Tritone: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
Mike Battaglia (talk | contribs)
autopatrolled, Bureaucrats, confirmaccount-notify, emailconfirmed, Interface administrators, Sysops
1,399 edits
formatting
m →In edos: Add link to semioctave
 
(48 intermediate revisions by 11 users not shown)
Line 1: Line 1:
The '''tritone''' is the name of a musical interval that is made up of three whole tones, which in [[12edo]] is 600 cents. In non-12 tunings, and particularly in JI, there are a number of different tritones which have subtly different flavors, such as [[7/5]] and [[10/7]]. The temperament eliminating [[50/49]] is of particular interest in that it equates these two tritones, and provides a JI basis for the 12-EDO concept of the "tritone substitution."
{{Infobox interval region
| Name = Tritone, augmented fourth, diminished fifth
| Cents lower = 570
| Cents lower wide = 540
| Cents upper = 630
| Cents upper wide = 660
| JI intervals = 7/5, 10/7
| MOSes = [[2L 5s]], [[2L 7s]]
| Complement = [[Tritone]]
| Lower region = [[Semi-augmented&nbsp;fourth]] <br> [[Perfect&nbsp;fourth]]
| Higher region = [[Semidiminished&nbsp;fifth]] <br> [[Perfect&nbsp;fifth]]
}}
{{Wikipedia}}
A '''tritone''' is an interval that spans approximately half an octave (logarithmically). There are two different types of tritones in the diatonic scale:


[[Category:Stub]]
* '''augmented fourths (A4)''' span 4 degrees or 3 steps of the diatonic scale; an augmented fourth is found between the 4th and 7th notes of the major scale
* '''diminished fifths (d5)''' span 5 degrees or 4 steps of the diatonic scale; a diminished fifth is found between the 2nd and 6th notes of the minor scale
 
More generally, an interval close to 600 cents in size can be called a tritone.
 
== As an interval region ==
As an [[interval region]], a tritone is typically near 600 [[cent]]s in size. A rough tuning range for the tritone is about 560 to 640 cents according to [[Margo Schulter]]'s theory of interval regions. ''Tritone'' in this sense 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 [[2edo|1\2edo]].
 
For the sake of fully covering the range of intervals within the octave, this page also covers '''semiaugmented fourths''' of about 550 cents, and '''semidiminished fifths''' of about 650 cents. Note that these are not conventionally considered tritones, and are included here for simplicity and for having more than one pair of simple ratios to use in the edo section. More info may be found at [[semiaugmented fourth]] and [[semidiminished fifth]]. As such, this article covers intervals from 560 to 640 cents, but intervals between 540-560 and 640-660 cents have been "grandfathered in" due to the fact that superfourths and subfifths were not originally given their own articles.
 
=== In MOS scales ===
Intervals between 545 and 654 cents generate the following [[mos]] scales:
 
These tables start from the last monolarge [[mos]] generated by the interval range.
 
Scales with more than 12 notes are not included.
 
{| class="wikitable"
|-
! Range
! colspan="6" | Mos
|-
| 545–654{{c}}
| [[1L 1s]]
| [[2L 1s]]
| [[2L 3s]]
| [[2L 5s]]
| [[2L 7s]]
| [[2L 9s]]
|}
 
== As a diatonic interval category ==
As a diatonic interval category, an augmented fourth is an interval that spans three scale steps in the [[5L 2s|diatonic]] scale with the augmented (wider) quality. It is generated by stacking 6 fifths [[Octave reduction|octave reduced]], and depending on the specific tuning, it ranges from 514 to 720 [[Cent|¢]] ([[7edo|3\7]] to [[5edo|3\5]]).
 
A diminished fifth is its enharmonic equivalent and octave complement, and spans four scale steps in the diatonic scale with the diminished (narrower) quality. It is generated by stacking 6 fourths octave reduced, and ranges from 480 to 686 [[Cent|¢]] ([[5edo|2\5]] to [[7edo|4\7]]).
 
In [[just intonation]], an interval may be classified as a tritone if it is reasonably mapped to 6 steps of the chromatic scale. Formally, this is 12\24, which is used as opposed to [[12edo]]'s 6\12 to better capture the characteristics of many intervals in the [[11-limit|11-]] and [[13-limit]]. Augmented fourths are further mapped to 3 steps of the diatonic scale (3\7) and diminished fifths are mapped to 4 steps of the diatonic scale (4\7). 
 
Both tritones can be stacked with each other to form an octave.
 
In TAMNAMs, the tritones are called the '''augmented 3-diastep''' and '''diminished 4-diastep''' for the augmented fourth and diminished fifth respectively.
 
=== Scale info ===
The diatonic scale contains one augmented fourth and one diminished fifth. In the Ionian mode, the augmented fourth is found on the fourth degree, and the diminished fifth is found on the seventh degree.
 
== In just intonation ==
Due to being close to 600{{c}}, 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]]'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:
 
* 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{{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{{c}} and 617{{c}} respectively.
* 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 ==
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]]s. Note that many edos map 7/5 and 10/7 to the [[semioctave]].
 
{| class="wikitable"
|-
! Edo
! 11/8
! 7/5
! 16/11
! 10/7
! Other tritones
|-
| 12
| colspan="4" | 600{{c}}
|
|-
| 15
| colspan="2" | 560{{c}}
| colspan="2" | 640{{c}}
|
|-
| 16
| 525{{c}}
| 600{{c}}
| 675{{c}}
| 600{{c}}
|
|-
| 17
| colspan="2" | 565{{c}}
| colspan="2" | 635{{c}}
|
|-
| 19
| colspan="2" | 568{{c}}
| colspan="2" | 632{{c}}
|
|-
| 22
| 545{{c}}
| 600{{c}}
| 655{{c}}
| 600{{c}}
|
|-
| 24
| 550{{c}}
| 600{{c}}
| 650{{c}}
| 600{{c}}
|
|-
| 25
| *
| 576{{c}}
| *
| 624{{c}}
|
|-
| 26
| 554{{c}}
| 600{{c}}
| 646{{c}}
| 600{{c}}
|
|-
| 27
| *
| 578{{c}}
| *
| 622{{c}}
|
|-
| 29
| *
| 579{{c}}
| *
| 621{{c}}
|
|-
| 31
| 542{{c}}
| 581{{c}}
| 658{{c}}
| 619{{c}}
|
|-
| 34
| 565{{c}}
| 600{{c}}
| 635{{c}}
| 600{{c}}
|
|-
| 41
| 556{{c}}
| 585{{c}}
| 644{{c}}
| 615{{c}}
|
|-
| 53
| 543{{c}}
| 589{{c}}
| 657{{c}}
| 611{{c}}
| {{nowrap|634{{c}} ≈ 36/25|566{{c}} ≈ 25/18}}
|}
 
== In regular temperaments ==
 
=== 1/2-octave temperaments ===
Temperaments involving tritones often involve tempering a pair of tritones together. As such, each pair of tritones has a corresponding temperament, which equates both tritones to the semioctave:
 
Note that in these temperaments, the tritone is usually considered as the period, even though it is technically also a generator.
{| class="wikitable"
|-
! Pair of tritones
! Temperament
|-
| 45/32, 64/45
| [[Diaschismic]]
|-
| 25/18, 36/25
| [[Diminished (temperament)|Diminished]]
|-
| 7/5, 10/7
| [[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:
 
=== Other temperaments ===
 
==== 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.
 
{{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]])}}
 
== 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"
|-
! class="unsortable" | Ratios
! Prime<br>limit
! Distance<br>from 600{{c}}
! Comma
|-
| [[729/512]], [[1024/729]]
| 3
| 11.730
| [[Pythagorean comma|531441/524288]]
|-
| [[45/32]], [[64/45]]
| 5
| 9.776
| [[Diaschisma|2048/2025]]
|-
| [[7/5]], [[10/7]]
| 7
| 17.488
| [[50/49]]
|-
| [[99/70]], [[140/99]]
| 11
| 0.088
| [[9801/9800]]
|-
| [[13/9]], [[18/13]]
| 13
| 36.618
| [[169/162]]
|-
| [[24/17]], [[17/12]]
| 17
| 3.000
| [[289/288]]
|-
| [[27/19]], [[38/27]]
| 19
| 8.352
| [[729/722]]
|-
| [[23/16]], [[32/23]]
| 23
| 28.274
| [[544/529]]
|-
| [[41/29]], [[58/41]]
| 41
| 0.515
| [[1682/1681]]
|}
 
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]].{{Navbox intervals}}
 
[[Category:Tritone]] <!-- Main article -->

Latest revision as of 08:46, 18 July 2025

← Semi-augmented fourth
Perfect fourth 		Tritone, augmented fourth, diminished fifth Semidiminished fifth
Perfect fifth →
Name Tritone, augmented fourth, diminished fifth
Lower bound 540¢ – 570¢
Upper bound 630¢ – 660¢
Generated MOSes 2L 5s, 2L 7s
Example JI intervals
Intervals 7/5 (582.5¢)
10/7 (617.5¢)
Related regions
Complement Tritone
English Wikipedia has an article on:

A tritone is an interval that spans approximately half an octave (logarithmically). There are two different types of tritones in the diatonic scale:

  • augmented fourths (A4) span 4 degrees or 3 steps of the diatonic scale; an augmented fourth is found between the 4th and 7th notes of the major scale
  • diminished fifths (d5) span 5 degrees or 4 steps of the diatonic scale; a diminished fifth is found between the 2nd and 6th notes of the minor scale

More generally, an interval close to 600 cents in size can be called a tritone.

As an interval region

As an interval region, a tritone is typically near 600 cents in size. A rough tuning range for the tritone is about 560 to 640 cents according to Margo Schulter's theory of interval regions. Tritone in this sense 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.

For the sake of fully covering the range of intervals within the octave, this page also covers semiaugmented fourths of about 550 cents, and semidiminished fifths of about 650 cents. Note that these are not conventionally considered tritones, and are included here for simplicity and for having more than one pair of simple ratios to use in the edo section. More info may be found at semiaugmented fourth and semidiminished fifth. As such, this article covers intervals from 560 to 640 cents, but intervals between 540-560 and 640-660 cents have been "grandfathered in" due to the fact that superfourths and subfifths were not originally given their own articles.

In MOS scales

Intervals between 545 and 654 cents generate the following mos scales:

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

Scales with more than 12 notes are not included.

Range Mos
545–654 ¢ 1L 1s 2L 1s 2L 3s 2L 5s 2L 7s 2L 9s

As a diatonic interval category

As a diatonic interval category, an augmented fourth is an interval that spans three scale steps in the diatonic scale with the augmented (wider) quality. It is generated by stacking 6 fifths octave reduced, and depending on the specific tuning, it ranges from 514 to 720 ¢ (3\7 to 3\5).

A diminished fifth is its enharmonic equivalent and octave complement, and spans four scale steps in the diatonic scale with the diminished (narrower) quality. It is generated by stacking 6 fourths octave reduced, and ranges from 480 to 686 ¢ (2\5 to 4\7).

In just intonation, an interval may be classified as a tritone if it is reasonably mapped to 6 steps of the chromatic scale. Formally, this is 12\24, which is used as opposed to 12edo's 6\12 to better capture the characteristics of many intervals in the 11- and 13-limit. Augmented fourths are further mapped to 3 steps of the diatonic scale (3\7) and diminished fifths are mapped to 4 steps of the diatonic scale (4\7).

Both tritones can be stacked with each other to form an octave.

In TAMNAMs, the tritones are called the augmented 3-diastep and diminished 4-diastep for the augmented fourth and diminished fifth respectively.

Scale info

The diatonic scale contains one augmented fourth and one diminished fifth. In the Ionian mode, the augmented fourth is found on the fourth degree, and the diminished fifth is found on the seventh degree.

In just intonation

Due to being close to 600 ¢, 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's, which is an interval of about 612 ¢. There is also the octave complement, the Pythagorean diminished fifth of 1024/729, which is about 588 ¢ in size.

Much simpler tritones exist in higher 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 ¢ and 610 ¢ 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 ¢ and 631 ¢ respectively.
  • The 7-limit narrow tritone and wide tritone are ratios of 7/5 and 10/7 respectively, and are about 583 ¢ and 617 ¢ respectively.
  • The 11-limit superfourth and subfifth are ratios of 11/8 and 16/11 respectively, and are about 551 ¢ and 649 ¢ respectively; they are listed here because they barely do not make the cutoff (550 ¢ and 650 ¢) to be included in the pages on fourths and fifths.

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 edos. Note that many edos map 7/5 and 10/7 to the semioctave.

Edo 11/8 7/5 16/11 10/7 Other tritones
12 600 ¢
15 560 ¢ 640 ¢
16 525 ¢ 600 ¢ 675 ¢ 600 ¢
17 565 ¢ 635 ¢
19 568 ¢ 632 ¢
22 545 ¢ 600 ¢ 655 ¢ 600 ¢
24 550 ¢ 600 ¢ 650 ¢ 600 ¢
25 * 576 ¢ * 624 ¢
26 554 ¢ 600 ¢ 646 ¢ 600 ¢
27 * 578 ¢ * 622 ¢
29 * 579 ¢ * 621 ¢
31 542 ¢ 581 ¢ 658 ¢ 619 ¢
34 565 ¢ 600 ¢ 635 ¢ 600 ¢
41 556 ¢ 585 ¢ 644 ¢ 615 ¢
53 543 ¢ 589 ¢ 657 ¢ 611 ¢ 634 ¢ ≈ 36/25, 566 ¢ ≈ 25/18

In regular temperaments

1/2-octave temperaments

Temperaments involving tritones often involve tempering a pair of tritones together. As such, each pair of tritones has a corresponding temperament, which equates both tritones to the semioctave:

Note that in these temperaments, the tritone is usually considered as the period, even though it is technically also a generator.

Pair of tritones Temperament
45/32, 64/45 Diaschismic
25/18, 36/25 Diminished
7/5, 10/7 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:

Other temperaments

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.

Todo: complete section

Temperaments that use 7/5 as a generator

Todo: complete section

(see List of rank two temperaments by generator and period #Generator .7E7.2F5)

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:

Ratios Prime
limit 		Distance
from 600 ¢ 		Comma
729/512, 1024/729 3 11.730 531441/524288
45/32, 64/45 5 9.776 2048/2025
7/5, 10/7 7 17.488 50/49
99/70, 140/99 11 0.088 9801/9800
13/9, 18/13 13 36.618 169/162
24/17, 17/12 17 3.000 289/288
27/19, 38/27 19 8.352 729/722
23/16, 32/23 23 28.274 544/529
41/29, 58/41 41 0.515 1682/1681

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.

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
Retrieved from "https://en.xen.wiki/index.php?title=Tritone&oldid=204728"