_2’ \; w_3’ \vert N[/math]

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 a size and complexity commensurate with that of the basis commas, are pulled in radially and away from the JI zero plane to populate the space near the origin.

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 (represented here by its rebased tuning vector) proves useful and is illustrated in the linked files:

[math]\qquad \langle w \vert = \langle \, c \;\; \sigma \;\; k \, \vert[/math]

where c = syntonic comma, σ = schisma, k = kleisma.

Its change-of-basis matrices are

[math]\qquad W = \left[ \; \vert c \rangle \; \vert \sigma \rangle \; \vert k \rangle \; \right] = \left[ \begin{array}{rr} -4 & -15 & -6\\ 4 & 8 & -5\\ -1 & 1 & 6\\ \end{array} \right] \qquad \qquad N = W^{-1} = \left[ \begin{array}{rr} 53 & 84 &123\\ -19 & -30 & -44\\ 12 & 19 & 28\\ \end{array} \right][/math]

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 below, close to the JI zero line: the atom of Kirnberger and a 665edo comma, ‘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...

This 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 53edo, 19edo and 12edo, 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: leading diagonals
• aristoxenean temperaments: trailing 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.

In a given prime limit there is an infinite number of comma basis sets to choose from, and any such set can be transformed in simple ways to generate others. For example, starting from the set described above, any comma in the nk = 1 plane can substitute for the kleisma as the third basis comma.

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