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'''Interval size measure''' means the ''distance'' between pitches. Intervals can be measured logarithmically or by frequency ratios.
'''Interval size measure''' or '''interval size unit''' means the ''distance'' between pitches. Intervals can be measured [[#logarithmic|logarithmic]] or by frequency [[#ratio|ratios]].


==Logarithmic==
== Logarithmic ==
All logarithmic measures can be combined by adding and subtracting them.
All logarithmic measures can be combined by adding and subtracting them.


===Gross===
=== Backslash notation ===
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.
A common shorthand in use in the microtonal community is ''k''\''N'', written with a backslash (\) instead of a forwardslash (/), to refer to an interval with a frequency ratio of 2<sup>''k''/''N''</sup>. ''k''\''N'' is pronounced "''k'' steps of ''N'' [[edo]]", and can be derived from the meaning of "[[step]]s" in the context of edos (unless talking about steps of specific subsets/scales of some edo).  


For "atonal" music it was replaced by the number of 12edo-semitones.
Steps are linear in the log-frequency domain, so expressions like {{nowrap|11\19 − 6\19 {{=}} 5\19}} hold. In general, we have
: {{nowrap|''a''\''N'' + ''b''\''N'' {{=}} (''a'' + ''b'')\''N''}}


Proposal: The '''relative interval measure''' is the number of steps between two pitches of an [[Equal-step tuning|equal]] tuning, sometimes called [[Degree|degree]]s (of an edo). For generators, the backslash notation is used d\edo, but it is also a ratio (of a logarithmic measure).
which expresses the same thing as {{nowrap|2<sup>''a''/''N''</sup> × 2<sup>''b''/''N''</sup> {{=}} 2<sup>(''a'' + ''b'')/''N''</sup>.}}


===Fine===
Or equivalently, for subtraction/division:
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).
: {{nowrap|''a''\''N'' − ''b''\''N'' {{=}} (''a'' − ''b'')\''N''}}


See [http://www.huygens-fokker.org/docs/measures.html Logarithmic Interval Measures]
which expresses the same thing as {{nowrap|2<sup>''a''/''N''</sup> / 2<sup>''b''/''N''</sup> {{=}} 2<sup>(''a'' - ''b'')/''N''</sup>.}}


Within a given [[Equal-step tuning|equal]]-stepped tonal system, the [[Relative_cent|relative cent]] (rct, r¢) can be used to describe properties of pitches (for instance the approximation of [[Just intonation|JI]] intervals). It is defined as on 100th (or 1 percent) of the interval between two neighbouring pitches in the used equal tuning.
Backslash notation can be extended to support [[nonoctave]] [[equal tuning]]s by writing the tuning in full after the backslash. For example, 11\13edt means 11 steps of [[13edt]], 14\9edf means 14 steps of [[9edf]], and 7\12ed12/5 means 7 steps of [[12ed12/5]].


see also: Kirnberger Atom http://arxiv.org/abs/0907.5249
=== Gross ===
The [[octave]] and the [[decade]] are common coarse units for interval sizes. The {{w|decibel}}, being a relative logarithmic-scale unit for power or root-power quantities, is inappropriate for measuring intervals; the decade is used instead. Similarly, the {{w|neper}} (Np) and the dineper (dNp), like the decibel, should not be used. However, in the absence of a substitute, dinepers have an application in [[logarithmic approximants]].


==Ratio==
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. An early unit for measuring intervals is the "[[tone]]" which dates back to classic Greece.
Intervals can be measured also giving their [http://en.wikipedia.org/wiki/Interval_ratio (frequency) ratio]. For instance the major third as [[5/4|5/4]] or the pure fifth [[3/2|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,
In serial music, all intervals were measured by the number of 12edo [[semitone (interval size measure)|semitone]]s. In analogy, the '''relative interval measure''' is the number of steps between two pitches of an [[equal tuning]], sometimes called "[[degree]]s". These measures can be written using [[#Backslash notation|backslash notation]] if the degree itself isn't sufficiently clear in context.


which is a diatonic semitone below an octave (2/1)/(15/8) = 2/1*8/15 = 16/15.
=== 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.  


Another notation for ratios is a vector of prime factor exponents, often called a [[monzo|monzo]], such as |-4 4 -1&gt; (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.
==== Octave-based fine measures ====
[[Category:interval]]
The following table demonstrates a list of measures derived from the logarithmic division of the octave: {{todo|complete table|research|comment=Add all missing citations.}}
[[Category:interval_size]]
 
[[Category:interval_size_measure]]
{| class="wikitable sortable"
[[Category:measure]]
|+ style="font-size: 105%;" | List of octave-based fine measures (logarithmic)
[[Category:proposal]]
|-
[[Category:size]]
! Unit name (symbol):
[[Category:theory]]
! Divisions of octave
[[Category:todo:review]]
! Prime factors
! Origin/significance
|-
| [[Eka]]
| [[16edo|16]]
| 2<sup>4</sup>
| From Sanskrit ''eka'': one, unit; chromatic unit of Armodue 16edo theory<ref>[http://www.armodue.com/risorse.htm Armodue: le risorse di un nuovo sistema musicale]</ref>.
|-
| [[Normal shruti]]
| [[22edo|22]]
| 2 × 11
| Proposed by [[User:Tristanbay|Tristan Bay]] (2025) in reference to the Indian tradition of dividing the octave into 22 unequal parts.
|-
| [[Normal diesis]]
| [[31edo|31]]
| 31 (prime)
| See the dedicated page.
|-
| [[Dea]]
| [[41edo|41]]
| 41 (prime)
| Proposed by [[User:Tristanbay|Tristan Bay]] (2025) to reflect that a mina is a "minute" (1/60 the width) of a 1\41 "degree".
|-
| [[Méride]]
| [[43edo|43]]
| 43 (prime)
| Proposed by [[Joseph Sauveur]], as 7 heptaméride units<ref name="measure">[http://www.huygens-fokker.org/docs/measures.html Stichting Huygens–Fokker: Logarithmic Interval Measures]</ref><ref>[http://tonalsoft.com/enc/m/meride.aspx Tonalsoft | ''Méride / 43-ed2 / 43-edo / 43-ET / 43-tone equal-temperament'']</ref>.
|-
| [[Holdrian comma]]
| [[53edo|53]]
| 53 (prime)
| See the dedicated page.
|-
| [[Holdrian comma|Mercator's old comma]]
| [[55edo|55]]
| 5 × 11
| Not to be confused with [[Mercator's comma]].
|-
| [[Decitone]]
| [[60edo|60]]
| 2<sup>2</sup> × 3 × 5
| Standard SI prefix + 12edo tone
|-
| [[Morion]]
| [[72edo|72]]
| 2<sup>3</sup> × 3<sup>2</sup>
| See the dedicated page.
|-
| [[Farab]]
| [[144edo|144]]
| 2<sup>4</sup> × 3<sup>2</sup>
| 1/12 of [[12edo]] semitone; proposed by [[al-Farabi]] in 10th century<ref name="measure"/><ref>[http://tonalsoft.com/enc/f/farab.aspx Tonalsoft | ''Farab''].</ref>.
|-
| [[Mem]]
| [[205edo|205]]
| 5 × 41
| Unit used by H-Pi Instruments<ref name="measure"/><ref>[http://musictheory.zentral.zone/huntsystem1.html H-Pi Instruments | Hunt Theoretical System]</ref><ref>[http://tonalsoft.com/enc/m/mem.aspx Tonalsoft | ''Mem, 205-edo'']</ref>.
|-
| [[Tredek]]
| [[270edo|270]]
| 2 × 3<sup>3</sup> × 5
| Proposed by [[Joseph Monzo]] (2013)<ref>[http://tonalsoft.com/enc/t/tredek.aspx Tonalsoft | ''Tredek, 270-edo'']</ref>.
|-
| [[Savart]]*
| [[300edo|300]]
| 2<sup>2</sup> × 3 × 5<sup>2</sup>
| [[Alexander Wood]]'s definition of the Savart<ref>''[https://books.google.com.au/books?id=NWZ8CgAAQBAJ&lpg=PT50&vq=savart&pg=PT51 The Physics of Music]'', Alexander Wood, 1944.</ref>, containing [[12edo]]. 
|-
| [[Heptaméride]] / [[eptaméride]] / [[savart]]*
| [[301edo|301]]
| 7 × 43
| 301 ≃ 1,000 × log<sub>10</sub>2; 1/7 of Méride unit; proposed by Joseph Sauveur (1701), advocated by [[Félix Savart]]<ref name="measure"/><ref>[http://tonalsoft.com/enc/h/heptameride.aspx Tonalsoft | ''Heptaméride'']</ref>.
|-
| [[Gene]]
| [[311edo|311]]
| 311 (prime)
| Proposed by Joseph Monzo (2007)<ref>[http://tonalsoft.com/enc/g/gene.aspx Tonalsoft | ''Gene, 311-edo'']</ref>.
|-
| [[Dröbisch Angle]]
| [[360edo|360]]
| 2<sup>3</sup> × 3<sup>2</sup> × 5
| Proposed as ''angle'' by [[Moritz Dröbisch]] in the 19th century, later by [[Andrew Pikler]] as the current name in ''Logarithmic Frequency Systems'' (1966)<ref name="measure"/>.
|-
| [[Squb]]
| [[494edo|494]]
| 2 × 13 × 19
| Named after [[729/728]], the squbema, due to its similar size.
|-
| [[Great iring]] / [[great centitone|centitone]]
| [[500edo|500]]
| 2<sup>2</sup> × 5<sup>3</sup>
| {{Citation needed}}
|-
| Dexl
| [[540edo|540]]
| 2<sup>2</sup> × 3<sup>3</sup> × 5
| Proposed by Joseph Monzo (2023)<ref>[http://tonalsoft.com/enc/d/dexl.aspx Tonalsoft | ''Dexl, 540-edo'']</ref>.
|-
| [[Iring]] / [[centitone]]
| [[600edo|600]]
| 2<sup>3</sup> × 3 × 5<sup>2</sup>
| [[Relative cent]] of [[6edo]]; proposed by [[Widogast Iring]] (1898), later by [[Joseph Yasser]] as a "centitone", a standard SI prefix + 12edo tone (1932)<ref name="measure"/><ref>[http://www.tonalsoft.com/enc/c/centitone.aspx Tonalsoft | ''Centitone, iring'']</ref>.
|-
| [[Nil]] / [[skisma]] (Sk)
| [[612edo|612]]
| 2<sup>2</sup> × 3<sup>2</sup> × 17
| Proposed by [[James Paul White]] (1894) as ''nil'', and by Gene Ward Smith (2007) as ''skisma''<ref name="measure"/><ref>[http://tonalsoft.com/enc/s/sk.aspx Tonalsoft | ''Sk, 612-edo'']</ref>. Edo representation of [[Sagittal notation|Sagittal]]'s Ultra (Herculean) precision level JI notation (58eda), where it is known as an "ultrina".
|-
| [[Delfi]]
| [[665edo|665]]
| 5 × 7 × 19
| <ref name="measure"/>
|-
| [[Small iring]] / [[small centitone|centitone]]
| [[700edo|700]]
| 2<sup>2</sup> × 5<sup>2</sup> x 7
| {{Citation needed}}
|-
| [[Woolhouse unit]]
| [[730edo|730]]
| 2 × 5 × 73
| Proposed by [[Wesley S.B. Woolhouse]] (1835)<ref>[https://archive.org/details/essayonmusicali00woolgoog/page/n34/mode/2up ''Essay on musical intervals, harmonics, and the temperament of the musical scale, &c''], Wesley S.B. Woolhouse. </ref>.
|-
| [[Millioctave]] (moct)
| [[1000edo|1000]]
| 2<sup>3</sup> × 5<sup>3</sup>
| See the dedicated page.
|-
| [[Cent]] (¢)
| 1200
| 2<sup>4</sup> × 3 × 5<sup>2</sup>
| See the dedicated page.
|-
| Dingle
| [[1395edo|1395]]
| 3<sup>2</sup> × 5 × 31
| Proposed by [[User:Tristanbay|Tristan Bay]] (2026) as a 31edo-friendly fine-grain measure, shortened from "'''di'''esis a'''ngle'''".
|-
| Decifarab
| [[1440edo|1440]]
| 2<sup>5</sup> × 3<sup>2</sup> × 5
| Standard SI prefix + [[farab]]<ref name="measure"/>.
|-
| Heptamu (7mu)
| [[1536edo|1536]]
| 2<sup>9</sup> × 3
| Seventh MIDI-resolution unit, 1/128 (1/(2<sup>7</sup>)) of [[12edo]] semitone<ref>[http://tonalsoft.com/enc/number/7mu.aspx Tonalsoft | ''7mu / heptamu'']</ref>
|-
| śata
| [[1600edo|1600]]
| 2<sup>6</sup> × 5<sup>2</sup>
| From Sanskrit ''śatam'': hundred; [[relative cent]] of Armodue 16edo theory{{Citation needed}}
|-
| [[Iota]]
| [[1700edo|1700]]
| 2<sup>2</sup> × 5<sup>2</sup> × 17
| [[Relative cent]] of [[17edo]]; proposed by [[Margo Schulter]] (2002) and [[George Secor]]<ref name="measure"/>.
|-
| [[Harmos]]
| [[1728edo|1728]]
| 2<sup>6</sup> × 3<sup>3</sup>
| 1728 = 12<sup>3</sup>; 1/144 of [[12edo]] semitone; Proposed by [[Paul Beaver]]<ref name="measure"/><ref name="equal">[http://tonalsoft.com/enc/e/equal-temperament.aspx Tonalsoft | ''Equal temperaments'']</ref>.
|-
| Hind śat / Indian cent
| 2200
| 2<sup>3</sup> × 11 × 5<sup>2</sup>
| {{Citation needed}}
|-
| [[Mina]]
| [[2460edo|2460]]
| 2<sup>2</sup> × 3 × 5 × 41
| Abbreviation of "schismina", edo representation of [[Sagittal notation|Sagittal]]'s Extreme (Olympian) precision level JI notation (233eda)<ref name="measure"/><ref>[http://tonalsoft.com/enc/m/mina.aspx Tonalsoft | ''Mina'']</ref>.
|-
| Centidiesis
| 3100
| 2<sup>2</sup> × 5<sup>2</sup> x 31
| {{Citation needed}}
|-
| Centiméride
| 4300
| 2<sup>2</sup> × 5<sup>2</sup> x 43
| {{Citation needed}}
|-
| [[4320edo|Click]]
| [[4320edo|4320]]
| 2<sup>5</sup> × 3<sup>3</sup> × 5
| Proposed by [[User:Eliora|Eliora]]. See the dedicated page.
|-
| [[Major tina]]
| [[8269edo|8269]]
| 8269 (prime)
| Proposed by [[Flora Canou]] (2021)<ref>[https://forum.sagittal.org/viewtopic.php?f=4&t=515 The Sagittal Forum | ''Definition of the tina reviewed'']</ref>.
|-
| [[Tina]]
| [[8539edo|8539]]
| 8539 (prime)
| Provides good approximations for 41-limit primes except 37; named by [[Dave Keenan]] and [[George Secor]]; edo representation of [[Sagittal notation|Sagittal]]'s Insane (Magrathean) precision level JI notation (809eda)<ref name="measure"/><ref>[http://tonalsoft.com/enc/t/tina.aspx Tonalsoft | ''Tina'']</ref>.
|-
| [[Purdal]]
| [[9900edo|9900]]
| 2<sup>2</sup> × 3<sup>2</sup> × 5<sup>2</sup> × 11
| [[Relative cent]] of [[99edo]]; suggested by [[Osmiorisbendi]], advocated by [[Tútim Dennsuul Wafiil]]. See the dedicated page.
|-
| [[Türk sent]] / [[Turkish cent]]
| [[10600edo|10600]]
| 2<sup>3</sup> × 5<sup>2</sup> × 53
| [[Relative cent]] of [[106edo]], 1/200 of [[53edo]]; invented by [[M. Ekrem Karadeniz]] (1965), influenced by [[Abdülkadir Töre]]<ref name="measure"/><ref>[http://www.tonalsoft.com/enc/t/turk-sent.aspx Tonalsoft | ''Türk-sent'']</ref><ref>[http://www.ozanyarman.com/files/doctorate_thesis.pdf ''79-Tone Tuning & Theory for Turkish Maqam Music''], Ozan Yarman. </ref>.
|-
| [[Prima]]
| [[12276edo|12276]]
| 2<sup>2</sup> × 3<sup>2</sup> × 11 × 31
| Proposed by [[Erv Wilson]], [[Gene Ward Smith]] and [[Gavin Putland]]<ref name="measure"/>.
|-
| [[Jinn]]
| [[16808edo|16808]]
| 2<sup>3</sup> × 11 × 191
| See the dedicated page.
|-
| [[Jot]]
| [[30103edo|30103]]
| 30103 (prime)
| 30103 ≃ 100,000 × log<sub>10</sub>2; proposed by [[Augustus de Morgan]] (1864)<ref name="measure"/><ref>[http://www.tonalsoft.com/enc/j/jot.aspx Tonalsoft | ''Jot'']</ref><ref name="equal"/>.
|-
| [[Imp]]
| [[31920edo|31920]]
| 2<sup>4</sup> × 3 × 5 × 7 × 19
| <ref name="measure"/>
|-
| [[Flu]]
| [[46032edo|46032]]
| 2<sup>4</sup> × 3 × 7 × 137
| Proposed by Gene Ward Smith (2005)<ref name="measure"/><ref>[http://tonalsoft.com/enc/f/flu.aspx Tonalsoft | ''Flu'']</ref>.
|-
| [[Normal atom]]
| [[78005edo|78005]]
| 5 × 15601
| Proposed by Tristan Bay (2023); 78005edo consistently maps Kirnberger's atom to 1 edostep and is a very strong 5-limit system.
|-
| [[MIDI Tuning Standard unit]] (14mu)
| [[196608edo|196608]]
| 2<sup>16</sup> × 3
| Fourteenth MIDI-resolution unit, 1/16384 (1/(2<sup>14</sup>)) of [[12edo]] semitone<ref name="measure"/>.
|}
<nowiki />* More to be added regarding the Heptaméride/Savart units
 
==== Non-octave fine measures ====
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:
 
{| class="wikitable sortable"
|+ style="font-size: 105%;" | List of non-octave fine measures (logarithmic)
|-
! Unit name (symbol)
! Base interval
! Divisions of base interval
! Origin/significance
|-
| Hekt
| 3/1 (twelfth)
| 1300
| 1/100 of 13edt (Bohlen–Pierce) scale step
|-
| Euhekt
| 3/1 (twelfth)
| 3900
| 1/100 of 39edt (Triboh) scale step
|-
| Grad
| [[Pythagorean comma|531441/524288]] (Pythagorean comma)
| 12
| [[12edo]] flattens [[3/2]] by this amount
|-
| Tuning unit
| [[531441/524288]] (Pythagorean comma)
| 720
|
|}
 
To ''convert hekts'', which is quite common in EDT systems, ''into cents'', use following formula: <code> c = h*12/13*math.log(3)/math.log(2) </code>
 
=== Relative measures ===
Within a given [[equal-step tuning|equal-stepped tuning system]], the [[relative cent]] (rct, r¢) can be used to describe properties of pitches (for instance the approximation of [[Just intonation|JI]] intervals). It is defined as on 100th (or 1 percent) of the interval between two neighbouring pitches in the used equal tuning.
 
== Ratio ==
Intervals can be measured also giving their [[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 {{nowrap|3/2 × 5/4 {{=}} [[15/8]]}},
 
which is a diatonic semitone below an octave {{nowrap|([[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 {{monzo| -4 4 -1 }} (for the syntonic comma, {{nowrap|81/80 = 2<sup>−4</sup> × 3<sup>4</sup> × 5<sup>−1</sup>}}), which builds a bridge back to the logarithmic measure: intervals can be combined by component-wise addition or subtraction of their vectors.
 
== See also ==
* [[Interval span]]
 
== Articles ==
* [http://arxiv.org/abs/0907.5249 ''Why the Kirnberger Kernel Is So Small''] by [[Don N. Page]]
 
== References ==
<references />
 
[[Category:Interval]]

Latest revision as of 12:00, 4 April 2026

Interval size measure or interval size unit means the distance between pitches. Intervals can be measured logarithmic or by frequency ratios.

Logarithmic

All logarithmic measures can be combined by adding and subtracting them.

Backslash notation

A common shorthand in use in the microtonal community is k\N, written with a backslash (\) instead of a forwardslash (/), to refer to an interval with a frequency ratio of 2k/N. k\N is pronounced "k steps of N edo", and can be derived from the meaning of "steps" in the context of edos (unless talking about steps of specific subsets/scales of some edo).

Steps are linear in the log-frequency domain, so expressions like 11\19 − 6\19 = 5\19 hold. In general, we have

a\N + b\N = (a + b)\N

which expresses the same thing as 2a/N × 2b/N = 2(a + b)/N.

Or equivalently, for subtraction/division:

a\N − b\N = (a − b)\N

which expresses the same thing as 2a/N / 2b/N = 2(a - b)/N.

Backslash notation can be extended to support nonoctave equal tunings by writing the tuning in full after the backslash. For example, 11\13edt means 11 steps of 13edt, 14\9edf means 14 steps of 9edf, and 7\12ed12/5 means 7 steps of 12ed12/5.

Gross

The octave and the decade are common coarse units for interval sizes. The decibel, being a relative logarithmic-scale unit for power or root-power quantities, is inappropriate for measuring intervals; the decade is used instead. Similarly, the neper (Np) and the dineper (dNp), like the decibel, should not be used. However, in the absence of a substitute, dinepers have an application in logarithmic approximants.

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. An early unit for measuring intervals is the "tone" which dates back to classic Greece.

In serial music, all intervals were measured by the number of 12edo semitones. In analogy, the relative interval measure is the number of steps between two pitches of an equal tuning, sometimes called "degrees". These measures can be written using backslash notation if the degree itself isn't sufficiently clear in context.

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.

Octave-based fine measures

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 16edo theory[1].
Normal shruti 22 2 × 11 Proposed by Tristan Bay (2025) in reference to the Indian tradition of dividing the octave into 22 unequal parts.
Normal diesis 31 31 (prime) See the dedicated page.
Dea 41 41 (prime) Proposed by Tristan Bay (2025) to reflect that a mina is a "minute" (1/60 the width) of a 1\41 "degree".
Méride 43 43 (prime) Proposed by Joseph Sauveur, as 7 heptaméride units[2][3].
Holdrian comma 53 53 (prime) See the dedicated page.
Mercator's old comma 55 5 × 11 Not to be confused with Mercator's comma.
Decitone 60 22 × 3 × 5 Standard SI prefix + 12edo tone
Morion 72 23 × 32 See the dedicated page.
Farab 144 24 × 32 1/12 of 12edo semitone; proposed by al-Farabi in 10th century[2][4].
Mem 205 5 × 41 Unit used by H-Pi Instruments[2][5][6].
Tredek 270 2 × 33 × 5 Proposed by Joseph Monzo (2013)[7].
Savart* 300 22 × 3 × 52 Alexander Wood's definition of the Savart[8], containing 12edo.
Heptaméride / eptaméride / savart* 301 7 × 43 301 ≃ 1,000 × log102; 1/7 of Méride unit; proposed by Joseph Sauveur (1701), advocated by Félix Savart[2][9].
Gene 311 311 (prime) Proposed by Joseph Monzo (2007)[10].
Dröbisch Angle 360 23 × 32 × 5 Proposed as angle by Moritz Dröbisch in the 19th century, later by Andrew Pikler as the current name in Logarithmic Frequency Systems (1966)[2].
Squb 494 2 × 13 × 19 Named after 729/728, the squbema, due to its similar size.
Great iring / centitone 500 22 × 53 [citation needed]
Dexl 540 22 × 33 × 5 Proposed by Joseph Monzo (2023)[11].
Iring / centitone 600 23 × 3 × 52 Relative cent of 6edo; proposed by Widogast Iring (1898), later by Joseph Yasser as a "centitone", a standard SI prefix + 12edo tone (1932)[2][12].
Nil / skisma (Sk) 612 22 × 32 × 17 Proposed by James Paul White (1894) as nil, and by Gene Ward Smith (2007) as skisma[2][13]. Edo representation of Sagittal's Ultra (Herculean) precision level JI notation (58eda), where it is known as an "ultrina".
Delfi 665 5 × 7 × 19 [2]
Small iring / centitone 700 22 × 52 x 7 [citation needed]
Woolhouse unit 730 2 × 5 × 73 Proposed by Wesley S.B. Woolhouse (1835)[14].
Millioctave (moct) 1000 23 × 53 See the dedicated page.
Cent (¢) 1200 24 × 3 × 52 See the dedicated page.
Dingle 1395 32 × 5 × 31 Proposed by Tristan Bay (2026) as a 31edo-friendly fine-grain measure, shortened from "diesis angle".
Decifarab 1440 25 × 32 × 5 Standard SI prefix + farab[2].
Heptamu (7mu) 1536 29 × 3 Seventh MIDI-resolution unit, 1/128 (1/(27)) of 12edo semitone[15]
śata 1600 26 × 52 From Sanskrit śatam: hundred; relative cent of Armodue 16edo theory[citation needed]
Iota 1700 22 × 52 × 17 Relative cent of 17edo; proposed by Margo Schulter (2002) and George Secor[2].
Harmos 1728 26 × 33 1728 = 123; 1/144 of 12edo semitone; Proposed by Paul Beaver[2][16].
Hind śat / Indian cent 2200 23 × 11 × 52 [citation needed]
Mina 2460 22 × 3 × 5 × 41 Abbreviation of "schismina", edo representation of Sagittal's Extreme (Olympian) precision level JI notation (233eda)[2][17].
Centidiesis 3100 22 × 52 x 31 [citation needed]
Centiméride 4300 22 × 52 x 43 [citation needed]
Click 4320 25 × 33 × 5 Proposed by Eliora. See the dedicated page.
Major tina 8269 8269 (prime) Proposed by Flora Canou (2021)[18].
Tina 8539 8539 (prime) Provides good approximations for 41-limit primes except 37; named by Dave Keenan and George Secor; edo representation of Sagittal's Insane (Magrathean) precision level JI notation (809eda)[2][19].
Purdal 9900 22 × 32 × 52 × 11 Relative cent of 99edo; suggested by Osmiorisbendi, advocated by Tútim Dennsuul Wafiil. See the dedicated page.
Türk sent / Turkish cent 10600 23 × 52 × 53 Relative cent of 106edo, 1/200 of 53edo; invented by M. Ekrem Karadeniz (1965), influenced by Abdülkadir Töre[2][20][21].
Prima 12276 22 × 32 × 11 × 31 Proposed by Erv Wilson, Gene Ward Smith and Gavin Putland[2].
Jinn 16808 23 × 11 × 191 See the dedicated page.
Jot 30103 30103 (prime) 30103 ≃ 100,000 × log102; proposed by Augustus de Morgan (1864)[2][22][16].
Imp 31920 24 × 3 × 5 × 7 × 19 [2]
Flu 46032 24 × 3 × 7 × 137 Proposed by Gene Ward Smith (2005)[2][23].
Normal atom 78005 5 × 15601 Proposed by Tristan Bay (2023); 78005edo consistently maps Kirnberger's atom to 1 edostep and is a very strong 5-limit system.
MIDI Tuning Standard unit (14mu) 196608 216 × 3 Fourteenth MIDI-resolution unit, 1/16384 (1/(214)) of 12edo semitone[2].

* More to be added regarding the Heptaméride/Savart units

Non-octave fine measures

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) Base interval Divisions of base interval Origin/significance
Hekt 3/1 (twelfth) 1300 1/100 of 13edt (Bohlen–Pierce) scale step
Euhekt 3/1 (twelfth) 3900 1/100 of 39edt (Triboh) scale step
Grad 531441/524288 (Pythagorean comma) 12 12edo flattens 3/2 by this amount
Tuning unit 531441/524288 (Pythagorean comma) 720

To convert hekts, which is quite common in EDT systems, into cents, use following formula: c = h*12/13*math.log(3)/math.log(2)

Relative measures

Within a given equal-stepped tuning 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.

Ratio

Intervals can be measured also giving their 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, 2−4 × 34 × 5−1), which builds a bridge back to the logarithmic measure: intervals can be combined by component-wise addition or subtraction of their vectors.

See also

Articles

References

  1. ↑ Armodue: le risorse di un nuovo sistema musicale
  2. ↑ 2.00 2.01 2.02 2.03 2.04 2.05 2.06 2.07 2.08 2.09 2.10 2.11 2.12 2.13 2.14 2.15 2.16 2.17 2.18 Stichting Huygens–Fokker: Logarithmic Interval Measures
  3. ↑ Tonalsoft | Méride / 43-ed2 / 43-edo / 43-ET / 43-tone equal-temperament
  4. ↑ Tonalsoft | Farab.
  5. ↑ H-Pi Instruments | Hunt Theoretical System
  6. ↑ Tonalsoft | Mem, 205-edo
  7. ↑ Tonalsoft | Tredek, 270-edo
  8. ↑ The Physics of Music, Alexander Wood, 1944.
  9. ↑ Tonalsoft | Heptaméride
  10. ↑ Tonalsoft | Gene, 311-edo
  11. ↑ Tonalsoft | Dexl, 540-edo
  12. ↑ Tonalsoft | Centitone, iring
  13. ↑ Tonalsoft | Sk, 612-edo
  14. ↑ Essay on musical intervals, harmonics, and the temperament of the musical scale, &c, Wesley S.B. Woolhouse.
  15. ↑ Tonalsoft | 7mu / heptamu
  16. ↑ 16.0 16.1 Tonalsoft | Equal temperaments
  17. ↑ Tonalsoft | Mina
  18. ↑ The Sagittal Forum | Definition of the tina reviewed
  19. ↑ Tonalsoft | Tina
  20. ↑ Tonalsoft | Türk-sent
  21. ↑ 79-Tone Tuning & Theory for Turkish Maqam Music, Ozan Yarman.
  22. ↑ Tonalsoft | Jot
  23. ↑ Tonalsoft | Flu
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