Interval size measure: Difference between revisions

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

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=== 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.

==== Octave-based fine measures ====

The following table demonstrates a list of measures derived from the logarithmic division of the octave<ref>[http://www.huygens-fokker.org/docs/measures.html Stichting Huygens-Fokker: Logarithmic Interval Measures]</ref>:

~~The following table demonstrates a list of measures derived from the logarithmic division of the octave:~~

{| class="wikitable sortable"

|+ List of Octave-Based Fine Measures (Logarithmic)

|-

! Unit ~~name~~ (~~symbol~~):

! Unit Name (Symbol):

! Divisions of Octave

! Prime Factors

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| [[16edo|16]]

| 24

| From Sanskrit ''eka'': one, unit; chromatic unit of Armodue ~~16ED2~~ Theory

| From Sanskrit ''eka'': one, unit; chromatic unit of Armodue 16edo Theory

|-

| [[Normal diesis]]

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| [[144edo|144]]

| 24 × 32

| 1/12 of [[~~12ED2~~]] semitone; Proposed by al-Farabi in 10th century ([http://www.huygens-fokker.org/docs/measures.html source])

| 1/12 of [[12edo]] semitone; Proposed by al-Farabi in 10th century ([http://www.huygens-fokker.org/docs/measures.html source])

|-

| [[Mem]]

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| [[300edo|300]]

| 22 × 3 × 52

| Alexander Wood's definition of the Savart (''[https://books.google.com.au/books?id=NWZ8CgAAQBAJ&lpg=PT50&vq=savart&pg=PT51 The Physics of Music]'', 1944), compatible with [[~~12ED2~~]] system

| Alexander Wood's definition of the Savart (''[https://books.google.com.au/books?id=NWZ8CgAAQBAJ&lpg=PT50&vq=savart&pg=PT51 The Physics of Music]'', 1944), compatible with [[12edo]] system

|-

| [[Heptaméride]]/[[Eptaméride]]/[[Savart]]*

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|

|-

| [[~~Dröbisch_Angle|~~Dröbisch Angle]]

| [[Dröbisch Angle]]

| [[360edo|360]]

| 23 × 32 × 5

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|

|-

|Great [[Iring]]/[[Centitone]]

| Great [[Iring]]/[[Centitone]]

|[[500edo|500]]

| [[500edo|500]]

|22 × 53

| 22 × 53

|

|-

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| [[600edo|600]]

| 23 × 3 × 52

| [[Relative cent]] of [[~~6ED2~~]] ([[~~12ED2~~]] tone); Proposed by Widogast Iring (1898), later by Joseph Yasser as a "centitone" (1932). ([http://www.tonalsoft.com/enc/c/centitone.aspx source])

| [[Relative cent]] of [[6edo]] ([[12edo]] tone); Proposed by Widogast Iring (1898), later by Joseph Yasser as a "centitone" (1932). ([http://www.tonalsoft.com/enc/c/centitone.aspx source])

|-

| [[Skisma]]

| [[612edo|612]]

| 22 × 32 × 17

| ~~ED2~~ representation of [[Sagittal notation|Sagittal]]'s Ultra (Herculean) precision level JI notation (~~58-EDA~~), where it is known as an "ultrina"

| 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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|

|-

|Small [[Iring]]/[[Centitone]]

| Small [[Iring]]/[[Centitone]]

|[[700edo|700]]

| [[700edo|700]]

|22 × 52 x 7

| 22 × 52 x 7

|

|-

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| [[1000edo|1000]]

| 23 × 53

| [~~https~~:~~//en.wikipedia.org/wiki/Metric_prefix~~ SI-prefix] division of octave

| [[Wikipedia: Metric prefix|SI-prefix]] division of octave

|-

| [[cent]] (¢)

| 1200

| 24 × 3 × 52

| 1/100 of [[~~12ED2~~]] semitone

| 1/100 of [[12edo]] semitone

|-

| greater muon

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|

|-

|śata

| śata

|[[1600edo|1600]]

| [[1600edo|1600]]

|26 × 52

| 26 × 52

|From Sanskrit ''śatam'': hundred; [[Relative cent]] of Armodue ~~16ED2~~ Theory

| From Sanskrit ''śatam'': hundred; [[Relative cent]] of Armodue 16edo Theory

|-

| tile

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| [[1\1700_octave|1700]]

| 22 × 52 × 17

| [[Relative cent]] of [[~~17ED2~~]]; proposed by [[Margo Schulter]] ([http://www.huygens-fokker.org/docs/measures.html source]) and [[George Secor]] ([[~~Relative_cent~~|source]])

| [[Relative cent]] of [[17edo]]; proposed by [[Margo Schulter]] ([http://www.huygens-fokker.org/docs/measures.html source]) and [[George Secor]] ([[Relative cent|source]])

|-

| [[Harmos]]

| [[1728edo|1728]]

| 26 × 33

| 1728 = 123; 1/144 of [[~~12ED2~~]] semitone; Proposed by Paul Beaver ([http://www.tonalsoft.com/enc/e/equal-temperament.aspx source])

| 1728 = 123; 1/144 of [[12edo]] semitone; Proposed by Paul Beaver ([http://www.tonalsoft.com/enc/e/equal-temperament.aspx source])

|-

|Hind śat / Indian cent

| Hind śat / Indian cent

|2200

| 2200

|23 × 11 × 52

| 23 × 11 × 52

|

|-

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| [[2460edo|2460]]

| 22 × 3 × 5 × 41

| Abbreviation of "schismina", ~~ED2~~ representation of [[Sagittal notation|Sagittal]]'s Extreme (Olympian) precision level JI notation (~~233-EDA~~)

| Abbreviation of "schismina", edo representation of [[Sagittal notation|Sagittal]]'s Extreme (Olympian) precision level JI notation (233eda)

|-

|Centidiesis

| Centidiesis

|3100

| 3100

|22 × 52 x 31

| 22 × 52 x 31

|

|-

|Centiméride

| Centiméride

|4300

| 4300

|22 × 52 x 43

| 22 × 52 x 43

|

|-

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| [[8539edo|8539]]

| PRIME

| Provides good approximations for 41-limit primes except 37 ([http://www.tonalsoft.com/enc/t/tina.aspx source]); named by [[Dave Keenan]] and [[George Secor]]; ~~ED2~~ representation of [[Sagittal notation|Sagittal]]'s Insane (Magrathean) precision level JI notation (~~809-EDA~~)

| Provides good approximations for 41-limit primes except 37 ([http://www.tonalsoft.com/enc/t/tina.aspx source]); named by [[Dave Keenan]] and [[George Secor]]; edo representation of [[Sagittal notation|Sagittal]]'s Insane (Magrathean) precision level JI notation (809eda)

|-

| [[Purdal]]

| [[9900edo|9900]]

| 22 × 32 × 52 × 11

| [[Relative cent]] of [[99edo~~|99ED2~~]]; Suggested by [[Osmiorisbendi]], advocated by [[Tútim Dennsuul Wafiil]]

| [[Relative cent]] of [[99edo]]; Suggested by [[Osmiorisbendi]], advocated by [[Tútim Dennsuul Wafiil]]

|-

| [[Türk sent]] / [[Turkish cent]]

| [[10600edo|10600]]

| 23 × 52 × 53

| [[Relative cent]] of [[~~106ED2~~]], 1/200 of [[~~53ED2~~]]; invented by [http://www.tonalsoft.com/enc/t/turk-sent.aspx M. Ekrem Karadeniz] (1965), influenced by [https://core.ac.uk/download/pdf/76124322.pdf Abdülkadir Töre]

| [[Relative cent]] of [[106edo]], 1/200 of [[53edo]]; invented by [http://www.tonalsoft.com/enc/t/turk-sent.aspx M. Ekrem Karadeniz] (1965), influenced by [https://core.ac.uk/download/pdf/76124322.pdf Abdülkadir Töre]

|-

| [[Prima]]

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|

|-

| [[~~MIDI_Tuning_Standard_unit|~~MIDI Tuning Standard unit]]

| [[MIDI Tuning Standard unit]]

| [[196608edo|196608]]

| 216 × 3

| 14mu (14th MIDI unit), fourteenth MIDI-resolution unit, 1/16384 (1/(214)) of [[12edo|12ED2]] semitone

|-

|[[Mean free path]]

| [[Mean free path]]

|~216,608,494

| ~216,608,494

|2×41×2641567

| 2×41×2641567

|

|}

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==== 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:

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|+ List of Non-Octave Fine Measures (Logarithmic)

|-

! Unit ~~name~~ (~~symbol~~):

! Unit Name (Symbol):

! Base ~~interval~~:

! Base Interval:

! Parts of ~~base interval~~:

! Parts of Base Interval:

! Origin/Significance

|-

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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>

~~See [http://www.huygens-fokker.org/docs/measures.html Logarithmic Interval Measures]~~

=== 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~~ 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.

see also: Kirnberger Atom http://arxiv.org/abs/0907.5249

== Ratio ==

Intervals can be measured also giving their [~~http~~:~~//en.wikipedia.org/wiki/Interval_ratio~~ (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:

Intervals can be measured also giving their [[Wikipedia: Interval ratio (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]],

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]].

~~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 {{monzo| -4 4 -1 }} (for the syntonic comma, 81/80 = 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.

~~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, 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.~~

== Notes ==

[[Category:Interval size]]

@@ Line 2: / Line 2: @@
 == 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. An early unit for measuring intervals is the "[[tone]]" which dates back to classic Greece.
@@ Line 12: / Line 10: @@
 === 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.
 ==== Octave-based fine measures ====
+The following table demonstrates a list of measures derived from the logarithmic division of the octave<ref>[http://www.huygens-fokker.org/docs/measures.html Stichting Huygens-Fokker: Logarithmic Interval Measures]</ref>:
-The following table demonstrates a list of measures derived from the logarithmic division of the octave:
 {| class="wikitable sortable"
 |+ List of Octave-Based Fine Measures (Logarithmic)
 |-
-! Unit name (symbol):
+! Unit Name (Symbol):
 ! Divisions of Octave
 ! Prime Factors
@@ Line 29: / Line 26: @@
 | [[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 16edo Theory
 |-
 | [[Normal diesis]]
@@ Line 54: / Line 51: @@
 | [[144edo|144]]
 | 2<sup>4</sup> × 3<sup>2</sup>
-| 1/12 of [[12ED2]] semitone; Proposed by al-Farabi in 10th century ([http://www.huygens-fokker.org/docs/measures.html source])
+| 1/12 of [[12edo]] semitone; Proposed by al-Farabi in 10th century ([http://www.huygens-fokker.org/docs/measures.html source])
 |-
 | [[Mem]]
@@ Line 69: / Line 66: @@
 | [[300edo|300]]
 | 2<sup>2</sup> × 3 × 5<sup>2</sup>
-| Alexander Wood's definition of the Savart (''[https://books.google.com.au/books?id=NWZ8CgAAQBAJ&lpg=PT50&vq=savart&pg=PT51 The Physics of Music]'', 1944), compatible with [[12ED2]] system
+| Alexander Wood's definition of the Savart (''[https://books.google.com.au/books?id=NWZ8CgAAQBAJ&lpg=PT50&vq=savart&pg=PT51 The Physics of Music]'', 1944), compatible with [[12edo]] system
 |-
 | [[Heptaméride]]/[[Eptaméride]]/[[Savart]]*
@@ Line 81: / Line 78: @@
 |
 |-
-| [[Dröbisch_Angle|Dröbisch Angle]]
+| [[Dröbisch Angle]]
 | [[360edo|360]]
 | 2<sup>3</sup> × 3<sup>2</sup> × 5
@@ Line 91: / Line 88: @@
 |
 |-
-|Great [[Iring]]/[[Centitone]]
+| Great [[Iring]]/[[Centitone]]
-|[[500edo|500]]
+| [[500edo|500]]
-|2<sup>2</sup> × 5<sup>3</sup>
+| 2<sup>2</sup> × 5<sup>3</sup>
 |
 |-
@@ Line 99: / Line 96: @@
 | [[600edo|600]]
 | 2<sup>3</sup> × 3 × 5<sup>2</sup>
-| [[Relative cent]] of [[6ED2]] ([[12ED2]] tone); Proposed by Widogast Iring (1898), later by Joseph Yasser as a "centitone" (1932). ([http://www.tonalsoft.com/enc/c/centitone.aspx source])
+| [[Relative cent]] of [[6edo]] ([[12edo]] tone); Proposed by Widogast Iring (1898), later by Joseph Yasser as a "centitone" (1932). ([http://www.tonalsoft.com/enc/c/centitone.aspx source])
 |-
 | [[Skisma]]
 | [[612edo|612]]
 | 2<sup>2</sup> × 3<sup>2</sup> × 17
-| ED2 representation of [[Sagittal notation|Sagittal]]'s Ultra (Herculean) precision level JI notation (58-EDA), where it is known as an "ultrina"
+| Edo representation of [[Sagittal notation|Sagittal]]'s Ultra (Herculean) precision level JI notation (58eda), where it is known as an "ultrina"
 |-
 | [[Delfi]]
@@ Line 111: / Line 108: @@
 |
 |-
-|Small [[Iring]]/[[Centitone]]
+| Small [[Iring]]/[[Centitone]]
-|[[700edo|700]]
+| [[700edo|700]]
-|2<sup>2</sup> × 5<sup>2</sup> x 7
+| 2<sup>2</sup> × 5<sup>2</sup> x 7
 |
 |-
@@ Line 124: / Line 121: @@
 | [[1000edo|1000]]
 | 2<sup>3</sup> × 5<sup>3</sup>
-| [https://en.wikipedia.org/wiki/Metric_prefix SI-prefix] division of octave
+| [[Wikipedia: Metric prefix|SI-prefix]] division of octave
 |-
 | [[cent]] (¢)
 | 1200
 | 2<sup>4</sup> × 3 × 5<sup>2</sup>
-| 1/100 of [[12ED2]] semitone
+| 1/100 of [[12edo]] semitone
 |-
 | greater muon
@@ Line 191: / Line 188: @@
 |
 |-
-|śata
+| śata
-|[[1600edo|1600]]
+| [[1600edo|1600]]
-|2<sup>6</sup> × 5<sup>2</sup>
+| 2<sup>6</sup> × 5<sup>2</sup>
-|From Sanskrit ''śatam'': hundred; [[Relative cent]] of Armodue 16ED2 Theory
+| From Sanskrit ''śatam'': hundred; [[Relative cent]] of Armodue 16edo Theory
 |-
 | tile
@@ Line 204: / Line 201: @@
 | [[1\1700_octave|1700]]
 | 2<sup>2</sup> × 5<sup>2</sup> × 17
-| [[Relative cent]] of [[17ED2]]; proposed by [[Margo Schulter]] ([http://www.huygens-fokker.org/docs/measures.html source]) and [[George Secor]] ([[Relative_cent|source]])
+| [[Relative cent]] of [[17edo]]; proposed by [[Margo Schulter]] ([http://www.huygens-fokker.org/docs/measures.html source]) and [[George Secor]] ([[Relative cent|source]])
 |-
 | [[Harmos]]
 | [[1728edo|1728]]
 | 2<sup>6</sup> × 3<sup>3</sup>
-| 1728 = 12<sup>3</sup>; 1/144 of [[12ED2]] semitone; Proposed by Paul Beaver ([http://www.tonalsoft.com/enc/e/equal-temperament.aspx source])
+| 1728 = 12<sup>3</sup>; 1/144 of [[12edo]] semitone; Proposed by Paul Beaver ([http://www.tonalsoft.com/enc/e/equal-temperament.aspx source])
 |-
-|Hind śat / Indian cent
+| Hind śat / Indian cent
-|2200
+| 2200
-|2<sup>3</sup> × 11 × 5<sup>2</sup>
+| 2<sup>3</sup> × 11 × 5<sup>2</sup>
 |
 |-
@@ Line 219: / Line 216: @@
 | [[2460edo|2460]]
 | 2<sup>2</sup> × 3 × 5 × 41
-| Abbreviation of "schismina", ED2 representation of [[Sagittal notation|Sagittal]]'s Extreme (Olympian) precision level JI notation (233-EDA)
+| Abbreviation of "schismina", edo representation of [[Sagittal notation|Sagittal]]'s Extreme (Olympian) precision level JI notation (233eda)
 |-
-|Centidiesis
+| Centidiesis
-|3100
+| 3100
-|2<sup>2</sup> × 5<sup>2</sup> x 31
+| 2<sup>2</sup> × 5<sup>2</sup> x 31
 |
 |-
-|Centiméride
+| Centiméride
-|4300
+| 4300
-|2<sup>2</sup> × 5<sup>2</sup> x 43
+| 2<sup>2</sup> × 5<sup>2</sup> x 43
 |
 |-
@@ Line 234: / Line 231: @@
 | [[8539edo|8539]]
 | PRIME
-| Provides good approximations for 41-limit primes except 37 ([http://www.tonalsoft.com/enc/t/tina.aspx source]); named by [[Dave Keenan]] and [[George Secor]]; ED2 representation of [[Sagittal notation|Sagittal]]'s Insane (Magrathean) precision level JI notation (809-EDA)
+| Provides good approximations for 41-limit primes except 37 ([http://www.tonalsoft.com/enc/t/tina.aspx source]); named by [[Dave Keenan]] and [[George Secor]]; edo representation of [[Sagittal notation|Sagittal]]'s Insane (Magrathean) precision level JI notation (809eda)
 |-
 | [[Purdal]]
 | [[9900edo|9900]]
 | 2<sup>2</sup> × 3<sup>2</sup> × 5<sup>2</sup> × 11
-| [[Relative cent]] of [[99edo|99ED2]]; Suggested by [[Osmiorisbendi]], advocated by [[Tútim Dennsuul Wafiil]]
+| [[Relative cent]] of [[99edo]]; Suggested by [[Osmiorisbendi]], advocated by [[Tútim Dennsuul Wafiil]]
 |-
 | [[Türk sent]] / [[Turkish cent]]
 | [[10600edo|10600]]
 | 2<sup>3</sup> × 5<sup>2</sup> × 53
-| [[Relative cent]] of [[106ED2]], 1/200 of [[53ED2]]; invented by [http://www.tonalsoft.com/enc/t/turk-sent.aspx M. Ekrem Karadeniz] (1965), influenced by  [https://core.ac.uk/download/pdf/76124322.pdf Abdülkadir Töre]
+| [[Relative cent]] of [[106edo]], 1/200 of [[53edo]]; invented by [http://www.tonalsoft.com/enc/t/turk-sent.aspx M. Ekrem Karadeniz] (1965), influenced by  [https://core.ac.uk/download/pdf/76124322.pdf Abdülkadir Töre]
 |-
 | [[Prima]]
@@ Line 271: / Line 268: @@
 |
 |-
-| [[MIDI_Tuning_Standard_unit|MIDI Tuning Standard unit]]
+| [[MIDI Tuning Standard unit]]
 | [[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
 |-
-|[[Mean free path]]
+| [[Mean free path]]
-|~216,608,494
+| ~216,608,494
-|2×41×2641567
+| 2×41×2641567
 |
 |}
@@ Line 285: / Line 282: @@
 ==== 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:
@@ Line 292: / Line 288: @@
 |+ List of Non-Octave Fine Measures (Logarithmic)
 |-
-! Unit name (symbol):
+! Unit Name (Symbol):
-! Base interval:
+! Base Interval:
-! Parts of base interval:
+! Parts of Base Interval:
 ! Origin/Significance
 |-
@@ Line 324: / Line 320: @@
 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>
-See [http://www.huygens-fokker.org/docs/measures.html Logarithmic Interval Measures]
 === 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 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.
 see also: Kirnberger Atom http://arxiv.org/abs/0907.5249
 == Ratio ==
-Intervals can be measured also giving their [http://en.wikipedia.org/wiki/Interval_ratio (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:
+Intervals can be measured also giving their [[Wikipedia: Interval ratio (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]],
-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]].
-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 {{monzo| -4 4 -1 }} (for the syntonic comma, 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.
-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, 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.
+== Notes ==
+<references/>
 [[Category:Interval size]]