Note-layout spreadsheet

I just posted, to the iGetItMusic.com website, a spreadsheet that's useful for exploring isomorphic note-layouts: JIMS_Note.xls. There has been some discussion of isomorphic note-layouts in the Music Notation Project's forums, which prompted this blog posting.

The spreadsheet workbook starts with a "Variables" sheet, which contains six highlighted cells (variables):
Alpha (Period)
Beta (Generator)
H_Alpha
V_Alpha
H_Beta
V_Beta

Some years ago, I posted a document that described isomorphic note-layouts in terms of the adjacency intervals H & V. That document defined H and V in semitones. That's fine for 12-tone equal temperament (12-tet), but the "semitone" has no musical meaning outside of that one tuning. ("Augmented unison" has meaning; "minor second" has meaning; but the "semitone" is, in 12-tet, a conflation of those two meaningful terms into one meaningless artifact.)

To make the description of adjacency intervals more general, the JIMS_Note spreadsheet defines adjacency intervals as vectors [a, b], where a is the number of alphas, and b is the number of betas. The interval [a, b] is ((a*alpha) + (b*beta)) cents wide. Alpha and beta are the two intervals that define a rank-2 tuning of p-limit just intonation. In the syntonic temperament, alpha is the tempered octave (c. 1200 cents), and beta is the tempered perfect fifth (c. 702 cents). 12-tet is just one specific tuning of the syntonic temperament, in which alpha is exactly 1200 cents and beta is exactly 700 cents.

Every note in such a rank-2 temperament can be defined as a note[a, b] where both a and b are integers. This two-dimensional definition of notes is a good fit with two-dimensional hexagonal-grid keyboards, exactly as the one-dimensional (i.e., stack of semitones) definition of notes is a good fit with the one-dimensional piano-style keyboard.

In the JIMS_Note spreadsheet, one can
- specify values for alpha and beta (thus specifying the tuning), and also
- specify the adjacency intervals H & V using the generalized [a, b] intervals rather than semitone counts.

Using this spreadsheet, you can plug any adjacency intervals you like into the "Variable" sheet's definitions of H & V, and see the resulting isomorphic note-layouts.

The "Variables" sheet also contains a table of adjacency intervals for common isomorphic keyboards, including the Wicki, Janko, Chromatic Button Accordion (both type B and type C), Wesley, and Triad (aka Harmonic Table).

When plugging new values into the spreadsheet, it helps to know how to express common tonal intervals in [a, b] form.
A1: [-4, 7]
m2: [ 3, -5]
M2: [-1, 2]
m3: [ 2, -3]
M3: [-2, 4]
d4: [ 5, -8]
P4: [ 1, -1]
A4: [-3, 6]
d5: [ 4, -6]
P5: [ 0, 1]
A5: [-4, 8]
...and P8: [1, 0]

Many of these intervals are enharmonic in 12-tet, such as A1/m2, d4/M3, and A4/d5. However, that's an artifact of 12-tet; in all other tunings, these pairs are NOT enharmonic. Defining these intervals using the [a, b] approach makes Dynamic tonality possible.