Semitone (interval region): Difference between revisions

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A '''semitone''' is an interval that spans exactly one step of a 12-tone [[chromatic]] scale. In [[just intonation]], an interval may be classified as a semitone if it is reasonably mapped to [[24edo|2\24]]. The use of 24edo's 2\24 as the mapping criteria here rather than [[12edo]]'s 1\12 better captures the characteristics of many intervals in the [[11-limit|11-]] and [[13-limit]]. Semitones come in two functional categories based on their number of steps in the [[5L 2s|diatonic]] scale:
{{Infobox interval region|Name=Semitone, minor second, augmented unison|Cents lower=75|Cents lower wide=60|Cents upper=125|Cents upper wide=140|JI intervals=16/15, 25/24|MOSes=1L 8s, 9L 1s, 1L 9s|Complement=[[Major seventh]]|Lower region=[[Comma and diesis]]|Higher region=[[Neutral second]]}}{{Wikipedia|Semitone}}
* [[Diatonic semitone]]s, minor seconds (m2) or limmas,
A '''semitone''', as a concrete [[interval region]], is typically near 100{{cent}} in size, distinct from [[commas and dieses]] (less than 60{{c}}), and from [[neutral second]]s (about 150{{c}}). A rough tuning range for the semitone is about 60{{c}} to 125{{c}} according to [[Margo Schulter]]'s theory of interval regions.
* [[Chromatic semitone]]s, augmented unisons (A1) or chromas.
Functionally, a semitone is an interval that makes up part of a [[tone]], often as one step of a 12-tone chromatic scale, which is a possible criterion for the classification of an interval as a semitone in [[just intonation]].  


As a concrete [[interval region]], it is typically near 100 [[cent]]s in size, distinct from [[commas and dieses]] (less than 60 cents), and from [[neutral second]]s (about 150 cents). A rough tuning range for the semitone is about 60 cents to 125 cents according to [[Margo Schulter]]'s theory of interval regions.  
Semitones come in two functional categories based on their number of steps in the [[5L 2s|diatonic]] scale:
*[[Diatonic semitone]]s, minor seconds (m2), or limmas,
*[[Chromatic semitone]]s, augmented unisons (A1), or chromas.
 
The intervals covered in this article range from 50{{c}} to 140{{c}}.  


== In just intonation ==
== In just intonation ==
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* In the 3-limit:
* In the 3-limit:
** The ''limma'', or ''Pythagorean diatonic semitone'', is a ratio of [[256/243]], and is about 90 cents.
** The ''limma'', or ''Pythagorean diatonic semitone'', is a ratio of [[256/243]], and is about 90{{c}}.
** The ''apotome'', or ''Pythagorean chromatic semitone'', is a ratio of [[2187/2048]], and is about 114 cents.
** The ''apotome'', or ''Pythagorean chromatic semitone'', is a ratio of [[2187/2048]], and is about 114{{c}}.
* In the 5-limit:
* In the 5-limit:
** The ''classical diatonic semitone'' is a ratio of [[16/15]], and is about 112 cents.
** The ''classical diatonic semitone'' is a ratio of [[16/15]], and is about 112{{c}}.
** The ''classical chromatic semitone'' is a ratio of [[25/24]], and is about 71 cents.
** The ''classical chromatic semitone'' is a ratio of [[25/24]], and is about 71{{c}}.
*** There is also a ''ptolemaic chromatic semitone'', which is a ratio of [[135/128]], and is about 92 cents.
*** There is also a ''ptolemaic chromatic semitone'', which is a ratio of [[135/128]], and is about 92{{c}}.
* In higher limits:
* In higher limits:
** The 7-limit ''third-tone'' is a ratio of [[28/27]], and is about 63 cents.
** The 7-limit ''third-tone'' is a ratio of [[28/27]], and is about 63{{c}}.
** The 7-limit ''minor semitone'' is a ratio of [[21/20]], and is about 84 cents.
** The 7-limit ''minor semitone'' is a ratio of [[21/20]], and is about 84{{c}}.
** The 7-limit ''major semitone'' is a ratio of [[15/14]], and is about 119 cents.
** The 7-limit ''major semitone'' is a ratio of [[15/14]], and is about 119{{c}}.
** The 11-limit ''minor semitone'' is a ratio of [[22/21]], and is about 81 cents.
** The 11-limit ''minor semitone'' is a ratio of [[22/21]], and is about 81{{c}}.
** The 13-limit ''sinaic'' is a ratio of [[14/13]], and is about 128 cents.
** The 13-limit ''sinaic'' is a ratio of [[14/13]], and is about 128{{c}}.
** The 13-limit ''greater 2/3-tone'' is a ratio of [[13/12]], and is about 139 cents.
** The 13-limit ''greater 2/3-tone'' is a ratio of [[13/12]], and is about 139{{c}}.
** The 17-limit ''large semitone'' is a ratio of [[17/16]], and is about 104 cents.
** The 17-limit ''large semitone'' is a ratio of [[17/16]], and is about 104{{c}}.
** The 17-limit ''small semitone'' is a ratio of [[18/17]], and is about 99 cents.
** The 17-limit ''small semitone'' is a ratio of [[18/17]], and is about 99{{c}}.


=== By delta ===
=== By delta ===
This table lists just semitones by [[Delta-N|delta]]:
This table lists just semitones by [[Delta-N|delta]]; simple semitone ratios tend to be [[Superparticular ratio|superparticular]].
{| class="wikitable"
{| class="wikitable"
|+
!Delta 1 (Superparticular)
!Cents
|-
|-
|[[13/12]]
! Delta 1 (Superparticular)
|139c
! Cents
|-
| [[13/12]]
| 139{{c}}
|-
|-
|[[14/13]]
| [[14/13]]
|128c
| 128{{c}}
|-
|-
|[[15/14]]
| [[15/14]]
|119c
| 119{{c}}
|-
|-
|[[16/15]]
| [[16/15]]
|112c
| 112{{c}}
|-
|-
|[[17/16]]
| [[17/16]]
|104c
| 104{{c}}
|-
|-
|[[18/17]]
| [[18/17]]
|99c
| 99{{c}}
|-
|-
|[[19/18]]
| [[19/18]]
|94c
| 94{{c}}
|-
|-
|[[20/19]]
| [[20/19]]
|89c
| 89{{c}}
|-
|-
|[[21/20]]
| [[21/20]]
|85c
| 85{{c}}
|-
|-
|[[22/21]]
| [[22/21]]
|81c
| 81{{c}}
|-
|-
|[[23/22]]
| [[23/22]]
|77c
| 77{{c}}
|-
|-
|[[24/23]]
| [[24/23]]
|74c
| 74{{c}}
|-
|-
|[[25/24]]
| [[25/24]]
|71c
| 71{{c}}
|-
|-
|[[26/25]]
| [[26/25]]
|68c
| 68{{c}}
|-
|-
|[[27/26]]
| [[27/26]]
|65c
| 65{{c}}
|-
|-
|[[28/27]]
| [[28/27]]
|63c
| 63{{c}}
|-
|-
|[[29/28]]
| [[29/28]]
|61c
| 61{{c}}
|-
|-
|[[30/29]]
| [[30/29]]
|59c
| 59{{c}}
|-
|-
|[[31/30]]
| [[31/30]]
|57c
| 57{{c}}
|-
|-
|[[32/31]]
| [[32/31]]
|55c
| 55{{c}}
|-
|-
|[[33/32]]
| [[33/32]]
|53c
| 53{{c}}
|-
|-
|[[34/33]]
| [[34/33]]
|52c
| 52{{c}}
|-
|-
|[[35/34]]
| [[35/34]]
|50c
| 50{{c}}
|}
|}


== In edos ==
== In EDOs ==
The following table lists the best tuning of 16/15, 25/24, and other semitones if present, in various significant [[edo]]s.
The following table lists the best tuning of 16/15, 25/24, and other semitones if present, in various significant [[edo|EDO]]s.
{| class="wikitable"
{| class="wikitable"
|+
!Edo
!16/15
!25/24
!Other semitones
|-
|-
|12
! EDO
| colspan="2" |100c
! 16/15
|
! 25/24
! Other semitones
|-
| 12
| colspan="2" | 100{{c}}
|  
|-
|-
|15
| 15
| colspan="2" |80c
| colspan="2" | 80{{c}}
|
|  
|-
|-
|16
| 16
| colspan="2" |75c
| colspan="2" | 75{{c}}
|
|  
|-
|-
|17
| 17
|141c
| 141{{c}}
|71c
| 71{{c}}
|
|  
|-
|-
|19
| 19
|126c
| 126{{c}}
|63c
| 63{{c}}
|
|  
|-
|-
|22
| 22
|109c
| 109{{c}}
|55c
| 55{{c}}
|
|  
|-
|-
|24
| 24
|100c
| 100{{c}}
|50c
| 50{{c}}
|
|  
|-
|-
|25
| 25
|96c
| 96{{c}}
|*
| *
|
|  
|-
|-
|26
| 26
| colspan="2" |92c
| colspan="2" | 92{{c}}
|
|  
|-
|-
|27
| 27
|133c
| 133{{c}}
|89c
| 89{{c}}
|
|  
|-
|-
|29
| 29
|124c
| 124{{c}}
|83c
| 83{{c}}
|
|  
|-
|-
|31
| 31
|116c
| 116{{c}}
|77c
| 77{{c}}
|
|  
|-
|-
|34
| 34
|106c
| 106{{c}}
|71c
| 71{{c}}
|
|  
|-
|-
|41
| 41
|117c
| 117{{c}}
|59c
| 59{{c}}
|88c ≈ 256/243
| {{nowrap|88{{c}} ≈ 256/243}}
|-
|-
|53
| 53
|113c
| 113{{c}}
|68c
| 68{{c}}
|91c ≈ 256/243
| {{nowrap|91{{c}} ≈ 256/243}}
|}
|}


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=== Temperaments that use 25/24 as a generator ===
=== Temperaments that use 25/24 as a generator ===
* [[Vishnu]], which stacks seven 25/24s to make a just [[perfect fourth]] of [[4/3]]
* [[Valentine]], which divides [[3/2]] into nine small semitones, five of which make [[5/4]]. See also the related [[Carlos Alpha]].
* [[Chlorine]], equivalent to [[17edo]], stacking seventeen 25/24s to make an octave
* [[Vishnu]], which stacks seven 25/24s to make a just [[perfect fourth]] of [[4/3]].
* [[Chlorine]], based on [[17edo]], stacking seventeen 25/24s to make an octave.


=== Temperaments that use 16/15 as a generator ===
=== Temperaments that use 16/15 as a generator ===
Line 201: Line 207:


When 16/15 is tempered out, it leads to [[father]] temperament.
When 16/15 is tempered out, it leads to [[father]] temperament.
== In moment-of-symmetry scales ==
Intervals between 100 and 133{{c}} generate the following [[MOS]] scales:
These tables start from the last monolarge [[MOS]] generated by the interval range.
MOSes with more than 12 notes are not included.
{| class="wikitable"
|-
! Range
! colspan="2" | MOS
|-
| 100–109{{c}}
| [[1L 10s]]
| [[11L 1s]]
|-
| 109–120{{c}}
| [[1L 9s]]
| [[10L 1s]]
|-
| 120–133{{c}}
| [[1L 8s]]
| [[9L 1s]]
|}


== See also ==
== See also ==
Line 206: Line 237:


{{Navbox intervals}}
{{Navbox intervals}}
[[Category:12edo]]
[[Category:12edo]]

Latest revision as of 10:08, 14 March 2025

← Comma and diesis Semitone, minor second, augmented unison Neutral second →
Name Semitone, minor second, augmented unison
Lower bound 60¢ – 75¢
Upper bound 125¢ – 140¢
Generated MOSes 1L 8s, 9L 1s, 1L 9s
Example JI intervals
Intervals 16/15 (111.7¢)
25/24 (70.7¢)
Related regions
Complement Major seventh
English Wikipedia has an article on:

A semitone, as a concrete interval region, is typically near 100 ¢ in size, distinct from commas and dieses (less than 60 ¢), and from neutral seconds (about 150 ¢). A rough tuning range for the semitone is about 60 ¢ to 125 ¢ according to Margo Schulter's theory of interval regions.

Functionally, a semitone is an interval that makes up part of a tone, often as one step of a 12-tone chromatic scale, which is a possible criterion for the classification of an interval as a semitone in just intonation.

Semitones come in two functional categories based on their number of steps in the diatonic scale:

The intervals covered in this article range from 50 ¢ to 140 ¢.

In just intonation

By prime limit

In the low prime limits, up to the 5-limit, in which the West has developed a formal system of diatonic harmony, the distinction between diatonic and chromatic semitones is the clearest, so a pair of 2 semitones will be provided for each. However, higher than the 5-limit, function as diatonic vs. chromatic tends to become less clear, and larger intervals can be seen as belonging to neither category.

  • In the 3-limit:
    • The limma, or Pythagorean diatonic semitone, is a ratio of 256/243, and is about 90 ¢.
    • The apotome, or Pythagorean chromatic semitone, is a ratio of 2187/2048, and is about 114 ¢.
  • In the 5-limit:
    • The classical diatonic semitone is a ratio of 16/15, and is about 112 ¢.
    • The classical chromatic semitone is a ratio of 25/24, and is about 71 ¢.
      • There is also a ptolemaic chromatic semitone, which is a ratio of 135/128, and is about 92 ¢.
  • In higher limits:
    • The 7-limit third-tone is a ratio of 28/27, and is about 63 ¢.
    • The 7-limit minor semitone is a ratio of 21/20, and is about 84 ¢.
    • The 7-limit major semitone is a ratio of 15/14, and is about 119 ¢.
    • The 11-limit minor semitone is a ratio of 22/21, and is about 81 ¢.
    • The 13-limit sinaic is a ratio of 14/13, and is about 128 ¢.
    • The 13-limit greater 2/3-tone is a ratio of 13/12, and is about 139 ¢.
    • The 17-limit large semitone is a ratio of 17/16, and is about 104 ¢.
    • The 17-limit small semitone is a ratio of 18/17, and is about 99 ¢.

By delta

This table lists just semitones by delta; simple semitone ratios tend to be superparticular.

Delta 1 (Superparticular) Cents
13/12 139 ¢
14/13 128 ¢
15/14 119 ¢
16/15 112 ¢
17/16 104 ¢
18/17 99 ¢
19/18 94 ¢
20/19 89 ¢
21/20 85 ¢
22/21 81 ¢
23/22 77 ¢
24/23 74 ¢
25/24 71 ¢
26/25 68 ¢
27/26 65 ¢
28/27 63 ¢
29/28 61 ¢
30/29 59 ¢
31/30 57 ¢
32/31 55 ¢
33/32 53 ¢
34/33 52 ¢
35/34 50 ¢

In EDOs

The following table lists the best tuning of 16/15, 25/24, and other semitones if present, in various significant EDOs.

EDO 16/15 25/24 Other semitones
12 100 ¢
15 80 ¢
16 75 ¢
17 141 ¢ 71 ¢
19 126 ¢ 63 ¢
22 109 ¢ 55 ¢
24 100 ¢ 50 ¢
25 96 ¢ *
26 92 ¢
27 133 ¢ 89 ¢
29 124 ¢ 83 ¢
31 116 ¢ 77 ¢
34 106 ¢ 71 ¢
41 117 ¢ 59 ¢ 88 ¢ ≈ 256/243
53 113 ¢ 68 ¢ 91 ¢ ≈ 256/243

In regular temperaments

Two important, simple semitone ratios are 16/15 and 25/24. The following notable temperaments are generated by them:

Temperaments that use 25/24 as a generator

Temperaments that use 16/15 as a generator

  • Miracle, which splits 3/2 into six semitones, each representing both 15/14 and 16/15.
  • Negri, which splits 4/3 into four semitones, such that three of them represent 5/4.
  • Diaschismic, which is usually described as having a fifth as its second generator, but can alternatively be generated by a half-octave and a semitone.

Compton has one step of 12edo as its first generator, representing 256/243.

When 25/24 is tempered out, it leads to dicot temperament.

When 16/15 is tempered out, it leads to father temperament.

In moment-of-symmetry scales

Intervals between 100 and 133 ¢ generate the following MOS scales:

These tables start from the last monolarge MOS generated by the interval range.

MOSes with more than 12 notes are not included.

Range MOS
100–109 ¢ 1L 10s 11L 1s
109–120 ¢ 1L 9s 10L 1s
120–133 ¢ 1L 8s 9L 1s

See also


View • Talk • EditInterval classification
Interval regions
Unison and octave Unison • Comma and diesis • Octave
Seconds Minor second • Neutral second • Major second
Thirds Minor third • Neutral third • Major third
Fourths and fifths Perfect fourth • Superfourth • Tritone • Subfifth • Perfect fifth
Sixths Minor sixth • Neutral sixth • Major sixth
Sevenths Minor seventh • Neutral seventh • Major seventh
Interseptimal intervals Interseptimal 2nd-3rd • Interseptimal 3rd-4th • Interseptimal 5th-6th • Interseptimal 6th-7th
Interval qualities
Diatonic qualities Diminished • Minor • Perfect • Major • Augmented
Tuning ranges Neutral (interval quality) • Submajor and supraminor • Pental major and minor • Novamajor and novaminor • Neogothic major and minor • Supermajor and subminor • Ultramajor and inframinor
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