Interval size measure

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Interval size measure or interval size unit means the distance between pitches. Intervals can be measured logarithmic 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. 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:

List of Octave-Based Fine Measures (Logarithmic)
Unit name (symbol): Divisions of Octave Prime Factors Origin / Significance
Eka 16 24 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 23 × 32
Farab 144 24 × 32 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 × 33 × 5
Savart* 300 22 × 3 × 52 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 * log102; 1/7 of Méride unit; Proposed by Joseph Sauveur (1701), advocated by Félix Savart
Gene 311 PRIME
Dröbisch Angle 360 23 × 32 × 5
Squb 494 2 × 13 × 19
Iring/Centitone 600 23 × 3 × 52 Relative cent of 6ED2 (12ED2 tone); Proposed by Widogast Iring (1898), later by Joseph Yasser as a "centitone" (1932). (source)
Skisma 612 22 × 32 × 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 23 × 53 SI-prefix division of octave
cent (¢) 1200 24 × 3 × 52 1/100 of 12ED2 semitone
greater muon 1224 23 × 32 × 17
triangular cent 1260 22 × 32 × 5 × 7
pion 1272 23 × 3 × 53
pound 1344 26 × 3 × 7
neutron 1392 24 × 3 × 29
lesser muon 1428 22 × 3 × 7 × 17
deciFarab 1440 25 × 32 × 5 1/10 of Farab unit
quadratic cent 1452 22 × 3 × 112
ksion 1476 22 × 32 × 41
cubic cent 1500 22 × 3 × 53
7mu 1536 29 × 3 (7th MIDI unit), seventh MIDI-resolution unit, 1/128 (1/(27)) of 12ED2 semitone
rhoon 1560 23 × 3 × 5 × 13
tile 1632 25 × 3 × 17
Iota 1700 22 × 52 × 17 Relative cent of 17ED2; proposed by Margo Schulter (source) and George Secor (source)
Harmos 1728 26 × 33 1728 = 123; 1/144 of 12ED2 semitone; Proposed by Paul Beaver (source)
Mina 2460 22 × 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 22 × 32 × 52 × 11 Relative cent of 99ED2; Suggested by Osmiorisbendi, advocated by Tútim Dennsuul Wafiil
Türk sent / Turkish cent 10600 23 × 52 × 53 Relative cent of 106ED2, 1/200 of 53ED2; invented by M. Ekrem Karadeniz (1965), influenced by Abdülkadir Töre
Prima 12276 22 × 32 × 11 × 31
Jinn 16808 23 × 11 × 191
Jot 30103 PRIME 30103 ≃ 100,000 * log102; Proposed by Augustus de Morgan(1864)
Imp 31920 24 × 3 × 5 × 7 × 19
Flu 46032 24 × 3 × 7 × 137
MIDI Tuning Standard unit 196608 216 × 3 14mu (14th MIDI unit), fourteenth MIDI-resolution unit, 1/16384 (1/(214)) 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:

List of Non-Octave Fine Measures (Logarithmic)
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.