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A '''tritone''' is an interval that is near 600 [[Cent|cents]] in size, distinct from the [[perfect fifth]] of roughly 700 cents and the [[perfect fourth]] of roughly 500 cents. A rough tuning range for the tritone is about 540 to 660 cents, however people tend to narrow that range to around 570 to 630 cents in order to treat superfourths and subfifths as distinct categories. In this case, for the sake of conciseness, however, they are treated as tritones.
A '''tritone''' is an interval that is near 600{{cent}} in size, distinct from the [[perfect fifth]] of roughly 700{{c}} and the [[perfect fourth]] of roughly 500{{c}}. A rough tuning range for the tritone is about 540 to 660{{c}}, however people tend to narrow that range to around 570 to 630{{c}} in order to treat superfourths and subfifths as distinct categories. In this case, for the sake of conciseness, however, they are treated as tritones.


 
The term "tritone" can also refer to the semi-octave, a tritone of exactly 600{{c}} found in every even [[EDO]], due to the fact that it is 1\[[2edo]]. This is not the main subject of this page, but the semi-octave is significant to the nature of tritones so it will be referenced further.
The term "tritone" can also refer to the semi-octave, a tritone of exactly 600 cents found in every even [[EDO]], due to the fact that it is 1\[[2edo]]. This is not the main subject of this page, but the semi-octave is significant to the nature of tritones so it will be referenced further.


== In just intonation ==
== In just intonation ==
Due to being close to 600{{cent}}, tritones come in octave-complementary pairs. For low-limit harmony, these pairs are often referred to as "augmented fourth" (A4) and "diminished fifth" (d5) based on their function in diatonic harmony, but in higher limits, the tritones are usually just distinguished by size.
Due to being close to 600{{cent}}, tritones come in octave-complementary pairs. For low-limit harmony, these pairs are often referred to as "augmented fourth" (A4) and "diminished fifth" (d5) based on their function in diatonic harmony, but in higher limits, the tritones are usually just distinguished by size.


Historically, the term "tritone" referred to the '''Pythagorean augmented fourth,''' the ratio of 729/512 reached by stacking three Pythagorean whole tones (hence "tri-tone"), or equivalently, six [[3/2]]<nowiki/>s, which is an interval of about 612{{cent}}. There is also the octave complement, the '''Pythagorean diminished fifth''' of 1024/729, which is about 588{{cent}} in size.
Historically, the term "tritone" referred to the '''Pythagorean augmented fourth,''' the ratio of 729/512 reached by stacking three Pythagorean whole tones (hence "tri-tone"), or equivalently, six [[3/2]]<nowiki/>s, which is an interval of about 612{{c}}. There is also the octave complement, the '''Pythagorean diminished fifth''' of 1024/729, which is about 588{{c}} in size.


Much [[Odd limit|simpler]] tritones exist in higher [[Prime limit|limits]], however, for example:
Much [[Odd limit|simpler]] tritones exist in higher [[Prime limit|limits]], however, for example:


* The 5-limit '''ptolemaic augmented fourth''' and '''ptolemaic diminished fifth''' are ratios of 45/32 and 64/45 respectively, and are about 590{{cent}} and 610{{cent}} respectively.
* The 5-limit '''ptolemaic augmented fourth''' and '''ptolemaic diminished fifth''' are ratios of 45/32 and 64/45 respectively, and are about 590{{c}} and 610{{c}} respectively.
** There are also the '''classical augmented fourth''' and '''classical diminished fifth,''' which are ratios of 25/18 and 36/25 respectively, and are about 569{{cent}} and 631{{cent}} respectively.
** There are also the '''classical augmented fourth''' and '''classical diminished fifth,''' which are ratios of 25/18 and 36/25 respectively, and are about 569{{c}} and 631{{c}} respectively.
* The 7-limit '''narrow tritone''' and '''wide tritone''' are ratios of 7/5 and 10/7 respectively, and are about 583{{cent}} and 617{{cent}} respectively.
* The 7-limit '''narrow tritone''' and '''wide tritone''' are ratios of 7/5 and 10/7 respectively, and are about 583{{c}} and 617{{c}} respectively.
* The 11-limit '''superfourth''' and '''subfifth''' are ratios of 11/8 and 16/11 respectively, and are about 551{{cent}} and 649{{cent}} respectively; they are listed here because they barely do not make the cutoff (550{{cent}} and 650{{cent}}) to be included in the pages on fourths and fifths.
* The 11-limit '''superfourth''' and '''subfifth''' are ratios of 11/8 and 16/11 respectively, and are about 551{{c}} and 649{{c}} respectively; they are listed here because they barely do not make the cutoff (550{{c}} and 650{{c}}) to be included in the pages on fourths and fifths.


== In EDOs ==
== In EDOs ==
The following table lists the tunings of 11/8, 7/5, and their octave complements, as well as other tritones if present, in various significant [[EDO|EDOs]]. Note that many EDOs map 7/5 and 10/7 to the semioctave.  
The following table lists the tunings of 11/8, 7/5, and their octave complements, as well as other tritones if present, in various significant [[EDO|EDOs]]. Note that many EDOs map 7/5 and 10/7 to the semioctave.  
{| class="wikitable"
{| class="wikitable"
!EDO
!11/8
!7/5
!16/11
!10/7
!Other tritones
|-
|-
|12
! EDO
| colspan="4" |600{{cent}}
! 11/8
|
! 7/5
! 16/11
! 10/7
! Other tritones
|-
|-
|15
| 12
| colspan="2" |560{{cent}}
| colspan="4" | 600{{c}}
| colspan="2" |640{{cent}}
|  
|
|-
|-
|16
| 15
|525{{cent}}
| colspan="2" | 560{{c}}
|600{{cent}}
| colspan="2" | 640{{c}}
|675{{cent}}
|  
|600{{cent}}
|
|-
|-
|17
| 16
| colspan="2" |565{{cent}}
| 525{{c}}
| colspan="2" |635{{cent}}
| 600{{c}}
|
| 675{{c}}
| 600{{c}}
|  
|-
|-
|19
| 17
| colspan="2" |568{{cent}}
| colspan="2" | 565{{c}}
| colspan="2" |632{{cent}}
| colspan="2" | 635{{c}}
|
|  
|-
|-
|22
| 19
|545{{cent}}
| colspan="2" | 568{{c}}
|600{{cent}}
| colspan="2" | 632{{c}}
|655{{cent}}
|  
|600{{cent}}
|
|-
|-
|24
| 22
|550{{cent}}
| 545{{c}}
|600{{cent}}
| 600{{c}}
|650{{cent}}
| 655{{c}}
|600{{cent}}
| 600{{c}}
|
|  
|-
|-
|25
| 24
|*
| 550{{c}}
|576{{cent}}
| 600{{c}}
|*
| 650{{c}}
|624{{cent}}
| 600{{c}}
|
|  
|-
|-
|26
| 25
|554{{cent}}
| *
|600{{cent}}
| 576{{c}}
|646{{cent}}
| *
|600{{cent}}
| 624{{c}}
|
|  
|-
|-
|27
| 26
|*
| 554{{c}}
|578{{cent}}
| 600{{c}}
|*
| 646{{c}}
|622{{cent}}
| 600{{c}}
|
|  
|-
|-
|29
| 27
|*
| *
|579{{cent}}
| 578{{c}}
|*
| *
|621{{cent}}
| 622{{c}}
|
|  
|-
|-
|31
| 29
|542{{cent}}
| *
|581{{cent}}
| 579{{c}}
|658{{cent}}
| *
|619{{cent}}
| 621{{c}}
|
|  
|-
|-
|34
| 31
|565{{cent}}
| 542{{c}}
|600{{cent}}
| 581{{c}}
|635{{cent}}
| 658{{c}}
|600{{cent}}
| 619{{c}}
|
|  
|-
|-
|41
| 34
|556{{cent}}
| 565{{c}}
|585{{cent}}
| 600{{c}}
|644{{cent}}
| 635{{c}}
|615{{cent}}
| 600{{c}}
|
|  
|-
|-
|53
| 41
|543{{cent}}
| 556{{c}}
|589{{cent}}
| 585{{c}}
|657{{cent}}
| 644{{c}}
|611{{cent}}
| 615{{c}}
|634{{cent}} ≈ 36/25, 566{{cent}} ≈ 25/18
|
|-
| 53
| 543{{c}}
| 589{{c}}
| 657{{c}}
| 611{{c}}
| 634{{c}} ≈ 36/25, 566{{c}} ≈ 25/18
|}
|}
[[Category:Tritone]]
[[Category:Tritone]]
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Note that these technically do not have the semioctave as a ''generator'', since making it a fraction of an octave causes it to become a ''period.''
Note that these technically do not have the semioctave as a ''generator'', since making it a fraction of an octave causes it to become a ''period.''
{| class="wikitable"
{| class="wikitable"
!Pair of tritones
!Temperament
|-
|-
|45/32, 64/45
! Pair of tritones
|[[Diaschismic]]
! Temperament
|-
|-
|25/18, 36/25
| 45/32, 64/45
|[[Diminished (temperament)|Diminished]]
| [[Diaschismic]]
|-
|-
|7/5, 10/7
| 25/18, 36/25
|[[Jubilismic clan|Jubilismic]]
| [[Diminished (temperament)|Diminished]]
|-
|-
|11/8, 16/11
| 7/5, 10/7
|Temperament of [[128/121]]
| [[Jubilismic clan|Jubilismic]]
|-
| 11/8, 16/11
| Temperament of [[128/121]]
|}
|}
Note that sometimes, tritones ''are'' used as generators, utilizing the small commas between the tritone pairs to approximate some other interval. The two simplest tritones, 11/8 and 7/5, also happen to be rather far from the semioctave, and as such are rather useful for this purpose:
Note that sometimes, tritones ''are'' used as generators, utilizing the small commas between the tritone pairs to approximate some other interval. The two simplest tritones, 11/8 and 7/5, also happen to be rather far from the semioctave, and as such are rather useful for this purpose:
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{| class="wikitable sortable center-all right-3"
{| class="wikitable sortable center-all right-3"
|-
! class="unsortable" | Ratios  
! class="unsortable" | Ratios  
! Prime<br>limit
! Prime<br>limit
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|}
|}


{{Navbox intervals}}<!-- main article -->
{{Navbox intervals}} <!-- Main article -->
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