Tritone: Difference between revisions

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== Tritones as approximations of the semioctave ==

In some tuning systems having an even number of divisions of the octave (equal or well-tempered), the tritone (defined as three whole tones) is the same as the half-octave; if the divisions of the octave are equal and the octave is tuned pure, the tritone will therefore be exactly the square root of 2. The following table compares selected JI tritone pairs that approximate the half-octave and the commas separating them:

{| class="wikitable sortable center-all right-3"

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However, it is possible for a tuning system to have an even number of notes and a pair of tritones that is not the same as the semioctave.

An example of a tuning system having an even number of notes and a pair of tritones that is not the same as the semioctave is [[26edo]]. In [[flattone]] tuning systems such as 26edo, 11/8 is ~~C-F♯~~, which is the lesser tritone mapping to 12\26 (not the semioctave 13\26); 16/11 is ~~C-G♭~~, which is the greater tritone mapping to 14\26. Note that these tritones are not necessarily valid generators for tuning systems analogous to flattone but substituting 11/8 for [[4/3]]; for instance, these are not valid generators for 26edo with a whole-octave period, because they instead produce [[13edo]].

An example of a tuning system having an even number of notes and a pair of tritones that is not the same as the semioctave is [[26edo]]. In [[flattone]] tuning systems such as 26edo, 11/8 is C–F♯, which is the lesser tritone mapping to 12\26 (not the semioctave 13\26); 16/11 is C–G♭, which is the greater tritone mapping to 14\26. Note that these tritones are not necessarily valid generators for tuning systems analogous to flattone but substituting 11/8 for [[4/3]]; for instance, these are not valid generators for 26edo with a whole-octave period, because they instead produce [[13edo]].

== Temperaments that can use tritones as generators ==

=== Temperaments that use 11/8 as a generator ===

In 13edo, 11/8 and 16/11 map to the tritones 6\13 and 7\13, respectively, and are valid generators for it. Furthermore, although 12\26 is not valid as a generator for 26edo with a whole-octave period, it ''is'' valid with a half-octave period, for a temperament analogous to [[Injera]] but substituting 11/8 for 4/3.

=== Temperaments that use 7/5 as a generator ===

{{todo|complete section|inline=1|text=(see [[List_of_rank_two_temperaments_by_generator_and_period#Generator_.7E7.2F5]])}}

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MOSes with more than 12 notes are not included.

{| class="wikitable"

|+

~~!Range~~

~~! colspan="6" |MOS~~

|-

|~~545~~-~~654c~~

! Range

|1L 1s

! colspan="6" | MOS

|2L 1s

|-

|2L 3s

| 545–654{{c}}

|2L 5s

| 1L 1s

|2L 7s

| 2L 1s

|2L 9s

| 2L 3s

| 2L 5s

| 2L 7s

| 2L 9s

|}

{{Navbox intervals}}

[[Category:Tritone]]

@@ Line 153: / Line 153: @@
 == Tritones as approximations of the semioctave ==
+In some tuning systems having an even number of divisions of the octave (equal or well-tempered), the tritone (defined as three whole tones) is the same as the half-octave; if the divisions of the octave are equal and the octave is tuned pure, the tritone will therefore be exactly the square root of 2. The following table compares selected JI tritone pairs that approximate the half-octave and the commas separating them:
-In some tuning systems having an even number of divisions of the octave (equal or well-tempered), the tritone (defined as three whole tones) is the same as the half-octave; if the divisions of the octave are equal and the octave is tuned pure, the tritone will therefore be exactly the square root of 2.  The following table compares selected JI tritone pairs that approximate the half-octave and the commas separating them:
 {| class="wikitable sortable center-all right-3"
@@ Line 211: / Line 210: @@
 However, it is possible for a tuning system to have an even number of notes and a pair of tritones that is not the same as the semioctave.
-An example of a tuning system having an even number of notes and a pair of tritones that is not the same as the semioctave is [[26edo]].  In [[flattone]] tuning systems such as 26edo, 11/8 is C-F♯, which is the lesser tritone mapping to 12\26 (not the semioctave 13\26); 16/11 is C-G♭, which is the greater tritone mapping to 14\26.  Note that these tritones are not necessarily valid generators for tuning systems analogous to flattone but substituting 11/8 for [[4/3]]; for instance, these are not valid generators for 26edo with a whole-octave period, because they instead produce [[13edo]].
+An example of a tuning system having an even number of notes and a pair of tritones that is not the same as the semioctave is [[26edo]]. In [[flattone]] tuning systems such as 26edo, 11/8 is C–F♯, which is the lesser tritone mapping to 12\26 (not the semioctave 13\26); 16/11 is C–G♭, which is the greater tritone mapping to 14\26. Note that these tritones are not necessarily valid generators for tuning systems analogous to flattone but substituting 11/8 for [[4/3]]; for instance, these are not valid generators for 26edo with a whole-octave period, because they instead produce [[13edo]].
 == Temperaments that can use tritones as generators ==
 === Temperaments that use 11/8 as a generator ===
+In 13edo, 11/8 and 16/11 map to the tritones 6\13 and 7\13, respectively, and are valid generators for it. Furthermore, although 12\26 is not valid as a generator for 26edo with a whole-octave period, it ''is'' valid with a half-octave period, for a temperament analogous to [[Injera]] but substituting 11/8 for 4/3.
-In 13edo, 11/8 and 16/11 map to the tritones 6\13 and 7\13, respectively, and are valid generators for it.  Furthermore, although 12\26 is not valid as a generator for 26edo with a whole-octave period, it ''is'' valid with a half-octave period, for a temperament analogous to [[Injera]] but substituting 11/8 for 4/3.
 {{todo|complete section|inline=1}}
 === Temperaments that use 7/5 as a generator ===
 {{todo|complete section|inline=1|text=(see [[List_of_rank_two_temperaments_by_generator_and_period#Generator_.7E7.2F5]])}}
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 MOSes with more than 12 notes are not included.
 {| class="wikitable"
-|+
-!Range
-! colspan="6" |MOS
 |-
-|545-654c
+! Range
-|1L 1s
+! colspan="6" | MOS
-|2L 1s
+|-
-|2L 3s
+| 545–654{{c}}
-|2L 5s
+| 1L&nbsp;1s
-|2L 7s
+| 2L&nbsp;1s
-|2L 9s
+| 2L&nbsp;3s
+| 2L&nbsp;5s
+| 2L&nbsp;7s
+| 2L&nbsp;9s
 |}
 {{Navbox intervals}} <!-- Main article -->
 [[Category:Tritone]]