Abstracted Tuning

Aaron Brick, 2002

In preparing to compose or perform music, we must elect a set of notes, or scale, with which to work. Our criteria are two: that the frequency intervals available to a player equal, or are acceptably close to, the small integer ratios which form the basis of harmony; and that the scale be easy to linearly traverse. Each of these movements is integral to the construction of a melody. Two basic kinds of scales have emerged: those with integer-ratio intervals, which facilitate jumping across harmonic ratios, and those with a uniformly-sized step, which facilitate stepping between notes.

Pythagoras was the first to derive a scale from intervals. He used 3:2 as the basis from which to calculate his scale; this method yielded a seven-note scale with two intervals, whole- and half-steps. This arrangement came to be known as a diatonic scale. Any harmonic ratio other than 3:2 and 4:3 can only be played by approximation, in some cases insufficiently accurate; and the successive traversal of intervals amplifies inaccuracy. The chromatic scale, a 16th century superset of the Pythagorean, encompasses five more semitones, producing a 12-note octave, each separated by a single semitone.

Dominant at the current day is equally-tempered tuning, in which the intervals between all adjacent notes are identical (1). The steps can describe fair approximations of most harmonic intervals, and since the error is spread out among all notes, so there are no glaringly wrong ratios. The calculation of frequencies in this scale depends upon an understanding of irrational numbers, so it took two millenia after Pythagoras to come around.

Chromatic interval-based scales (2) and equally tempered scales each have one key advantage, exact ratios or equally sized steps, and no scale has been designed to provide the one without making the other impossible. It is easy to see that harmonic ratios are not equally spaced, so the only way to take equal steps around them, or to represent larger integer ratios, is to approximate. Doing so requires extensive memorization, which is a significant burden for students of music.

    Scales and Notation
The twelve-note scale, derived and designed in many fashions, has persisted in Western music. Twelve is a manipulable, round number, and so arguably a good choice (3). Many other, less auspicious numbers of notes per octave are used in various scales around the world, just as ordinal counting exists in many number bases; these numbers, though similar in size, do not usually coincide. Generally, octaves with more than 12 notes are called microtonal.

The Western names for the notes are less convenient than they could be. They are labeled with a modified Roman majuscule alphabet from A to G (usually, somewhat perversely, enumerated from one C to the next). The scale (C, C# / Db, D, D# / Eb, E, F, F# / Gb, G, G# / Ab, A, A# / Bb, B, C) and intervals must be memorized, since it is difficult to perform arithmetic with them. Our grammar for this scale is imprecise enough that sharps and flats are actually the same thing, and each such note has two names.

A more convenient notation would require no memorization and facilitate arithmetic, for purposes of both composition and analysis. To that end, working with numbers is more appropriate than with letters, as well as allowing arbitrary precision. Intervallic arithmetic is easy with frequency values, and therefore also with their base 2 logarithms. The latter is a linear measure of octaves, not oscillations, and its integer component is an easy index for the octave into which a note falls. To convert such a value back to hertz, take 2 to its power. The scale of human hearing in octavic notation is roughly 4 through 14, or 2^4 though 2^14 Hz.

    Microtonalism and Dynamic Tuning
Microtonalism yields more precise intervals, but the large numbers of notes per octave can be unwieldy. A seventy-two note scale is common; curiously, its error in approximating 3:2 is 0.11%, the same as for a twelve-note scale. In the middle of our hearing range (near the note called concert A, relatively recently standardized at 440 Hz), this error represents less than one cycle per second. The relation is clearly accurate enough to avoid the discord called beating, and a layperson would not know the difference, but a trained ear can discern its error. It is believed that Westerners are culturally acclimated to the errors in playing intervals in 12-tone equal temperament scales.

These partial solutions leave perfectionists with a dilemma, subject to constant debate for the last several hundred years. The recent development of computing systems has now provided us the ability to automate complex and abstract calculations; such help allows us to produce an arbitrarily precise approximative solution. If we are willing to perform one algebraic calculation per transition, all intervals can be much more precisely represented.

The two kinds of traversal can be unified with an approach called dynamic tuning, where every new note is calculated according to the previous one and the interval between them. This scheme provides maximal accuracy for both stepping and jumping. It also makes melodic transposition trivial, since only a small number of notes, usually one, are explicitly defined. The ideal instrument for dynamic tuning is a computer, as the intervallic mathematics can be automated.

Algebra to calculate one next from the last is just as easy with the logarithmic values as without. In each mode, one sort of interval's value is irrational (4). Moving around by octaves is as simple as adding ones to a note.

Making precise calculations is no trouble for a computer or a sufficiently skilled player, but it suggests that instruments with a fixed number of discrete notes would not be able to play such music. A fretless stringed instrument or slide trombone would be able to do so, subject to the dexterity and arithmetic skill of their player. Such an approach would be extremely technically challenging, because this abstraction can define a superset of the melodies available in conventional staff notation. A more powerful graphical notation could be (and probably has been) devised, containing precise intervallic data.

Trained listeners may have an adverse reaction to notes they hear as off-key or halfway between two others. I have not yet verified this in practice.

The next step in my research is to study the derivation of modes from scales and conclude how any such abstractions could be useful. Then, trivially, I will develop a software layer to perform these calculations, and finally, use it to write music. Since these tasks will no doubt take some time, I elected to publish this essay rather than wait.

Any comments, suggestions, references, or corrections are welcome. Please write aaron@lithic.org.

1. The frequency coefficient for equally tempered notes in an octave with N notes is the Nth root of 2. in 12-tone equal temperament, the interval is about 1.06.
2. Besides the Pythagorean scale, this set includes the just intonational scale, mean temperament, and others.
3. It is worth noting that the difference between adjacent notes is generally several times smaller than a critical bandwidth; one cannot harmonize two adjacent notes, because the resulting sound will be dissonant.
4. In the frequency domain, jumping requires multiplication by an integer ratio (A/B), and stepping multiplication by an irrational root (see (1)). With logarithmic notes, stepping takes addition of an integer ratio (1/N), and jumping requires the addition of log2 (A/B).