Interval size measure: Difference between revisions

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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 [[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.

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.

=== Fine ===

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| [[16edo|16]]

| 24

| 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>.

| 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]] in reference to the Indian tradition of dividing the octave into 22 unequal parts.

| Proposed by [[User:Tristanbay|Tristan Bay]] (2025) in reference to the Indian tradition of dividing the octave into 22 unequal parts.

|-

| [[Normal diesis]]

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| 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]]

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| See the dedicated page.

|-

| [[Holdrian comma|~~Mercator’s~~ old comma]]

| [[Holdrian comma|Mercator's old comma]]

| [[55edo|55]]

| 5 × 11

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| [[60edo|60]]

| 22 × 3 × 5

|

| Standard SI prefix + 12edo tone

|-

| [[Morion]]

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| [[144edo|144]]

| 24 × 32

| 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>.

| 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]]

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| [[494edo|494]]

| 2 × 13 × 19

| ~~{{Citation needed}}~~

| Named after [[729/728]], the squbema, due to its similar size.

|-

| ~~Great~~ [[iring]] / [[centitone]]

| [[Great iring]] / [[great centitone|centitone]]

| [[500edo|500]]

| 22 × 53

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| [[600edo|600]]

| 23 × 3 × 52

| [[Relative cent]] of [[6edo]] ~~([[12edo]] tone)~~; ~~Proposed~~ by [[Widogast Iring]] (1898), later by [[Joseph Yasser]] as a "centitone" (1932)<ref name="measure"/><ref>[http://www.tonalsoft.com/enc/c/centitone.aspx Tonalsoft | ''Centitone, iring'']</ref>.

| [[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>.

|-

| [[~~Skisma~~]] (Sk)

| [[Nil]] / [[skisma]] (Sk)

| [[612edo|612]]

| 22 × 32 × 17

| ~~Edo representation of~~ [[~~Sagittal notation|Sagittal~~]]~~'s Ultra~~ (~~Herculean~~) ~~precision level JI notation~~ (~~58eda~~)~~, where it is known~~ as ~~an "ultrina"~~<ref name="measure"/><ref>[http://tonalsoft.com/enc/s/sk.aspx Tonalsoft | ''Sk, 612-edo'']</ref>.

| 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]]

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| <ref name="measure"/>

|-

| ~~Small~~ [[iring]] / [[centitone]]

| [[Small iring]] / [[small centitone|centitone]]

| [[700edo|700]]

| 22 × 52 x 7

| {{Citation needed}}

|-

| [[Woolhouse]]

| [[Woolhouse unit]]

| [[730edo|730]]

| 2 × 5 × 73

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| See the dedicated page.

|-

| ~~Greater muon~~

| Dingle

| [[~~1224edo~~|~~1224~~]]

| [[1395edo|1395]]

| ~~23 × 32 × 17~~

| 32 × 5 × 31

~~| {{Citation needed}}~~

| Proposed by [[User:Tristanbay|Tristan Bay]] (2026) as a 31edo-friendly fine-grain measure, shortened from "'''di'''esis a'''ngle'''".

|-

~~| Triangular cent~~

~~| [[1260edo|1260]]~~

~~| 22 ×~~ 32 × 5 × 7

~~| {{Citation needed}}~~

|-

~~| Pion~~

~~| [[1272edo|1272]]~~

~~| 23 × 3 × 53~~

~~| {{Citation needed}}~~

|-

~~| Pound~~

| [[~~1344edo~~|~~1344~~]]

~~| 26 × 3 × 7~~

~~| {{Citation needed}}~~

|-

~~| Neutron~~

~~| [[1392edo|1392]]~~

~~| 24 × 3 × 29~~

~~| {{Citation needed}}~~

|-

~~| Lesser muon~~

~~| [[1428edo|1428]]~~

~~| 22 × 3 × 7 × 17~~

~~| {{Citation needed}}~~

|-

| Decifarab

| [[1440edo|1440]]

| 25 × 32 × 5

| ~~1/10 of~~ [[~~Farab~~]] ~~unit~~<ref name="measure"/>.

| Standard SI prefix + [[farab]]<ref name="measure"/>.

|-

~~| Quadratic cent~~

~~| [[1452edo|1452]]~~

~~| 22 × 3 × 112~~

~~| {{Citation needed}}~~

|-

~~| Ksion~~

~~| [[1476edo|1476]]~~

~~| 22 × 32 × 41~~

~~| {{Citation needed}}~~

|-

~~| Cubic cent~~

~~| [[1500edo|1500]]~~

~~| 22 × 3 × 53~~

~~| {{Citation needed}}~~

|-

| Heptamu (7mu)

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| 29 × 3

| Seventh MIDI-resolution unit, 1/128 (1/(27)) of [[12edo]] semitone<ref>[http://tonalsoft.com/enc/number/7mu.aspx Tonalsoft | ''7mu / heptamu'']</ref>

|-

~~| Rhoon~~

~~| [[1560edo|1560]]~~

~~| 23 × 3 × 5 × 13~~

~~| {{Citation needed}}~~

|-

| śata

| [[1600edo|1600]]

| 26 × 52

| From Sanskrit ''śatam'': hundred; [[~~Relative~~ cent]] of Armodue 16edo ~~Theory{{Citation needed}}~~

| From Sanskrit ''śatam'': hundred; [[relative cent]] of Armodue 16edo theory{{Citation needed}}

|-

~~| Tile~~

~~| [[1632edo|1632]]~~

~~| 25 × 3 × 17~~

| {{Citation needed}}

|-

| [[Iota]]

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| 22 × 52 x 43

| {{Citation needed}}

|-

| [[4320edo|Click]]

| [[4320edo|4320]]

| 25 × 33 × 5

| Proposed by [[User:Eliora|Eliora]]. See the dedicated page.

|-

| [[Major tina]]

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| [[9900edo|9900]]

| 22 × 32 × 52 × 11

| [[Relative cent]] of [[99edo]]; ~~Suggested~~ by [[Osmiorisbendi]], advocated by [[Tútim Dennsuul Wafiil]]. See the dedicated page.

| [[Relative cent]] of [[99edo]]; suggested by [[Osmiorisbendi]], advocated by [[Tútim Dennsuul Wafiil]]. See the dedicated page.

|-

| [[Türk sent]] / [[Turkish cent]]

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| [[30103edo|30103]]

| 30103 (prime)

| 30103 ≃ 100,000 × log102; ~~Proposed~~ by [[Augustus de Morgan]] (1864)<ref name="measure"/><ref>[http://www.tonalsoft.com/enc/j/jot.aspx Tonalsoft | ''Jot'']</ref><ref name="equal"/>.

| 30103 ≃ 100,000 × log102; 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]]

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| [[78005edo|78005]]

| 5 × 15601

| ~~Name proposed~~ by Tristan Bay in 2023; 78005edo consistently maps Kirnberger's atom to 1 edostep and is a very strong 5-limit system. ~~{{Citation needed}}~~

| 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)

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| Euhekt

| 3/1 (twelfth)

| ~~1900~~

| 3900

| 1/100 of ~~19edt~~ (~~OnlyPure~~) scale step

| 1/100 of 39edt (Triboh) scale step

|-

| Grad

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=== Relative measures ===

Within a given [[~~Equal~~-step tuning|equal-stepped]] ~~tonal 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.

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 ==

@@ Line 25: / Line 25: @@
 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 [[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.
+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.
 === Fine ===
@@ Line 44: / Line 44: @@
 | [[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>.
+| 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]] in reference to the Indian tradition of dividing the octave into 22 unequal parts.
+| Proposed by [[User:Tristanbay|Tristan Bay]] (2025) in reference to the Indian tradition of dividing the octave into 22 unequal parts.
 |-
 | [[Normal diesis]]
@@ Line 55: / Line 55: @@
 | 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]]
@@ Line 66: / Line 71: @@
 | See the dedicated page.
 |-
-| [[Holdrian comma|Mercator’s old comma]]
+| [[Holdrian comma|Mercator's old comma]]
 | [[55edo|55]]
 | 5 × 11
@@ Line 74: / Line 79: @@
 | [[60edo|60]]
 | 2<sup>2</sup> × 3 × 5
-|
+| Standard SI prefix + 12edo tone
 |-
 | [[Morion]]
@@ Line 84: / Line 89: @@
 | [[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>.
+| 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]]
@@ Line 119: / Line 124: @@
 | [[494edo|494]]
 | 2 × 13 × 19
-| {{Citation needed}}
+| Named after [[729/728]], the squbema, due to its similar size.
 |-
-| Great [[iring]] / [[centitone]]
+| [[Great iring]] / [[great centitone|centitone]]
 | [[500edo|500]]
 | 2<sup>2</sup> × 5<sup>3</sup>
@@ Line 134: / Line 139: @@
 | [[600edo|600]]
 | 2<sup>3</sup> × 3 × 5<sup>2</sup>
-| [[Relative cent]] of [[6edo]] ([[12edo]] tone); Proposed by [[Widogast Iring]] (1898), later by [[Joseph Yasser]] as a "centitone" (1932)<ref name="measure"/><ref>[http://www.tonalsoft.com/enc/c/centitone.aspx Tonalsoft | ''Centitone, iring'']</ref>.
+| [[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>.
 |-
-| [[Skisma]] (Sk)
+| [[Nil]] / [[skisma]] (Sk)
 | [[612edo|612]]
 | 2<sup>2</sup> × 3<sup>2</sup> × 17
-| Edo representation of [[Sagittal notation|Sagittal]]'s Ultra (Herculean) precision level JI notation (58eda), where it is known as an "ultrina"<ref name="measure"/><ref>[http://tonalsoft.com/enc/s/sk.aspx Tonalsoft | ''Sk, 612-edo'']</ref>.
+| 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]]
@@ Line 146: / Line 151: @@
 | <ref name="measure"/>
 |-
-| Small [[iring]] / [[centitone]]
+| [[Small iring]] / [[small centitone|centitone]]
 | [[700edo|700]]
 | 2<sup>2</sup> × 5<sup>2</sup> x 7
 | {{Citation needed}}
 |-
-| [[Woolhouse]]
+| [[Woolhouse unit]]
 | [[730edo|730]]
 | 2 × 5 × 73
@@ Line 166: / Line 171: @@
 | See the dedicated page.
 |-
-| Greater muon
+| Dingle
-| [[1224edo|1224]]
+| [[1395edo|1395]]
-| 2<sup>3</sup> × 3<sup>2</sup> × 17
+| 3<sup>2</sup> × 5 × 31
-| {{Citation needed}}
+| Proposed by [[User:Tristanbay|Tristan Bay]] (2026) as a 31edo-friendly fine-grain measure, shortened from "'''di'''esis a'''ngle'''".
-|-
-| Triangular cent
-| [[1260edo|1260]]
-| 2<sup>2</sup> × 3<sup>2</sup> × 5 × 7
-| {{Citation needed}}
-|-
-| Pion
-| [[1272edo|1272]]
-| 2<sup>3</sup> × 3 × 53
-| {{Citation needed}}
-|-
-| Pound
-| [[1344edo|1344]]
-| 2<sup>6</sup> × 3 × 7
-| {{Citation needed}}
-|-
-| Neutron
-| [[1392edo|1392]]
-| 2<sup>4</sup> × 3 × 29
-| {{Citation needed}}
-|-
-| Lesser muon
-| [[1428edo|1428]]
-| 2<sup>2</sup> × 3 × 7 × 17
-| {{Citation needed}}
 |-
 | Decifarab
 | [[1440edo|1440]]
 | 2<sup>5</sup> × 3<sup>2</sup> × 5
-| 1/10 of [[Farab]] unit<ref name="measure"/>.
+| Standard SI prefix + [[farab]]<ref name="measure"/>.
-|-
-| Quadratic cent
-| [[1452edo|1452]]
-| 2<sup>2</sup> × 3 × 11<sup>2</sup>
-| {{Citation needed}}
-|-
-| Ksion
-| [[1476edo|1476]]
-| 2<sup>2</sup> × 3<sup>2</sup> × 41
-| {{Citation needed}}
-|-
-| Cubic cent
-| [[1500edo|1500]]
-| 2<sup>2</sup> × 3 × 5<sup>3</sup>
-| {{Citation needed}}
 |-
 | Heptamu (7mu)
@@ Line 220: / Line 185: @@
 | 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>
-|-
-| Rhoon
-| [[1560edo|1560]]
-| 2<sup>3</sup> × 3 × 5 × 13
-| {{Citation needed}}
 |-
 | śata
 | [[1600edo|1600]]
 | 2<sup>6</sup> × 5<sup>2</sup>
-| From Sanskrit ''śatam'': hundred; [[Relative cent]] of Armodue 16edo Theory{{Citation needed}}
+| From Sanskrit ''śatam'': hundred; [[relative cent]] of Armodue 16edo theory{{Citation needed}}
-|-
-| Tile
-| [[1632edo|1632]]
-| 2<sup>5</sup> × 3 × 17
-| {{Citation needed}}
 |-
 | [[Iota]]
@@ Line 265: / Line 220: @@
 | 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]]
@@ Line 279: / Line 239: @@
 | [[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.
+| [[Relative cent]] of [[99edo]]; suggested by [[Osmiorisbendi]], advocated by [[Tútim Dennsuul Wafiil]]. See the dedicated page.
 |-
 | [[Türk sent]] / [[Turkish cent]]
@@ Line 299: / Line 259: @@
 | [[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"/>.
+| 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]]
@@ Line 314: / Line 274: @@
 | [[78005edo|78005]]
 | 5 × 15601
-| Name proposed by Tristan Bay in 2023; 78005edo consistently maps Kirnberger's atom to 1 edostep and is a very strong 5-limit system. {{Citation needed}}
+| 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)
@@ Line 341: / Line 301: @@
 | Euhekt
 | 3/1 (twelfth)
-| 1900
+| 3900
-| 1/100 of 19edt (OnlyPure) scale step
+| 1/100 of 39edt (Triboh) scale step
 |-
 | Grad
@@ Line 358: / Line 318: @@
 === Relative measures ===
-Within a given [[Equal-step tuning|equal-stepped]] tonal 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.
+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 ==