Interval size measure: Difference between revisions

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Dw0391 (talk | contribs)
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m Sorry, this time I actually did remember to change "List of Non-Octave Fine Measures (Logarithmic)" table as indicated in the previous edit summary - mea culpa
Dw0391 (talk | contribs)
26 edits
Added "Prime Factors" column to table "List of Octave-Based Fine Measures (Logarithmic)"; populated column
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!Unit name (symbol):
!Unit name (symbol):
!Divisions of Octave
!Divisions of Octave
!Prime Factors
!Origin / Significance
!Origin / Significance
|-
|-
|[[Armodue_theory|Eka]]
|[[Armodue_theory|Eka]]
|[[16edo|16]]
|[[16edo|16]]
|2<sup>4</sup>
|From Sanskrit ''eka'': one, unit; chromatic unit of Armodue 16ED2 Theory
|From Sanskrit ''eka'': one, unit; chromatic unit of Armodue 16ED2 Theory
|-
|-
|[[Normal_diesis|Normal diesis]]
|[[Normal_diesis|Normal diesis]]
|[[31edo|31]]
|[[31edo|31]]
|PRIME
|
|
|-
|-
|[[Méride]]
|[[Méride]]
|[[43edo|43]]
|[[43edo|43]]
|PRIME
|Proposed by Joseph Sauveur as 7 heptaméride units
|Proposed by Joseph Sauveur as 7 heptaméride units
|-
|-
|[[Holdrian_comma|Holdrian comma]]
|[[Holdrian_comma|Holdrian comma]]
|[[53edo|53]]
|[[53edo|53]]
|PRIME
|
|
|-
|-
|[[Morion]]
|[[Morion]]
|[[72edo|72]]
|[[72edo|72]]
|2<sup>3</sup> × 3<sup>2</sup>
|
|
|-
|-
|[[Farab]]
|[[Farab]]
|[[144edo|144]]
|[[144edo|144]]
|2<sup>4</sup> × 3<sup>2</sup>
|Proposed by al-Farabi in 10th century; 1/12 of [[12edo|12ED2]] semitone
|Proposed by al-Farabi in 10th century; 1/12 of [[12edo|12ED2]] semitone
|-
|-
|[[Mem]]
|[[Mem]]
|[[205edo|205]]
|[[205edo|205]]
|5 × 41
|Unit used by [http://musictheory.zentral.zone/huntsystem1.html H-Pi Instruments]
|Unit used by [http://musictheory.zentral.zone/huntsystem1.html H-Pi Instruments]
|-
|-
|[[Tredek]]
|[[Tredek]]
|[[270edo|270]]
|[[270edo|270]]
|2 × 3<sup>3</sup> × 5
|
|
|-
|-
|[[Eptaméride|Heptaméride/Eptaméride]] or [[Savart]]
|[[Eptaméride|Heptaméride/Eptaméride]] or [[Savart]]
|[[301edo|301]]
|[[301edo|301]]
|7 × 43
|Proposed by Joseph Sauveur, advocated by Félix Savart; 301 ≃ 1,000 * log<sub>10</sub>2; 1/7 of Méride unit
|Proposed by Joseph Sauveur, advocated by Félix Savart; 301 ≃ 1,000 * log<sub>10</sub>2; 1/7 of Méride unit
|-
|-
|[[Gene]]
|[[Gene]]
|[[311edo|311]]
|[[311edo|311]]
|PRIME
|
|
|-
|-
|[[Dröbisch_Angle|Dröbisch Angle]]
|[[Dröbisch_Angle|Dröbisch Angle]]
|[[360edo|360]]
|[[360edo|360]]
|2<sup>3</sup> × 3<sup>2</sup> × 5
|
|
|-
|-
|[[Squb]]
|[[Squb]]
|[[494edo|494]]
|[[494edo|494]]
|2 × 13 × 19
|
|
|-
|-
|[[Iring]]
|[[Iring]]
|[[600edo|600]]
|[[600edo|600]]
|2<sup>3</sup> × 3 × 5<sup>2</sup>
|
|
|-
|-
|[[Skisma]]
|[[Skisma]]
|[[612edo|612]]
|[[612edo|612]]
|2<sup>2</sup> × 3<sup>2</sup> × 17
|
|
|-
|-
|[[Delfi]]
|[[Delfi]]
|[[665edo|665]]
|[[665edo|665]]
|5 × 7 × 19
|
|
|-
|-
|[[Woolhouse]]
|[[Woolhouse]]
|[[730edo|730]]
|[[730edo|730]]
|2 × 5 × 73
|Proposed by Wesley S.B. Woolhouse in ''Essay on musical intervals'' (1835)
|Proposed by Wesley S.B. Woolhouse in ''Essay on musical intervals'' (1835)
|-
|-
|[[millioctave]] (mO)
|[[millioctave]] (mO)
|[[1000edo|1000]]
|[[1000edo|1000]]
|2<sup>3</sup> × 5<sup>3</sup>
|[https://en.wikipedia.org/wiki/Metric_prefix SI-prefix] division of octave
|[https://en.wikipedia.org/wiki/Metric_prefix SI-prefix] division of octave
|-
|-
|[[cent]] (¢)
|[[cent]] (¢)
|1200
|1200
|2<sup>4</sup> × 3 × 5<sup>2</sup>
|1/100 of [[12edo|12ED2]] semitone
|1/100 of [[12edo|12ED2]] semitone
|-
|-
|greater muon
|greater muon
|[[1224edo|1224]]
|[[1224edo|1224]]
|2<sup>3</sup> × 3<sup>2</sup> × 17
|
|
|-
|-
|triangular cent
|triangular cent
|[[1260edo|1260]]
|[[1260edo|1260]]
|2<sup>2</sup> × 3<sup>2</sup> × 5 × 7
|
|
|-
|-
|pion
|pion
|[[1272edo|1272]]
|[[1272edo|1272]]
|2<sup>3</sup> × 3 × 53
|
|
|-
|-
|pound
|pound
|[[1344edo|1344]]
|[[1344edo|1344]]
|2<sup>6</sup> × 3 × 7
|
|
|-
|-
|neutron
|neutron
|[[1392edo|1392]]
|[[1392edo|1392]]
|2<sup>4</sup> × 3 × 29
|
|
|-
|-
|lesser muon
|lesser muon
|[[1428edo|1428]]
|[[1428edo|1428]]
|2<sup>2</sup> × 3 × 7 × 17
|
|
|-
|-
|deciFarab
|deciFarab
|[[1440edo|1440]]
|[[1440edo|1440]]
|2<sup>5</sup> × 3<sup>2</sup> × 5
|1/10 of Farab unit
|1/10 of Farab unit
|-
|-
|quadratic cent
|quadratic cent
|[[1452edo|1452]]
|[[1452edo|1452]]
|2<sup>2</sup> × 3 × 11<sup>2</sup>
|
|
|-
|-
|ksion
|ksion
|[[1476edo|1476]]
|[[1476edo|1476]]
|2<sup>2</sup> × 3<sup>2</sup> × 41
|
|
|-
|-
|cubic cent
|cubic cent
|[[1500edo|1500]]
|[[1500edo|1500]]
|2<sup>2</sup> × 3 × 5<sup>3</sup>
|
|
|-
|-
|7mu
|7mu
|[[1536edo|1536]]
|[[1536edo|1536]]
|2<sup>9</sup> × 3
|(7th MIDI unit), seventh MIDI-resolution unit, 1/128 (1/(2<sup>7</sup>)) of [[12edo|12ED2]] semitone
|(7th MIDI unit), seventh MIDI-resolution unit, 1/128 (1/(2<sup>7</sup>)) of [[12edo|12ED2]] semitone
|-
|-
|rhoon
|rhoon
|[[1560edo|1560]]
|[[1560edo|1560]]
|2<sup>3</sup> × 3 × 5 × 13
|
|
|-
|-
|tile
|tile
|[[1632edo|1632]]
|[[1632edo|1632]]
|2<sup>5</sup> × 3 × 17
|
|
|-
|-
|[[Iota]]
|[[Iota]]
|[[1\1700_octave|1700]]
|[[1\1700_octave|1700]]
|2<sup>2</sup> × 5<sup>2</sup> × 17
|
|
|-
|-
|[[Harmos]]
|[[Harmos]]
|[[1728edo|1728]]
|[[1728edo|1728]]
|2<sup>6</sup> × 3<sup>3</sup>
|Proposed by Paul Beaver; 1728 = 12<sup>3</sup>; 1/144 of [[12edo|12ED2]] semitone
|Proposed by Paul Beaver; 1728 = 12<sup>3</sup>; 1/144 of [[12edo|12ED2]] semitone
|-
|-
|[[Mina]]
|[[Mina]]
|[[2460edo|2460]]
|[[2460edo|2460]]
|2<sup>2</sup> × 3 × 5 × 41
|Abbreviation of "schismina", ED2 representation of the "Olympian Level" of [[Sagittal_notation|Sagittal notation system]]
|Abbreviation of "schismina", ED2 representation of the "Olympian Level" of [[Sagittal_notation|Sagittal notation system]]
|-
|-
|[[Tina]]
|[[Tina]]
|[[8539edo|8539]]
|[[8539edo|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]]
|[[9900edo|9900]]
|[[9900edo|9900]]
|2<sup>2</sup> × 3<sup>2</sup> × 5<sup>2</sup> × 11
|
|
|-
|-
|[[Türk_sent|Türk sent]]
|[[Türk_sent|Türk sent]]
|[[10600edo|10600]]
|[[10600edo|10600]]
|2<sup>3</sup> × 5<sup>2</sup> × 53
|
|
|-
|-
|[[Prima]]
|[[Prima]]
|[[12276edo|12276]]
|[[12276edo|12276]]
|2<sup>2</sup> × 3<sup>2</sup> × 11 × 31
|
|
|-
|-
|[[Jinn]]
|[[Jinn]]
|[[16808edo|16808]]
|[[16808edo|16808]]
|2<sup>3</sup> × 11 × 191
|
|
|-
|-
|[[Jot]]
|[[Jot]]
|[[30103edo|30103]]
|[[30103edo|30103]]
|PRIME
|
|
|-
|-
|[[Imp]]
|[[Imp]]
|[[31920edo|31920]]
|[[31920edo|31920]]
|2<sup>4</sup> × 3 × 5 × 7 × 19
|
|
|-
|-
|[[Flu]]
|[[Flu]]
|[[46032edo|46032]]
|[[46032edo|46032]]
|2<sup>4</sup> × 3 × 7 × 137
|
|
|-
|-
|[[MIDI_Tuning_Standard_unit|MIDI Tuning Standard unit]]
|[[MIDI_Tuning_Standard_unit|MIDI Tuning Standard unit]]
|[[196608edo|196608]]
|[[196608edo|196608]]
|2<sup>16</sup> × 3
|14mu (14th MIDI unit), fourteenth MIDI-resolution unit, 1/16384 (1/(2<sup>14</sup>)) of [[12edo|12ED2]] semitone
|14mu (14th MIDI unit), fourteenth MIDI-resolution unit, 1/16384 (1/(2<sup>14</sup>)) of [[12edo|12ED2]] semitone
|}
|}

Revision as of 00:21, 25 December 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:

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 16ED2 Theory
Normal diesis 31 PRIME
Méride 43 PRIME Proposed by Joseph Sauveur as 7 heptaméride units
Holdrian comma 53 PRIME
Morion 72 23 × 32
Farab 144 24 × 32 Proposed by al-Farabi in 10th century; 1/12 of 12ED2 semitone
Mem 205 5 × 41 Unit used by H-Pi Instruments
Tredek 270 2 × 33 × 5
Heptaméride/Eptaméride or Savart 301 7 × 43 Proposed by Joseph Sauveur, advocated by Félix Savart; 301 ≃ 1,000 * log102; 1/7 of Méride unit
Gene 311 PRIME
Dröbisch Angle 360 23 × 32 × 5
Squb 494 2 × 13 × 19
Iring 600 23 × 3 × 52
Skisma 612 22 × 32 × 17
Delfi 665 5 × 7 × 19
Woolhouse 730 2 × 5 × 73 Proposed by Wesley S.B. Woolhouse in Essay on musical intervals (1835)
millioctave (mO) 1000 23 × 53 SI-prefix division of octave
cent (¢) 1200 24 × 3 × 52 1/100 of 12ED2 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 1536 29 × 3 (7th MIDI unit), seventh MIDI-resolution unit, 1/128 (1/(27)) of 12ED2 semitone
rhoon 1560 23 × 3 × 5 × 13
tile 1632 25 × 3 × 17
Iota 1700 22 × 52 × 17
Harmos 1728 26 × 33 Proposed by Paul Beaver; 1728 = 123; 1/144 of 12ED2 semitone
Mina 2460 22 × 3 × 5 × 41 Abbreviation of "schismina", ED2 representation of the "Olympian Level" of Sagittal notation system
Tina 8539 PRIME Provides good approximations for 41-limit primes except 37; named by Keenan and Secor
Purdal 9900 22 × 32 × 52 × 11
Türk sent 10600 23 × 52 × 53
Prima 12276 22 × 32 × 11 × 31
Jinn 16808 23 × 11 × 191
Jot 30103 PRIME
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

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): Interval based on: Divisions of 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

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.

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