Interval size measure: Difference between revisions
Split Fine Measures table into two: one for octave-based measures, the other for non-octave measures |
Changed wording and added Wikipedia "Metric Prefix" link for millioctave unit; added "Significance / Origin" information to Harmos unit |
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|[[millioctave]] (mO) | |[[millioctave]] (mO) | ||
|[[1000edo|1\1000]] | |[[1000edo|1\1000]] | ||
| | |[https://en.wikipedia.org/wiki/Metric_prefix SI-prefix] division of octave | ||
|- | |- | ||
|[[cent]] (¢) | |[[cent]] (¢) | ||
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|[[Harmos]] | |[[Harmos]] | ||
|[[1728edo|1\1728]] | |[[1728edo|1\1728]] | ||
| | |Proposed by Paul Beaver; 1728 = 12<sup>3</sup>; 1/144 of [[12edo|12ED2]] semitone | ||
|- | |- | ||
|[[Mina]] | |[[Mina]] | ||
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|[[Tina]] | |[[Tina]] | ||
|[[8539edo|1\8539]] | |[[8539edo|1\8539]] | ||
|Prime; provides good approximations for 41-limit primes except 37; named by [[Dave_Keenan|Keenan]] and [[George_Secor|Secor]] | |Prime; provides good approximations for 41-limit primes except 37; named by [[Dave_Keenan|Keenan]] and [[George_Secor|Secor]] | ||
|- | |- | ||
|[[Purdal]] | |[[Purdal]] | ||
Revision as of 08:16, 13 October 2019
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 the octave:
| Unit name (symbol): | Division of Octave | Origin / Significance |
|---|---|---|
| Eka | 1\16 | From Sanskrit eka: one, unit; chromatic unit of Armodue 16ED2 Theory |
| Normal diesis | 1\31 | |
| Méride | 1\43 | Proposed by Joseph Sauveur as 7 heptaméride units |
| Holdrian comma | 1\53 | |
| Morion | 1\72 | |
| Farab | 1\144 | Proposed by al-Farabi in 10th century; 1/12 of 12ED2 semitone |
| Mem | 1\205 | Unit used by H-Pi Instruments |
| Tredek | 1\270 | |
| Heptaméride/Eptaméride or Savart | 1\301 | Proposed by Joseph Sauveur, advocated by Félix Savart; 301 ≃ 1,000 * log102; 1/7 of Méride unit |
| Gene | 1\311 | |
| Dröbisch Angle | 1\360 | |
| Squb | 1\494 | |
| Iring | 1\600 | |
| Skisma | 1\612 | |
| Delfi | 1\665 | |
| Woolhouse | 1\730 | Proposed by Wesley S.B. Woolhouse in Essay on musical intervals (1835) |
| millioctave (mO) | 1\1000 | SI-prefix division of octave |
| cent (¢) | 1\1200 | 1/100 of 12ED2 semitone |
| greater muon | 1\1224 | |
| triangular cent | 1\1260 | |
| pion | 1\1272 | |
| pound | 1\1344 | |
| neutron | 1\1392 | |
| lesser muon | 1\1428 | |
| deciFarab | 1\1440 | 1/10 of Farab unit |
| quadratic cent | 1\1452 | |
| ksion | 1\1476 | |
| cubic cent | 1\1500 | |
| 7mu | 1\1536 | (7th MIDI unit), seventh MIDI-resolution unit, 1/128 (1/(27)) of 12ED2 semitone |
| rhoon | 1\1560 | |
| tile | 1\1632 | |
| Iota | 1\1700 | |
| Harmos | 1\1728 | Proposed by Paul Beaver; 1728 = 123; 1/144 of 12ED2 semitone |
| Mina | 1\2460 | Abbreviation of "schismina", ED2 representation of the "Olympian Level" of Sagittal notation system |
| Tina | 1\8539 | Prime; provides good approximations for 41-limit primes except 37; named by Keenan and Secor |
| Purdal | 1\9900 | |
| Türk sent | 1\10600 | |
| Prima | 1\12276 | |
| Jinn | 1\16808 | |
| Jot | 1\30103 | |
| Imp | 1\31920 | |
| Flu | 1\46032 | |
| MIDI Tuning Standard unit | 1\196608 | 14mu (14th MIDI unit), fourteenth MIDI-resolution unit, 1/16384 (1/(214)) of 12ED2 semitone |
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:
| Unit name (symbol): | Interval based on: | Power: | Origin/Significance |
|---|---|---|---|
| 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.