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Tetrachord: Difference between revisions

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The ancient Greeks distinguished between three primary [[genera]] depending on the size of the largest interval, the ''characteristic interval'' (CI): the enharmonic, chromatic, and diatonic. Modern theorists have added a fourth genera, called hyperenharmonic.
The ancient Greeks distinguished between three primary [[genera]] depending on the size of the largest interval, the ''characteristic interval'' (CI): the enharmonic, chromatic, and diatonic. Modern theorists have added a fourth genera, called hyperenharmonic.


; hyperenharmonic genus: The CI is larger than 425 cents.
; Hyperenharmonic genus: The CI is larger than 425{{c}}.
; enharmonic genus: The CI approximates a major third, falling between 425 cents and 375 cents.
; Enharmonic genus: The CI approximates a major third, falling between 375–425{{c}}.
; chromatic genus: The CI approximates a minor or neutral third, falling between 375 cents and 250 cents.
; Chromatic genus: The CI approximates a minor or neutral third, falling between 250–375{{c}}.
; diatonic genus: The CI (and the other intervals) approximates a "[[tone]]", measuring less than 250 cents.
; Diatonic genus: The CI (and the other intervals) approximates a "[[tone]]", measuring less than 250{{c}}.


=== Ptolemy's catalog ===
=== Ptolemy's catalog ===
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{| class="wikitable"
{| class="wikitable"
|-
|+ style="font-size: 105%;" | Archytas's genera
|+ Archytas's Genera
|-
|-
| 28/27, 36/35, 5/4
| 28/27, 36/35, 5/4
| 63 + 49 + 386
| 63 + 49 + 386
| enharmonic
| Enharmonic
|-
|-
| 28/27, 243/224, 32/27
| 28/27, 243/224, 32/27
| 63 + 141 + 294
| 63 + 141 + 294
| chromatic
| Chromatic
|-
|-
| 28/27, 8/7, 9/8
| 28/27, 8/7, 9/8
| 63 + 231 + 204
| 63 + 231 + 204
| diatonic
| Diatonic
|}
|}


{| class="wikitable"
{| class="wikitable"
|-
|+ style="font-size: 105%;" | Eratosthenes's genera
|+ Eratosthenes's Genera
|-
|-
| 40/39, 39/38, 19/15
| 40/39, 39/38, 19/15
| 44 + 45 + 409
| 44 + 45 + 409
| enharmonic
| Enharmonic
|-
|-
| 20/19, 19/18, 6/5
| 20/19, 19/18, 6/5
| 89 + 94 + 316
| 89 + 94 + 316
| chromatic
| Chromatic
|-
|-
| 256/243, 9/8, 9/8
| 256/243, 9/8, 9/8
| 90 + 204 + 204
| 90 + 204 + 204
| diatonic
| Diatonic
|}
|}


{| class="wikitable"
{| class="wikitable"
|+ style="font-size: 105%;" | Didymos's genera
|-
|-
|+ Didymos's Genera
| 32/31, 31/30, 5/4
|-
| 55 + 57 + 386
| | 32/31, 31/30, 5/4
| Enharmonic
| | 55 + 57 + 386
| | enharmonic
|-
|-
| | 16/15, 25/24, 6/5
| 16/15, 25/24, 6/5
| | 112 + 74 + 316
| 112 + 74 + 316
| | chromatic
| Chromatic
|-
|-
| | 16/15, 10/9, 9/8
| 16/15, 10/9, 9/8
| | 112 + 182 + 204
| 112 + 182 + 204
| | diatonic
| Diatonic
|}
|}


{| class="wikitable"
{| class="wikitable"
|-
|+ style="font-size: 105%;" | Ptolemy's Tunings
|+ Ptolemy's Tunings
|-
|-
| 46/45, 24/23, 5/4
| 46/45, 24/23, 5/4
| 38 + 75 + 386
| 38 + 75 + 386
| enharmonic
| Enharmonic
|-
|-
| 28/27, 15/14, 6/5
| 28/27, 15/14, 6/5
| 63 + 119 + 316
| 63 + 119 + 316
| soft chromatic
| Soft chromatic
|-
|-
| 22/21, 12/11, 7/6
| 22/21, 12/11, 7/6
| 81 + 151 + 267
| 81 + 151 + 267
| intense chromatic
| Intense chromatic
|-
|-
| 21/20, 10/9, 8/7
| 21/20, 10/9, 8/7
| 85 + 182 + 231
| 85 + 182 + 231
| soft diatonic
| Soft diatonic
|-
|-
| 28/27, 8/7, 9/8
| 28/27, 8/7, 9/8
| 63 + 231 + 204
| 63 + 231 + 204
| diatonon toniaion
| Diatonon toniaion
|-
|-
| 256/243, 9/8, 9/8
| 256/243, 9/8, 9/8
| 90 + 204 + 204
| 90 + 204 + 204
| diatonon ditoniaion
| Diatonon ditoniaion
|-
|-
| 16/15, 9/8, 10/9
| 16/15, 9/8, 10/9
| 112 + 204 + 182
| 112 + 204 + 182
| intense diatonic
| Intense diatonic
|-
|-
| 12/11, 11/10, 10/9
| 12/11, 11/10, 10/9
| 151 + 165 + 182
| 151 + 165 + 182
| equable diatonic
| Equable diatonic
|}
|}


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== Generalized tetrachords ==
== Generalized tetrachords ==
All tetrachords share the interval of a perfect fourth, but they vary in the other two intervals. Assuming a just fourth, we can name the two variable intervals ''a'' & ''b'', & then write our generalized tetrachord like this:
All tetrachords share the interval of a perfect fourth, but they vary in the other two intervals. Assuming a just fourth, we can name the two variable intervals {{nowrap|''a'' & ''b''}}, and then write our generalized tetrachord like this:


1/1, a, b, 4/3
1/1, a, b, 4/3
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{| class="wikitable"
{| class="wikitable"
|-
|-
! mode 1
! Mode 1
| 1/1, a, b, 4/3, 3/2, 3a/2, 3b/2, 2/1
| 1/1, a, b, 4/3, 3/2, 3a/2, 3b/2, 2/1
|-
|-
! mode 2
! Mode 2
| 1/1, b/a, 4/3a, 3/2a, 3/2, 3b/2a, 2/a, 2/1
| 1/1, b/a, 4/3a, 3/2a, 3/2, 3b/2a, 2/a, 2/1
|-
|-
! mode 3
! Mode 3
| 1/1, 4b/3, 3b/2, 3a/2b, 3/2, 2/b, 2/b, 2/1
| 1/1, 4b/3, 3b/2, 3a/2b, 3/2, 2/b, 2/b, 2/1
|-
|-
! mode 4
! Mode 4
| 1/1, 9/8, 9a/8, 9b/8, 3/2, 3a, 3b, 2/1
| 1/1, 9/8, 9a/8, 9b/8, 3/2, 3a, 3b, 2/1
|-
|-
! mode 5
! Mode 5
| 1/1, a, b, 4/3, 4a/3, 4b/3, 16/9, 2/1
| 1/1, a, b, 4/3, 4a/3, 4b/3, 16/9, 2/1
|-
|-
! mode 6
! Mode 6
| 1/1, b/a, 4/3a, 4/3, 4b/3a, 16/9a, 2/1
| 1/1, b/a, 4/3a, 4/3, 4b/3a, 16/9a, 2/1
|-
|-
! mode 7
! Mode 7
| 1/1, 4/3b, 4a/3b, 4/3, 16/9b, 2/b, 2a/b, 2/1
| 1/1, 4/3b, 4a/3b, 4/3, 16/9b, 2/b, 2a/b, 2/1
|}
|}
Line 189: Line 185:
ssL, sLs, Lss
ssL, sLs, Lss


And, if you have only one step size (as is the case in [[Porcupine]] temperament, for instance), you have an evenly-divided fourth, and only one possible rotation. (A tetrachord of this type can be found in [[22edo]] - see [[22edo tetrachords]].)
And, if you have only one step size (as is the case in [[Porcupine]] temperament, for instance), you have an evenly-divided fourth, and only one possible rotation. (A tetrachord of this type can be found in [[22edo]]—see [[22edo tetrachords]].)


== Tetrachords in equal temperaments ==
== Tetrachords in equal temperaments ==
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{| class="wikitable"
{| class="wikitable"
|-
|-
! tetrachord notation
! Tetrachord notation
! cents between steps
! Cents between steps
! cents from 0
! Cents from 0
|-
|-
| 1-1-1
| 1-1-1
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{| class="wikitable"
{| class="wikitable"
|-
|-
! tetrachord notation
! Tetrachord notation
! cents between
! Cents between
! cents from 0
! Cents from 0
|-
|-
| 1-1-2
| 1-1-2
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