Interval arithmetic
Interval arithmetic systems are sets of rules for naming the notes or intervals produced by these operations:
- adding/subtracting an interval to/from a note, e.g. C + M2 = D and C – M2 = Bb
- adding/subtracting an interval to/from another interval, e.g. M2 + M2 = M3 and P4 – M2 = m3
- finding the interval between two notes. e.g. C to E is a M3
Example usages of each of these operations:
- finding which notes make up a D major chord
- finding which scale degrees are used by a IIm chord
- naming the intervals E–G and E–B, and identifying the type of chord E–G–B forms
A consistent interval arithmetic will produce 2nds larger than 3rds, 3rds larger than 4ths, etc. For example, 7/5 is logically a 5th because it's the sum of two 3rds, 7/6 and 6/5. This makes 10/7 a 4th. But since 10/7 is larger than 7/5, the 50/49 interval between them must be a negative 2nd. Some microtonalists notate 7/5 as a 4th, avoiding negative 2nds but abandoning consistency. Some notations allow one to rename 7/5 as a 4th via accidentals that raise/lower by a pythagorean comma, providing both freedom and consistency.
Degree vs stepspan
The stepspan of an interval is the number of steps it spans. The stepspan is 1 less than the degree: the stepspan of any type of 3rd (major, minor, etc.) is 2, the stepspan of any type of 4th is 3, and so on. Interval names based on stepspans are 2step, 3step, etc. The unison is a 0step.
For negative intervals, interpret "1 less than the degree" as "1 step closer to zero". For example the pythagorean comma is a negative dim 2nd, which has a stepspan of -1.
While degrees use 1-based indexing, stepspans use 0-based indexing. Many microtonalists prefer the latter because it makes interval arithmetic easier. This can be useful when working with a less-familiar framework like octotonic or nonotonic. This article mostly discusses heptatonic notations, and mostly uses 1-based indexing.
Rules
Stepspans and degrees
Stepspans add up directly, so a 2step interval plus a 3step interval produces a 5step interval.
Degrees add up directly, but one must subtract 1 afterwards: a 3rd plus a 4th makes a 6th because 3+4-1=6. (Subtracting 1 is equivalent to converting each degree to a stepspan by subtracting 1 twice, then adding stepspans, then converting the stepspan to a degree by adding 1.)
The degree of the interval between two notes is found by counting the notes in the interval. The interval from C to G covers C-D-E-F-G, which is 5 notes, making a 5th.
The stepspan between two notes is found by counting the steps between them. The interval C-G is made up of 4 steps: C-D, D-E, E-F and F-G, making a 4step.
To add an interval to a note: for stepspans, go up the note names a certain number of steps. For degrees, go up 2 steps for a 3rd, 3 steps for a 4th, etc.
Qualities
The process for adding up qualities such as major or perfect is more complex. Because conventional notation is generated by fifths, the quality is determined by the fifthspan of the interval.
| Fifthspan | Quality | Fifthspan | Quality |
|---|---|---|---|
| -1 to 1 | perfect | ||
| 2 to 5 | major | -2 to -5 | minor |
| 6 to 12 | augmented | -6 to -12 | diminished |
| 13 to 19 | doubly-augmented | -13 to -19 | doubly-diminished |
| etc. |
Fifthspans add up directly. So to find the quality of the sum of two intervals, add their fifthspans and look up the quality in the table above.
To find the quality of the interval between two notes, find the notes on the chain of fifths and find the fifthspan between them.
...Eb Bb F C G D A E B F# C#...
For example, from D to C is two steps fourthwards, so the fifthspan is -2, which indicates minor. Thus D to C is a minor 7th.
To add an interval to a note, find the fifthspan of the interval. Then find the note on the fifthchain and move by that fifthspan to the new note.
Microtonal inflections
Any microtonal inflections add up or cancel out directly.
| Addition | Cancellation |
|---|---|
| C + ^M2 = ^D | ^C + vM2 = D |
| M2 + vM2 = vM3 | ^M2 + vM2 = M3 |
| C to vE = vM3 | ^C to ^E = M3 |
Unconventional notations
A notation of N notes will usually have 3 perfect intervals, N-3 major ones, N-3 minor ones, and N of everything else, distributed symmetrically around P1. For example, the pentatonic fifthchain is:
...A3 A1 A4 M2 M5 P3 P1 P4 m2 m5 d3 d6 d4 d2…
The minor penta-2nd has a fifthspan of +2, and is 9/8. The major penta-2nd (fifthspan -3) is 32/27. The latter is major not minor simply because it is larger than 9/8 (the original meaning of major).
However, some notations have multiple fifthchains, such as Diaschismic/Saguguti or Porcupine/Triyoti. Notations for tunings of rank 3 or higher, including just intonation, has a theoretically infinite number of fifthchains. These fifthchains are usually distinguished by microtonal accidentals. Within each fifthchain, there will be 3 perfect intervals, N-3 major ones, etc.
The concept of fifthspan can be generalized to genspan, which also adds up directly. We can assign qualities in non-fifth-generated notations using a genchain. For example, Triyoti's heptatonic genchain of perfect 2nds is:
...A6 A7 A1 A2 M3 M4 M5 M6 P7 P1 P2 m3 m4 m5 m6 d7 d8 d2 d3…
The rules for interval arithmetic can be applied to these unconventional fifthchains and genchains as well. For example, fifthspans and genspans always add up directly, and the quality of an interval is always related directly to its fifthspan or genspan.
Enharmonic unisons
Adding together two intervals can make awkward names, e.g m3 + m2 = dim 4th. To make them less awkward, one can add or subtract an enharmonic unison, EU for short, which changes the spelling but not the tuning.
EUs can be essential for microtonal interval arithmetic. For example, the Porcupine/Triyoti genchain can instead be named using ups and downs. This preserves the three foundational properties of conventional notation: octave-equivalent, fifth-generated, and heptatonic.
…M6 vM7 ^1 M2 vM3 ^4 P5 vM6 ^m7 P1 vM2 ^m3 P4 v5 ^m6 m7 v8 ^m2 m3…
Because these three properties are preserved, interval arithmetic with such a genchain is exactly as usual, but with the ups and downs added in. But the genchain looks inconsistent. Shouldn't two steps from P1 be vM2 + vM2 = vvM3? It is, but since the EU for this notation is v3A1, vvM3 is just another way of spelling ^m3.
