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.

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[citation needed].
Normal diesis 31 PRIME See the dedicated page.
Méride 43 PRIME Proposed by Joseph Sauveur, as 7 heptaméride units[1][2].
Holdrian comma 53 PRIME [1]
Morion 72 23 × 32 See dedicated page.
Farab 144 24 × 32 1/12 of 12edo semitone; Proposed by al-Farabi in 10th century[1][3].
Mem 205 5 × 41 Unit used by H-Pi Instruments[1][4][5].
Tredek 270 2 × 33 × 5 Proposed by Joe Monzo (2013)[6].
Savart* 300 22 × 3 × 52 Alexander Wood's definition of the Savart[7], containing 12edo.
Heptaméride / eptaméride / savart* 301 7 × 43 301 ≃ 1,000 × log102; 1/7 of Méride unit; proposed by Joseph Sauveur (1701), advocated by Félix Savart[1][8].
Gene 311 PRIME Proposed by Joe Monzo (2007)[9].
Dröbisch Angle 360 23 × 32 × 5 Proposed as angle by Moritz Dröbisch in the 19th century, later by Andrew Pikler as the current name in Logarithmic Frequency Systems (1966)[1].
Squb 494 2 × 13 × 19 [citation needed]
Great iring / centitone 500 22 × 53 [citation needed]
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)[1][10].
Skisma (Sk) 612 22 × 32 × 17 Edo representation of Sagittal's Ultra (Herculean) precision level JI notation (58eda), where it is known as an "ultrina"[1][11].
Delfi 665 5 × 7 × 19 [1]
Small iring / centitone 700 22 × 52 x 7 [citation needed]
Woolhouse 730 2 × 5 × 73 Proposed by Wesley S.B. Woolhouse (1835)[12].
Millioctave (moct) 1000 23 × 53 See the dedicated page.
Cent (¢) 1200 24 × 3 × 52 See the dedicated page.
Greater muon 1224 23 × 32 × 17 [citation needed]
Triangular cent 1260 22 × 32 × 5 × 7 [citation needed]
Pion 1272 23 × 3 × 53 [citation needed]
Pound 1344 26 × 3 × 7 [citation needed]
Neutron 1392 24 × 3 × 29 [citation needed]
Lesser muon 1428 22 × 3 × 7 × 17 [citation needed]
Decifarab 1440 25 × 32 × 5 1/10 of Farab unit[1].
Quadratic cent 1452 22 × 3 × 112 [citation needed]
Ksion 1476 22 × 32 × 41 [citation needed]
Cubic cent 1500 22 × 3 × 53 [citation needed]
Heptamu (7mu) 1536 29 × 3 Seventh MIDI-resolution unit, 1/128 (1/(27)) of 12edo semitone[13]
Rhoon 1560 23 × 3 × 5 × 13 [citation needed]
śata 1600 26 × 52 From Sanskrit śatam: hundred; Relative cent of Armodue 16edo Theory[citation needed]
Tile 1632 25 × 3 × 17 [citation needed]
Iota 1700 22 × 52 × 17 Relative cent of 17edo; proposed by Margo Schulter (2002) and George Secor[1].
Harmos 1728 26 × 33 1728 = 123; 1/144 of 12edo semitone; Proposed by Paul Beaver[1][14].
Hind śat / Indian cent 2200 23 × 11 × 52 [citation needed]
Mina 2460 22 × 3 × 5 × 41 Abbreviation of "schismina", edo representation of Sagittal's Extreme (Olympian) precision level JI notation (233eda)[1][15].
Centidiesis 3100 22 × 52 x 31 [citation needed]
Centiméride 4300 22 × 52 x 43 [citation needed]
Tina 8539 PRIME Provides good approximations for 41-limit primes except 37; named by Dave Keenan and George Secor; edo representation of Sagittal's Insane (Magrathean) precision level JI notation (809eda)[1][16].
Purdal 9900 22 × 32 × 52 × 11 Relative cent of 99edo; Suggested by Osmiorisbendi, advocated by Tútim Dennsuul Wafiil. See the dedicated page.
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[1][17][18].
Prima 12276 22 × 32 × 11 × 31 Proposed by Erv Wilson, Gene Ward Smith and Gavin Putland[1].
Jinn 16808 23 × 11 × 191 See the dedicated page.
Jot 30103 PRIME 30103 ≃ 100,000 × log102; Proposed by Augustus de Morgan (1864)[1][19][14].
Imp 31920 24 × 3 × 5 × 7 × 19 [1]
Flu 46032 24 × 3 × 7 × 137 Proposed by Gene Ward Smith (2005)[1][20].
MIDI Tuning Standard unit (14mu) 196608 216 × 3 Fourteenth MIDI-resolution unit, 1/16384 (1/(214)) of 12edo semitone[1].
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]e[/math] ≈ 2.71828 1 the natural unit for logarithmic measurement
Dineper (dNp) [math]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.

Notes