Rational Notation

David Canright

This article first appeared in 1/1, the Journal of the Just Intonation Network, Volume 1, Number 1, p.8 (1985).

If you compose music in Just Intonation, you must have faced the problem of how to write it down, since standard musical notation implies standard, i.e. tempered, intonation. Some such composers use standard notation anyway, usually with some indication of how the tuning they want differs from tempered tuning. This is the obvious choice when writing for retuned keyboards, since any pianist can just sit down and play it. It also makes sense when using traditional harmony -- diatonic scales built from the prime numbers 3 and 5 -- the origin of standard notation.

Other composers, needing many more than twelve tones, resort to unusual accidentals or even special noteheads, resulting in idiosyncratic notation that's hard for anyone else to understand. And in any case, notating non-standard harmonies on a standard staff tends to obscure rather than clarify their harmonic role.

Harry Partch initially solved the notation problem, when he was writing only for adapted viola and voice, by simply writing down the sequence of ratios he wanted. He realized that ratios are the only completely general, unambiguous names for just intervals. However, seeing ratios in a row, one misses the visual sense of up and down, of melody. (By the way, Partch ended up developing a different notation for each of his instruments, based on the physical layout of the instrument rather than on the harmonic structure. As a result, trying to make harmonic sense out of one of his scores is a nightmare.)

My own response to the notation problem was to rethink the musical staff in a somewhat more scientific fashion. For one thing, it always bothered me that equally spaced lines on a standard staff represent unequal intervals (e.g. E to G is smaller than G to B). In my notation, the vertical spacing of the lines is directly proportional to the acoustical sizes of the intervals between the corresponding pitches, so the spaces are different sizes. The placement of a note within a space is also proportional to the intervals involved. Also, I choose which pitches should fall on lines (and thus how many lines and where) depending on the needs of the composition typically using four lines per 2/1 ("octave") for a simple scale.

To clarify what's what, I label each line and each space with the ratio of the corresponding pitch. (Sometimes I put two pitches in one large space, and so two labels, where the placement of the noteheads makes clear which pitch is intended.) So the stack of ratios at the left of the staff shows unambiguously exactly what scale is in use, and acts as a "key signature." Of course, a change of scales then likewise entails a change of staff. Also, pitches outside of the scale, e.g. passing tones, are written on "ledger lines" within the staff, each labeled with its ratio.

As for the time signature, to avoid confusion with all the ratios, I use the number of beats per measure followed by a note representing one beat, so 4/4 time becomes (4 quarternote), and 12/8 could be (4 dotted quarternote). The melodies can then be written down using regular notes with their normal metrical meanings.

At this point the examples should make sense. The first example (large GIF, 9KB) is the beginning of the Star Spangled Banner, as a common reference point. The staff covers two--2/1's (two "octaves"), and is vertically much more spread out than a standard staff, which allows for greater visual pitch discrimination. With care, one can distinguish the two different sizes of whole step in the staff labels. The second example (large GIF, 16KB) is a fragment of a duet that I wrote for my guitar. The staff covers the same range. The first scale is a type of dorian mode that I like, and the second is derived from an undertone scale on the 5/4 tone of the guitar. The two voices are distinguished by the note stems.

This system of notation has several distinct advantages. It is completely general; it can be used with any set of just intervals. (Of course, it could also be used for non-just intervals with different labeling.) It is unambiguous because all pitches are labeled with their harmonic ratios. The relative sizes of intervals are visually apparent. The harmonic relationships are clear to anyone familiar with ratios. And for the composer, once the staff is laid out, writing down melodies and harmonies is a cinch.

The main drawback is that one must make one's own music paper. For me, this isn't so bad because I have access to a computer with a good plotter, and I've written software to draw a staff given the scale. I used to write on tracing paper, tracing the staff from a reference page with many intervals plotted on it. That worked fine, but is not quite as accurate and not as fast as computer plotting. Usually I don't worry about visual accuracy on a rough draft, and just draw the lines by eye. This is quickest of all, and still clear because of the labels.

By the way, I should point out that when I say the vertical spacing is proportional to the acoustical size of the interval, that means the distance is proportional to the logarithm of the ratio. For example, if I choose 2/1 = 1 inch, then: 5/2 = 1 inch * log(3/2)/log(2/1) = .176/.301 = 0.585 inch.

(If using a ruler to plot this, it would be easier in metric.) This way the same interval shows up as the same distance anywhere in the staff.

For my purposes in writing just-intoned music, this notation's clarity, conceptual simplicity, and versatility far outweigh the slight hassle of drawing my own staff.


Since writing this article, I wrote an Encapsulated PostScript file that can be used to generate these acoustically-spaced music staves. This file can be edited for the particular scales, ledger lines, etc.; instructions are included as comments in the file. The resulting ASCII file can be used like any other EPS graphic (i.e., it can be printed on any PostScript printer, imported into word-processing or drawing programs, viewed with "ghostscript", etc.). This file is provided with an example (3 staves on a page, for a 2-"octave" major scale) and has also been modified to give the staves (including ledger lines and barlines, but not the notes) of Example 1 (EPS) (this example has a reduced Bounding Box) and Example 2 (EPS) (this example has a rotated page).

Example 1 (large GIF, 9KB)

(or try Example 1 embedded in HTML)

Example 2 (large GIF, 16KB)

(or try Example 2 embedded in HTML)

David Canright -- DCanright@NPS.edu