Interval size measure
Interval size measure or interval size unit means the distance between pitches. Intervals can be measured logarithmic or by frequency ratios.
Contents
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. An early unit for measuring intervals is the "tone" which dates back to classic Greece.
In serial music all intervals were measured by the number of 12edo-semitones. In analogy, 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 the octave:
Unit name (symbol): | Divisions of Octave | Prime Factors | Origin / Significance |
---|---|---|---|
Eka | 16 | 2^{4} | From Sanskrit eka: one, unit; chromatic unit of Armodue 16ED2 Theory |
Normal diesis | 31 | PRIME | |
Méride | 43 | PRIME | Proposed by Joseph Sauveur, as 7 heptaméride units (source) |
Holdrian comma | 53 | PRIME | |
Morion | 72 | 2^{3} × 3^{2} | |
Farab | 144 | 2^{4} × 3^{2} | 1/12 of 12ED2 semitone; Proposed by al-Farabi in 10th century (source) |
Mem | 205 | 5 × 41 | Unit used by H-Pi Instruments |
Tredek | 270 | 2 × 3^{3} × 5 | |
Savart* | 300 | 2^{2} × 3 × 5^{2} | Alexander Wood's definition of the Savart (The Physics of Music, 1944), compatible with 12ED2 system |
Heptaméride/Eptaméride/Savart* | 301 | 7 × 43 | 301 ≃ 1,000 * log_{10}2; 1/7 of Méride unit; Proposed by Joseph Sauveur (1701), advocated by Félix Savart |
Gene | 311 | PRIME | |
Dröbisch Angle | 360 | 2^{3} × 3^{2} × 5 | |
Squb | 494 | 2 × 13 × 19 | |
Iring/Centitone | 600 | 2^{3} × 3 × 5^{2} | Relative cent of 6ED2 (12ED2 tone); Proposed by Widogast Iring (1898), later by Joseph Yasser as a "centitone" (1932). (source) |
Skisma | 612 | 2^{2} × 3^{2} × 17 | |
Delfi | 665 | 5 × 7 × 19 | |
Woolhouse | 730 | 2 × 5 × 73 | Proposed by Wesley S.B. Woolhouse in Essay on musical intervals (1835) |
millioctave (mO) | 1000 | 2^{3} × 5^{3} | SI-prefix division of octave |
cent (¢) | 1200 | 2^{4} × 3 × 5^{2} | 1/100 of 12ED2 semitone |
greater muon | 1224 | 2^{3} × 3^{2} × 17 | |
triangular cent | 1260 | 2^{2} × 3^{2} × 5 × 7 | |
pion | 1272 | 2^{3} × 3 × 53 | |
pound | 1344 | 2^{6} × 3 × 7 | |
neutron | 1392 | 2^{4} × 3 × 29 | |
lesser muon | 1428 | 2^{2} × 3 × 7 × 17 | |
deciFarab | 1440 | 2^{5} × 3^{2} × 5 | 1/10 of Farab unit |
quadratic cent | 1452 | 2^{2} × 3 × 11^{2} | |
ksion | 1476 | 2^{2} × 3^{2} × 41 | |
cubic cent | 1500 | 2^{2} × 3 × 5^{3} | |
7mu | 1536 | 2^{9} × 3 | (7th MIDI unit), seventh MIDI-resolution unit, 1/128 (1/(2^{7})) of 12ED2 semitone |
rhoon | 1560 | 2^{3} × 3 × 5 × 13 | |
tile | 1632 | 2^{5} × 3 × 17 | |
Iota | 1700 | 2^{2} × 5^{2} × 17 | Relative cent of 17ED2; proposed by Margo Schulter (source) and George Secor (source) |
Harmos | 1728 | 2^{6} × 3^{3} | 1728 = 12^{3}; 1/144 of 12ED2 semitone; Proposed by Paul Beaver (source) |
Mina | 2460 | 2^{2} × 3 × 5 × 41 | Abbreviation of "schismina", ED2 representation of the "Olympian Level" of Sagittal notation system |
Tina | 8539 | PRIME | Provides good approximations for 41-limit primes except 37 (source); named by Dave Keenan and George Secor |
Purdal | 9900 | 2^{2} × 3^{2} × 5^{2} × 11 | Relative cent of 99ED2; Suggested by Osmiorisbendi, advocated by Tútim Dennsuul Wafiil |
Türk sent / Turkish cent | 10600 | 2^{3} × 5^{2} × 53 | Relative cent of 106ED2, 1/200 of 53ED2; invented by M. Ekrem Karadeniz (1965), influenced by Abdülkadir Töre |
Prima | 12276 | 2^{2} × 3^{2} × 11 × 31 | |
Jinn | 16808 | 2^{3} × 11 × 191 | |
Jot | 30103 | PRIME | 30103 ≃ 100,000 * log_{10}2; Proposed by Augustus de Morgan(1864) |
Imp | 31920 | 2^{4} × 3 × 5 × 7 × 19 | |
Flu | 46032 | 2^{4} × 3 × 7 × 137 | |
MIDI Tuning Standard unit | 196608 | 2^{16} × 3 | 14mu (14th MIDI unit), fourteenth MIDI-resolution unit, 1/16384 (1/(2^{14})) of 12ED2 semitone |
* More to be added regarding the Heptaméride/Savart units
There are other fine measurements based upon the logarithmic division of other intervals (e.g. 3/1 (twelfth)), a few of which are listed below:
Unit name (symbol): | Interval based on: | Divisions of Interval: | Origin/Significance |
---|---|---|---|
Hekt | 3/1 (twelfth) | 1300 | 1/100 of 13-ED3 (Bohlen-Pierce) scale step |
Grad | 531441/524288 (Pythagorean comma) | 12 | |
Tuning unit | 531441/524288 (Pythagorean comma) | 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.