Interval size measure

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.

Octave-based fine measures

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 16edo 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 12edo 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 12edo 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
Great Iring/Centitone	500	22 × 53
Iring/Centitone	600	23 × 3 × 52	Relative cent of 6edo (12edo tone); Proposed by Widogast Iring (1898), later by Joseph Yasser as a "centitone" (1932). (source)
Skisma	612	22 × 32 × 17	Edo representation of Sagittal's Ultra (Herculean) precision level JI notation (58eda), where it is known as an "ultrina"
Delfi	665	5 × 7 × 19
Small Iring/Centitone	700	22 × 52 x 7
Woolhouse	730	2 × 5 × 73	Proposed by Wesley S.B. Woolhouse in Essay on musical intervals (1835)
millioctave (moct)	1000	23 × 53	SI-prefix division of octave
cent (¢)	1200	24 × 3 × 52	1/100 of 12edo 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 or Heptamu	1536	29 × 3	(7th MIDI unit), seventh MIDI-resolution unit, 1/128 (1/(27)) of 12ED2 semitone (source)
rhoon	1560	23 × 3 × 5 × 13
śata	1600	26 × 52	From Sanskrit śatam: hundred; Relative cent of Armodue 16edo Theory
tile	1632	25 × 3 × 17
Iota	1700	22 × 52 × 17	Relative cent of 17edo; proposed by Margo Schulter (source) and George Secor (source)
Harmos	1728	26 × 33	1728 = 123; 1/144 of 12edo semitone; Proposed by Paul Beaver (source)
Hind śat / Indian cent	2200	23 × 11 × 52
Mina	2460	22 × 3 × 5 × 41	Abbreviation of "schismina", edo representation of Sagittal's Extreme (Olympian) precision level JI notation (233eda)
Centidiesis	3100	22 × 52 x 31
Centiméride	4300	22 × 52 x 43
Tina	8539	PRIME	Provides good approximations for 41-limit primes except 37 (source); named by Dave Keenan and George Secor; edo representation of Sagittal's Insane (Magrathean) precision level JI notation (809eda)
Purdal	9900	22 × 32 × 52 × 11	Relative cent of 99edo; Suggested by Osmiorisbendi, advocated by Tútim Dennsuul Wafiil
Türk sent / Turkish cent	10600	23 × 52 × 53	Relative cent of 106edo, 1/200 of 53edo; 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
Mean free path	~216,608,494	2×41×2641567

* More to be added regarding the Heptaméride/Savart units

Non-octave fine measures

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):	Base Interval:	Parts of Base 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
Neper (Np)	[math]\displaystyle{ e }[/math] ≈ 2.71828	1	the natural unit for logarithmic measurement
Dineper (dNp)	[math]\displaystyle{ e^2 }[/math] ≈ 7.38906	1	used for logarithmic approximants

To convert hekts, which is quite common in EDT systems, into cents, use following formula: c = h*12/13*math.log(3)/math.log(2)

Relative 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 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 × 3⁴ × 5^-1), which builds a bridge back to the logarithmic measure: intervals can be combined by component-wise addition or subtraction of their vectors.