Interval size measure
Interval size measure means the distance between pitches. Intervals can be measured logarithmically or by frequency ratios.
Logarithmic
All logarithmic measures can be combined by adding and subtracting them.
Gross
Intervals are sometimes expressed in the number of scale steps between them. These steps can be of different size, compare for example the names of the major scale in the classic music.
For "atonal" music it was replaced by the number of 12edo-semitones.
Proposal: The relative interval measure is the number of steps between two pitches of an equal tuning, sometimes called degrees (of an edo). For generators, the backslash notation is used d\edo, but it is also a ratio (of a logarithmic measure).
Fine
The cent (¢), 1\1200 octave, is the classic measure for intervals when more precision than 12edo is required. Some people object to it on the grounds that it is too (obviously) closely related to 12 equal.
The following table demonstrates a list of measures derived from the logarithmic division of a interval (e.g. octave, twelfth):
| Unit name (symbol): | Interval based on: | Power of interval: | Significance |
|---|---|---|---|
| Eka | 2/1 (octave) | 1\16 | From Sanskrit eka: one, unit; chromatic unit of 16ED2 Armodue Theory |
| Normal diesis | 2/1 (octave) | 1\31 | |
| Méride | 2/1 (octave) | 1\43 | |
| Holdrian comma | 2/1 (octave) | 1\53 | |
| Morion | 2/1 (octave) | 1\72 | |
| Farab | 2/1 (octave) | 1\144 | |
| Mem | 2/1 (octave) | 1\205 | Unit used by H-Pi Instruments |
| Tredek | 2/1 (octave) | 1\270 | |
| Eptaméride or Savart | 2/1 (octave) | 1\301 | |
| Gene | 2/1 (octave) | 1\311 | |
| Dröbisch Angle | 2/1 (octave) | 1\360 | |
| Squb | 2/1 (octave) | 1\494 | |
| Iring | 2/1 (octave) | 1\600 | |
| Skisma | 2/1 (octave) | 1\612 | |
| Delfi | 2/1 (octave) | 1\665 | |
| Woolhouse | 2/1 (octave) | 1\730 | |
| millioctave (mO) | 2/1 (octave) | 1\1000 | "Metric" / SI division of octave |
| cent (¢) | 2/1 (octave) | 1\1200 | 1/100 of 12ED2 semitone |
| greater muon | 2/1 (octave) | 1\1224 | |
| triangular cent | 2/1 (octave) | 1\1260 | |
| pion | 2/1 (octave) | 1\1272 | |
| pound | 2/1 (octave) | 1\1344 | |
| neutron | 2/1 (octave) | 1\1392 | |
| lesser muon | 2/1 (octave) | 1\1428 | |
| deciFarab | 2/1 (octave) | 1\1440 | 1/10 of Farab |
| quadratic cent | 2/1 (octave) | 1\1452 | |
| ksion | 2/1 (octave) | 1\1476 | |
| cubic cent | 2/1 (octave) | 1\1500 | |
| 7mu | 2/1 (octave) | 1\1536 | (7th MIDI unit), seventh MIDI-resolution unit, 1/128 (1/(27)) of 12ED2 semitone |
| rhoon | 2/1 (octave) | 1\1560 | |
| tile | 2/1 (octave) | 1\1632 | |
| Iota | 2/1 (octave) | 1\1700 | |
| Harmos | 2/1 (octave) | 1\1728 | |
| Mina | 2/1 (octave) | 1\2460 | |
| Tina | 2/1 (octave) | 1\8539 | |
| Purdal | 2/1 (octave) | 1\9900 | |
| Türk sent | 2/1 (octave) | 1\10600 | |
| Prima | 2/1 (octave) | 1\12276 | |
| Jinn | 2/1 (octave) | 1\16808 | |
| Jot | 2/1 (octave) | 1\30103 | |
| Imp | 2/1 (octave) | 1\31920 | |
| Flu | 2/1 (octave) | 1\46032 | |
| MIDI Tuning Standard unit | 2/1 (octave) | 1\196608 | 14mu (14th MIDI unit), fourteenth MIDI-resolution unit, 1/16384 (1/(214)) of 12ED2 semitone |
| Hekt | 3/1 (twelfth) | 1\1300 | 1/100 of 13-ED3 (Bohlen-Pierce) scale step |
| Grad | 531441/524288 (Pythagorean comma) | 1\12 | |
| Tuning unit | 531441/524288 (Pythagorean comma) | 1\720 |
See Logarithmic Interval Measures
Within a given equal-stepped tonal system, the relative cent (rct, r¢) can be used to describe properties of pitches (for instance the approximation of JI intervals). It is defined as on 100th (or 1 percent) of the interval between two neighbouring pitches in the used equal tuning.
see also: Kirnberger Atom http://arxiv.org/abs/0907.5249
Ratio
Intervals can be measured also giving their (frequency) ratio. For instance the major third as 5/4 or the pure fifth 3/2. When combining sizes given in ratios, you have to multiply or divide:
a pure fifth increased by a major third gives the major seventh 3/2*5/4 = 15/8,
which is a diatonic semitone below an octave (2/1)/(15/8) = 2/1*8/15 = 16/15.
Another notation for ratios is a vector of prime factor exponents, often called a monzo, such as |-4 4 -1> (for the syntonic comma, 81/80 = 2^(-4) * 3^4 * 5^(-1)), which builds a bridge back to the logarithmic measure: intervals can be combined by component-wise addition or subtraction of their vectors.