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):

List of Fine Measures (Logarithmic)

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	Abbreviation of "schismina", ED2 representation of the "Olympian Level" of Sagittal notation system
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.