Interval size measure: Difference between revisions

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m "http://www.h-pi.com/theory/measurement3.html" is now a dead link; changed external link at text "H-Pi Instruments" (also changed from "H-pi Instruments") to "http://musictheory.zentral.zone/huntsystem1.html".
Dw0391 (talk | contribs)
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Replaced large paragraph in "Logarithmic" section with a table displaying same information. Maintained proposed "backslash notation" in "Power of interval:" column, but I would have preferred to use forward slashes (see "Interval based on:" column).
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===Fine===
===Fine===
The [[cent]] (¢), [[1200edo|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 [[cent]] (¢), [[1200edo|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.  


Other measures include the [[Armodue_theory|Eka]], [[16edo|1\16 octave]], the [[Normal_diesis|Normal diesis]]: [[31edo|1\31 octave]]; the [[Méride]]: [[43edo|1\43 octave]]; the [[Holdrian_comma|Holdrian comma]]: [[53edo|1\53 octave]]; the [[Morion]]: [[72edo|1\72 octave]]; the [[Farab]]: [[144edo|1\144 octave]]; the [[Mem]]: [[205edo|1\205 octave]] (used by [http://musictheory.zentral.zone/huntsystem1.html H-Pi Instruments]); the [[Tredek]]: [[270edo|1\270 octave]]; the [[Eptaméride]] or [[Savart]]: [[301edo|1\301 of an octave]]; the [[Gene]]: [[311edo|1\311 octave]]; the [[Dröbisch_Angle|Dröbisch Angle]]: [[360edo|1\360 octave]]; the [[Squb]]: [[494edo|1\494 octave]]; the [[Iring]]: [[600edo|1\600 octave]]; the [[Skisma]]: [[612edo|1\612 octave]]; the [[Delfi]]: [[665edo|1\665 octave]]; the [[Woolhouse]]: [[730edo|1\730 octave]]; the [[millioctave]] (mO), [[1000edo|1\1000 octave]]; the [[fine_cent|fine cent]]s and fine cent-like units from [[1201edo|1\1201 octave]] down to [[1728edo|1\1728 octave]] (including the greater and lesser muons: [[1224edo|1\1224 octave]] and [[1428edo|1\1428 octave]]; the triangular, quadratic and cubic cents: [[1260edo|1\1260 octave]], [[1452edo|1\1452 octave]] and [[1500edo|1\1500 octave]]; the pion: [[1272edo|1\1272 octave]]; the pound: [[1344edo|1\1344 octave]]; the neutron: [[1392edo|1\1392 octave]]; the deciFarab: [[1440edo|1\1440 octave]]; the ksion: [[1476edo|1\1476 octave]]; the 7mu: [[1536edo|1\1536 octave]]; the rhoon: [[1560edo|1\1560 octave]]; the tile: [[1632edo|1\1632 octave]]; the [[Iota]]: [[1\1700_octave|1\1700 octave]] and finally the [[Harmos]]: [[1728edo|1\1728 octave]]); the [[Mina]]: [[2460edo|1\2460 octave]]; the [[Tina]]: [[8539edo|1\8539 octave]]; the [[Purdal]]: [[9900edo|1\9900 octave]]; the [[Türk_sent|Türk sent]]: [[10600edo|1\10600 octave]]; the [[Prima]]: [[12276edo|1\12276 octave]], the [[Jinn]]: [[16808edo|1\16808 octave]], the [[Jot]]: [[30103edo|1\30103 octave]]; the [[Imp]]: [[31920edo|1\31920 octave]]; the [[Flu]]: [[46032edo|1\46032 octave]]; and the [[MIDI_Tuning_Standard_unit|MIDI Tuning Standard unit]]: [[196608edo|1\196608 octave]]. Not based on the octave are the [[Grad]]: 1/12 of a Pythagorean comma, the [[Tuning unit]]: 1/720 of a Pythagorean comma and the [[Hekt]]: 1/1300 part of 3, ie 3^(1/1300).
The following table demonstrates a list of measures derived from the logarithmic division of a interval (e.g. octave, twelfth):
{| class="wikitable sortable"
|+List of Fine Measures (Logarithmic)
!Unit name (symbol):
!Interval based on:
!Power of interval:
!Significance
|-
|[[Armodue_theory|Eka]]
|2/1 (octave)
|[[16edo|1\16]]
|
|-
|[[Normal_diesis|Normal diesis]]
|2/1 (octave)
|[[31edo|1\31]]
|
|-
|[[Méride]]
|2/1 (octave)
|[[43edo|1\43]]
|
|-
|[[Holdrian_comma|Holdrian comma]]
|2/1 (octave)
|[[53edo|1\53]]
|
|-
|[[Morion]]
|2/1 (octave)
|[[72edo|1\72]]
|
|-
|[[Farab]]
|2/1 (octave)
|[[144edo|1\144]]
|
|-
|[[Mem]]
|2/1 (octave)
|[[205edo|1\205]]
|Unit used by [http://musictheory.zentral.zone/huntsystem1.html H-Pi Instruments]
|-
|[[Tredek]]
|2/1 (octave)
|[[270edo|1\270]]
|
|-
|[[Eptaméride]] or [[Savart]]
|2/1 (octave)
|[[301edo|1\301]]
|
|-
|[[Gene]]
|2/1 (octave)
|[[311edo|1\311]]
|
|-
|[[Dröbisch_Angle|Dröbisch Angle]]
|2/1 (octave)
|[[360edo|1\360]]
|
|-
|[[Squb]]
|2/1 (octave)
|[[494edo|1\494]]
|
|-
|[[Iring]]
|2/1 (octave)
|[[600edo|1\600]]
|
|-
|[[Skisma]]
|2/1 (octave)
|[[612edo|1\612]]
|
|-
|[[Delfi]]
|2/1 (octave)
|[[665edo|1\665]]
|
|-
|[[Woolhouse]]
|2/1 (octave)
|[[730edo|1\730]]
|
|-
|[[millioctave]] (mO)
|2/1 (octave)
|[[1000edo|1\1000]]
|"Metric" / SI division of octave
|-
|[[cent]] (¢)
|2/1 (octave)
|1\1200
|1/100 of 12-ED2 semitone
|-
|greater muon
|2/1 (octave)
|[[1224edo|1\1224]]
|
|-
|triangular cent
|2/1 (octave)
|[[1260edo|1\1260]]
|
|-
|pion
|2/1 (octave)
|[[1272edo|1\1272]]
|
|-
|pound
|2/1 (octave)
|[[1344edo|1\1344]]
|
|-
|neutron
|2/1 (octave)
|[[1392edo|1\1392]]
|
|-
|lesser muon
|2/1 (octave)
|[[1428edo|1\1428]]
|
|-
|deciFarab
|2/1 (octave)
|[[1440edo|1\1440]]
|1/10 of Farab
|-
|quadratic cent
|2/1 (octave)
|[[1452edo|1\1452]]
|
|-
|ksion
|2/1 (octave)
|[[1476edo|1\1476]]
|
|-
|cubic cent
|2/1 (octave)
|[[1500edo|1\1500]]
|
|-
|7mu
|2/1 (octave)
|[[1536edo|1\1536]]
|
|-
|rhoon
|2/1 (octave)
|[[1560edo|1\1560]]
|
|-
|tile
|2/1 (octave)
|[[1632edo|1\1632]]
|
|-
|[[Iota]]
|2/1 (octave)
|[[1\1700_octave|1\1700]]
|
|-
|[[Harmos]]
|2/1 (octave)
|[[1728edo|1\1728]]
|
|-
|[[Mina]]
|2/1 (octave)
|[[2460edo|1\2460]]
|
|-
|[[Tina]]
|2/1 (octave)
|[[8539edo|1\8539]]
|
|-
|[[Purdal]]
|2/1 (octave)
|[[9900edo|1\9900]]
|
|-
|[[Türk_sent|Türk sent]]
|2/1 (octave)
|[[10600edo|1\10600]]
|
|-
|[[Prima]]
|2/1 (octave)
|[[12276edo|1\12276]]
|
|-
|[[Jinn]]
|2/1 (octave)
|[[16808edo|1\16808]]
|
|-
|[[Jot]]
|2/1 (octave)
|[[30103edo|1\30103]]
|
|-
|[[Imp]]
|2/1 (octave)
|[[31920edo|1\31920]]
|
|-
|[[Flu]]
|2/1 (octave)
|[[46032edo|1\46032]]
|
|-
|[[MIDI_Tuning_Standard_unit|MIDI Tuning Standard unit]]
|2/1 (octave)
|[[196608edo|1\196608]]
|fourteenth MIDI-resolution unit, 2<sup>14</sup> = 1/16384 of [[12edo|12ED]]2 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 [http://www.huygens-fokker.org/docs/measures.html Logarithmic Interval Measures]
See [http://www.huygens-fokker.org/docs/measures.html Logarithmic Interval Measures]

Revision as of 08:28, 22 September 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 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
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 12-ED2 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
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
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 fourteenth MIDI-resolution unit, 214 = 1/16384 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.

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