Interval size measure

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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 closely related to 12 equal.

Other measures include the Eka, 1\16 octave, Normal diesis: 1\31 octave; the Méride: 1\43 octave; the Holdrian comma: 1\53 octave; the Morion: 1\72 octave; the Farab: 1\144 octave; the Mem: 1\205 octave (used by Hi-pi Instruments); the Tredek: 1\270 octave; the Eptaméride or Savart: 1\301 of an octave; the Gene: 1\311 octave; the Dröbisch Angle: 1\360 octave; the Squb: 1\494 octave; the Iring: 1\600 octave; the Skisma: 1\612 octave; the Delfi: 1\665 octave; the Woolhouse: 1\730 octave; the millioctave (mO), 1\1000 octave; the fine cent or deciFarab: 1\1440 octave; the Iota: 1\1700 octave; the Harmos: 1\1728 octave; the Mina: 1\2460 octave; the Tina: 1\8539 octave; the Purdal: 1\9900 octave; the Türk sent: 1\10600 octave; the Prima: 1\12276 octave, the Jinn: 1\16808 octave, the Jot: 1\30103 octave; the Imp: 1\31920 octave; the Flu: 1\46032 octave; and the MIDI Tuning Standard unit: 1\196608 octave. Not based on the octave are the Grad: 1/12 of a Pythagorean comma and the Hekt: 1/1300 part of 3, ie 3^(1/1300).

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.