Otonal heptad: Difference between revisions

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! oheptad.scl
{{Todo|inline=1|merge articles|comment= [[Otonal heptad]] → [[Harmonics 7–14]] }}
 
The '''otonal heptad''' is the seven-note [[just intonation]] scale 8:9:10:11:12:13:14:16: the harmonics 8 through 14, plus the octave. It is a common diatonic scale, and also sometimes called "the overtone scale" (see [[overtone scale]]). It is a rotation of [[Harmonics 7–14|harmonics 7–14]]: starting that scale on its second degree, the 8th harmonic. It is also [[Harmonics 8–16|harmonics 8–16]] without the 15th harmonic. As a [[13-limit]] scale built from consecutive harmonics, all of its notes share the same fundamental.
 
== Intervals ==
{| class="wikitable center-all"
! #
! Cents
! Ratio
! Interval name
! Audio
|-
| 0
| 0.000
| [[1/1]]
| unison
|
|-
| 1
| 203.910
| [[9/8]]
| major second
| [[File:Jid_9_8_pluck_adu_dr220.mp3]]
|-
| 2
| 386.314
| [[5/4]]
| just major third
| [[File:Jid_5_4_pluck_adu_dr220.mp3]]
|-
| 3
| 551.318
| [[11/8]]
| undecimal superfourth
| [[File:Jid_11_8_pluck_adu_dr220.mp3]]
|-
| 4
| 701.955
| [[3/2]]
| just perfect fifth
| [[File:Jid_3_2_pluck_adu_dr220.mp3]]
|-
| 5
| 840.528
| [[13/8]]
| tridecimal neutral sixth
| [[File:Jid_13_8_pluck_adu_dr220.mp3]]
|-
| 6
| 968.826
| [[7/4]]
| harmonic seventh
| [[File:Jid_7_4_pluck_adu_dr220.mp3]]
|-
| 7
| 1200.000
| [[2/1]]
| octave
| [[File:Jid_2_1_pluck_adu_dr220.mp3]]
|}
 
== Properties ==
The steps of the scale are [[9/8]], [[10/9]], [[11/10]], [[12/11]], [[13/12]], [[14/13]] and [[8/7]]. All but the last are [[superparticular]] steps between consecutive harmonics, and the last is the 14:16 step that closes the octave.
 
Above the root, the scale contains the [[just major triad]] (4:5:6), the [[harmonic seventh chord]] (4:5:6:7), as well as 4:5:6:7:9:11:13 (an extended harmonic 13th chord). It has no perfect fourth and no major seventh.


!
== Relation to the acoustic scale ==
The scale approximates the '''acoustic scale''' (also called the ''overtone scale'' or ''lydian dominant'' scale in jazz): a diatonic scale with a raised fourth and a lowered seventh, the fourth mode of the ascending melodic minor scale. The name is commonly said to come from the scale's resemblance to this segment of the harmonic series. In terms of the diatonic degrees, the acoustic scale is 1, 2, 3, {{sharp}}4, 5, 6, {{flat}}7. The common [[12edo]] version deviates significantly from the just intervals:
* [[5/4]] (386.3 cents) is 13.7 cents flat of the 12edo major third
* [[11/8]] (551.3 cents) is 48.7 cents flat of the augmented fourth
* [[13/8]] (840.5 cents) lies between the minor and major sixth, 40.5 cents above the minor sixth
* [[7/4]] (968.8 cents) is 31.2 cents flat of the minor seventh


Otonal heptad as a 54-wakalix
== Relation to the Carlos harmonic scale ==
The otonal heptad can be seen as the diatonic version of the [[Carlos harmonic scale]], which is a chromatic scale. The notes of the heptad are the even harmonics 16, 18, 20, 22, 24, 26 and 28 of that scale, which has five more: the harmonics 17, 19, 21, 27 and 30.


== Scala file ==
<pre>
! oheptad.scl
!
Otonal heptad 8:9:10:11:12:13:14:16
7
7
!
!
9/8
9/8
5/4
5/4
11/8
11/8
3/2
3/2
13/8
13/8
7/4
7/4
2/1
2/1
!
!
</pre>


! Fokblock([33/32, 22/21, 55/52, 15/14, 99/91], [6, 2, 3, 4, 2])
== See also ==
 
* [[Harmonics 8–16]]
! = Fokblock([33/32, 27/26, 22/21, 55/52, 15/14], [6, 2, 1, 3, 4])
* [[Harmonics 7–14]]
 
* [[Otonalpentad]], the corresponding scale made of the harmonics 8:9:10:12:14:16
! = Fokblock([33/32, 27/26, 55/52, 15/14, 99/91], [6, 4, 3, 4, 1])
* [[Carlos harmonic scale]]
 
* [[Overtone scale]]
! = Fokblock([36/35, 22/21, 96/91, 15/14, 99/91], [3, 2, 0, 4, 5])
 
! = Fokblock([40/39, 22/21, 55/52, 15/14, 99/91], [0, 2, 6, 4, 2])
 
! = Fokblock([40/39, 22/21, 96/91, 15/14, 99/91], [3, 2, 0, 4, 5])
 
! = Fokblock([40/39, 27/26, 22/21, 55/52, 15/14], [0, 2, 1, 6, 4])
 
! = Fokblock([40/39, 27/26, 55/52, 15/14, 99/91], [0, 4, 6, 4, 1])
 
! = Fokblock([40/39, 33/32, 22/21, 15/14, 99/91], [3, 6, 2, 4, 2])
 
! = Fokblock([40/39, 33/32, 22/21, 96/91, 15/14], [3, 5, 2, 2, 4])
 
! = Fokblock([40/39, 33/32, 27/26, 15/14, 99/91], [3, 6, 4, 4, 1])
 
! = Fokblock([40/39, 33/32, 27/26, 22/21, 15/14], [3, 6, 2, 1, 4])
 
! = Fokblock([40/39, 33/32, 80/77, 22/21, 96/91], [3, 6, 4, 2, 2])
 
! = Fokblock([40/39, 36/35, 22/21, 15/14, 99/91], [0, 0, 2, 4, 5])
 
! = Fokblock([40/39, 36/35, 22/21, 55/52, 15/14], [0, 2, 2, 5, 4])
 
! = Fokblock([40/39, 65/63, 80/77, 22/21, 15/14], [3, 4, 0, 1, 6])
 
! = Fokblock([40/39, 80/77, 22/21, 96/91, 15/14], [3, 1, 2, 2, 6])
 
! = Fokblock([55/54, 40/39, 22/21, 55/52, 15/14], [4, 0, 1, 5, 4])
 
! = Fokblock([55/54, 40/39, 27/26, 22/21, 15/14], [6, 0, 5, 1, 4])
 
! = Fokblock([55/54, 40/39, 36/35, 55/52, 15/14], [2, 0, 1, 5, 4])
 
! = Fokblock([55/54, 40/39, 65/63, 22/21, 15/14], [6, 0, 1, 1, 6])
 
! = Fokblock([55/54, 40/39, 65/63, 27/26, 22/21], [6, 0, 4, 6, 1])
 
! = Fokblock([66/65, 36/35, 22/21, 96/91, 15/14], [5, 3, 2, 0, 6])
 
! = Fokblock([66/65, 36/35, 22/21, 96/91, 99/91], [2, 3, 2, 0, 6])
 
! = Fokblock([66/65, 40/39, 22/21, 96/91, 15/14], [5, 3, 2, 0, 6])
 
! = Fokblock([66/65, 40/39, 22/21, 96/91, 99/91], [2, 3, 2, 0, 6])
 
! = Fokblock([66/65, 40/39, 36/35, 22/21, 15/14], [5, 0, 0, 2, 6])
 
! = Fokblock([66/65, 40/39, 36/35, 22/21, 99/91], [2, 0, 0, 2, 6])
 
! = Fokblock([66/65, 40/39, 65/63, 80/77, 15/14], [1, 3, 2, 0, 6])
 
! = Fokblock([66/65, 40/39, 80/77, 22/21, 15/14], [2, 3, 0, 2, 6])
 
! = Fokblock([78/77, 40/39, 36/35, 22/21, 15/14], [1, 0, 2, 2, 6])
 
! = Fokblock([78/77, 40/39, 36/35, 22/21, 55/52], [4, 0, 2, 2, 6])
 
! = Fokblock([78/77, 40/39, 65/63, 22/21, 15/14], [0, 0, 4, 1, 6])
 
! = Fokblock([78/77, 55/54, 40/39, 22/21, 15/14], [1, 4, 0, 1, 6])
 
! = Fokblock([78/77, 55/54, 40/39, 22/21, 55/52], [4, 4, 0, 1, 6])
 
! = Fokblock([78/77, 55/54, 40/39, 36/35, 15/14], [1, 2, 0, 1, 6])
 
! = Fokblock([78/77, 55/54, 40/39, 36/35, 55/52], [4, 2, 0, 1, 6])
 
! = Fokblock([78/77, 65/63, 80/77, 22/21, 15/14], [3, 4, 0, 1, 6])
 
! = Fokblock([78/77, 66/65, 40/39, 22/21, 15/14], [0, 2, 0, 2, 6])
 
! = Fokblock([78/77, 66/65, 40/39, 65/63, 15/14], [0, 1, 0, 2, 6])
 
! = Fokblock([78/77, 66/65, 65/63, 80/77, 15/14], [3, 1, 2, 0, 6])
 
! = Fokblock([78/77, 66/65, 80/77, 22/21, 15/14], [3, 2, 0, 2, 6])
 
! = Fokblock([144/143, 40/39, 27/26, 22/21, 15/14], [0, 3, 5, 1, 4])
 
! = Fokblock([144/143, 40/39, 33/32, 22/21, 15/14], [2, 3, 5, 1, 4])
 
! = Fokblock([144/143, 40/39, 33/32, 80/77, 22/21], [2, 3, 6, 4, 1])
 
! = Fokblock([144/143, 40/39, 33/32, 80/77, 96/91], [4, 3, 6, 4, 1])
 
! = Fokblock([144/143, 40/39, 33/32, 96/91, 15/14], [4, 3, 5, 1, 4])
 
! = Fokblock([144/143, 40/39, 65/63, 22/21, 15/14], [0, 3, 1, 1, 6])
 
! = Fokblock([144/143, 40/39, 65/63, 27/26, 22/21], [0, 3, 4, 6, 1])
 
! = Fokblock([144/143, 40/39, 80/77, 22/21, 15/14], [2, 3, 1, 1, 6])
 
! = Fokblock([144/143, 40/39, 80/77, 96/91, 15/14], [4, 3, 1, 1, 6])
 
! = Fokblock([144/143, 55/54, 27/26, 22/21, 15/14], [0, 3, 5, 1, 4])


! = Fokblock([144/143, 55/54, 65/63, 22/21, 15/14], [0, 3, 1, 1, 6])
{{Navbox harmonics}}


! = Fokblock([144/143, 55/54, 65/63, 27/26, 22/21], [0, 3, 4, 6, 1])
[[Category:7-tone scales]]
[[Category:13-limit]]
[[Category:Just intonation scales]]
[[Category:Harmonic]]
[[Category:Harmonic series]]
[[Category:Otonality]]
[[Category:Wakalixes]]
[[Category:Pages with Scala files]]

Latest revision as of 08:01, 11 October 2026

Todo: merge articles

Otonal heptad → Harmonics 7–14

The otonal heptad is the seven-note just intonation scale 8:9:10:11:12:13:14:16: the harmonics 8 through 14, plus the octave. It is a common diatonic scale, and also sometimes called "the overtone scale" (see overtone scale). It is a rotation of harmonics 7–14: starting that scale on its second degree, the 8th harmonic. It is also harmonics 8–16 without the 15th harmonic. As a 13-limit scale built from consecutive harmonics, all of its notes share the same fundamental.

Intervals

# Cents Ratio Interval name Audio
0 0.000 1/1 unison
1 203.910 9/8 major second
2 386.314 5/4 just major third
3 551.318 11/8 undecimal superfourth
4 701.955 3/2 just perfect fifth
5 840.528 13/8 tridecimal neutral sixth
6 968.826 7/4 harmonic seventh
7 1200.000 2/1 octave

Properties

The steps of the scale are 9/8, 10/9, 11/10, 12/11, 13/12, 14/13 and 8/7. All but the last are superparticular steps between consecutive harmonics, and the last is the 14:16 step that closes the octave.

Above the root, the scale contains the just major triad (4:5:6), the harmonic seventh chord (4:5:6:7), as well as 4:5:6:7:9:11:13 (an extended harmonic 13th chord). It has no perfect fourth and no major seventh.

Relation to the acoustic scale

The scale approximates the acoustic scale (also called the overtone scale or lydian dominant scale in jazz): a diatonic scale with a raised fourth and a lowered seventh, the fourth mode of the ascending melodic minor scale. The name is commonly said to come from the scale's resemblance to this segment of the harmonic series. In terms of the diatonic degrees, the acoustic scale is 1, 2, 3, 4, 5, 6, 7. The common 12edo version deviates significantly from the just intervals:

  • 5/4 (386.3 cents) is 13.7 cents flat of the 12edo major third
  • 11/8 (551.3 cents) is 48.7 cents flat of the augmented fourth
  • 13/8 (840.5 cents) lies between the minor and major sixth, 40.5 cents above the minor sixth
  • 7/4 (968.8 cents) is 31.2 cents flat of the minor seventh

Relation to the Carlos harmonic scale

The otonal heptad can be seen as the diatonic version of the Carlos harmonic scale, which is a chromatic scale. The notes of the heptad are the even harmonics 16, 18, 20, 22, 24, 26 and 28 of that scale, which has five more: the harmonics 17, 19, 21, 27 and 30.

Scala file

! oheptad.scl
!
Otonal heptad 8:9:10:11:12:13:14:16
7
!
9/8
5/4
11/8
3/2
13/8
7/4
2/1
!

See also

View • Talk • EditOvertone scales 
Small modes 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24
Larger modes 30 • 32 • 36 • 48 • 60 • 128
Families /2: 2 • 4 • 8 • 16 • 32 • 128
/3: 3 • 6 • 9 • 12 • 15 • 18 • 21 • 24
/5: 5 • 10 • 15 • 20 • 25 • 30 • 35 • 60 • 80
/7: 7 • 14 • 21 • 28 • 35 • 56
/11: 11 • 22 • 33
/13: 13 • 26
Related Harmonic series • Subharmonic series • Carlos harmonic scale • Ringer scale • Primodality
Retrieved from "https://en.xen.wiki/index.php?title=Otonal_heptad&oldid=239317"