Xenharmonic Wiki

Otonal heptad: Difference between revisions

VisualWikitext
Wikispaces>genewardsmith
**Imported revision 499334490 - Original comment: **
 
m Propose merging
 
(6 intermediate revisions by 3 users not shown)
Line 1: Line 1:
<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Todo|inline=1|merge articles|comment= [[Otonal heptad]] → [[Harmonics 7–14]] }}
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
 
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-03-30 11:00:28 UTC</tt>.<br>
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.
: The original revision id was <tt>499334490</tt>.<br>
 
: The revision comment was: <tt></tt><br>
== Intervals ==
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
{| class="wikitable center-all"
<h4>Original Wikitext content:</h4>
! #
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">! oheptad.scl
! 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
 
== 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 as a 54-wakalix
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
!
!
! Fokblock([33/32, 22/21, 55/52, 15/14, 99/91], [6, 2, 3, 4, 2])
</pre>
! = Fokblock([33/32, 27/26, 22/21, 55/52, 15/14], [6, 2, 1, 3, 4])
 
! = Fokblock([33/32, 27/26, 55/52, 15/14, 99/91], [6, 4, 3, 4, 1])
== See also ==
! = Fokblock([36/35, 22/21, 96/91, 15/14, 99/91], [3, 2, 0, 4, 5])
* [[Harmonics 8–16]]
! = Fokblock([40/39, 22/21, 55/52, 15/14, 99/91], [0, 2, 6, 4, 2])
* [[Harmonics 7–14]]
! = Fokblock([40/39, 22/21, 96/91, 15/14, 99/91], [3, 2, 0, 4, 5])
* [[Otonalpentad]], the corresponding scale made of the harmonics 8:9:10:12:14:16
! = Fokblock([40/39, 27/26, 22/21, 55/52, 15/14], [0, 2, 1, 6, 4])
* [[Carlos harmonic scale]]
! = Fokblock([40/39, 27/26, 55/52, 15/14, 99/91], [0, 4, 6, 4, 1])
* [[Overtone scale]]
! = 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])
{{Navbox harmonics}}
! = 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])
[[Category:7-tone scales]]
! = Fokblock([40/39, 33/32, 80/77, 22/21, 96/91], [3, 6, 4, 2, 2])
[[Category:13-limit]]
! = Fokblock([40/39, 36/35, 22/21, 15/14, 99/91], [0, 0, 2, 4, 5])
[[Category:Just intonation scales]]
! = Fokblock([40/39, 36/35, 22/21, 55/52, 15/14], [0, 2, 2, 5, 4])
[[Category:Harmonic]]
! = Fokblock([40/39, 65/63, 80/77, 22/21, 15/14], [3, 4, 0, 1, 6])
[[Category:Harmonic series]]
! = Fokblock([40/39, 80/77, 22/21, 96/91, 15/14], [3, 1, 2, 2, 6])
[[Category:Otonality]]
! = Fokblock([55/54, 40/39, 22/21, 55/52, 15/14], [4, 0, 1, 5, 4])
[[Category:Wakalixes]]
! = Fokblock([55/54, 40/39, 27/26, 22/21, 15/14], [6, 0, 5, 1, 4])
[[Category:Pages with Scala files]]
! = 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])
! = Fokblock([144/143, 55/54, 65/63, 27/26, 22/21], [0, 3, 4, 6, 1])
</pre></div>
<h4>Original HTML content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;oheptad&lt;/title&gt;&lt;/head&gt;&lt;body&gt;! oheptad.scl&lt;br /&gt;
!&lt;br /&gt;
Otonal heptad as a 54-wakalix&lt;br /&gt;
7&lt;br /&gt;
!&lt;br /&gt;
9/8&lt;br /&gt;
5/4&lt;br /&gt;
11/8&lt;br /&gt;
3/2&lt;br /&gt;
13/8&lt;br /&gt;
7/4&lt;br /&gt;
2/1&lt;br /&gt;
!&lt;br /&gt;
! Fokblock([33/32, 22/21, 55/52, 15/14, 99/91], [6, 2, 3, 4, 2])&lt;br /&gt;
! = Fokblock([33/32, 27/26, 22/21, 55/52, 15/14], [6, 2, 1, 3, 4])&lt;br /&gt;
! = Fokblock([33/32, 27/26, 55/52, 15/14, 99/91], [6, 4, 3, 4, 1])&lt;br /&gt;
! = Fokblock([36/35, 22/21, 96/91, 15/14, 99/91], [3, 2, 0, 4, 5])&lt;br /&gt;
! = Fokblock([40/39, 22/21, 55/52, 15/14, 99/91], [0, 2, 6, 4, 2])&lt;br /&gt;
! = Fokblock([40/39, 22/21, 96/91, 15/14, 99/91], [3, 2, 0, 4, 5])&lt;br /&gt;
! = Fokblock([40/39, 27/26, 22/21, 55/52, 15/14], [0, 2, 1, 6, 4])&lt;br /&gt;
! = Fokblock([40/39, 27/26, 55/52, 15/14, 99/91], [0, 4, 6, 4, 1])&lt;br /&gt;
! = Fokblock([40/39, 33/32, 22/21, 15/14, 99/91], [3, 6, 2, 4, 2])&lt;br /&gt;
! = Fokblock([40/39, 33/32, 22/21, 96/91, 15/14], [3, 5, 2, 2, 4])&lt;br /&gt;
! = Fokblock([40/39, 33/32, 27/26, 15/14, 99/91], [3, 6, 4, 4, 1])&lt;br /&gt;
! = Fokblock([40/39, 33/32, 27/26, 22/21, 15/14], [3, 6, 2, 1, 4])&lt;br /&gt;
! = Fokblock([40/39, 33/32, 80/77, 22/21, 96/91], [3, 6, 4, 2, 2])&lt;br /&gt;
! = Fokblock([40/39, 36/35, 22/21, 15/14, 99/91], [0, 0, 2, 4, 5])&lt;br /&gt;
! = Fokblock([40/39, 36/35, 22/21, 55/52, 15/14], [0, 2, 2, 5, 4])&lt;br /&gt;
! = Fokblock([40/39, 65/63, 80/77, 22/21, 15/14], [3, 4, 0, 1, 6])&lt;br /&gt;
! = Fokblock([40/39, 80/77, 22/21, 96/91, 15/14], [3, 1, 2, 2, 6])&lt;br /&gt;
! = Fokblock([55/54, 40/39, 22/21, 55/52, 15/14], [4, 0, 1, 5, 4])&lt;br /&gt;
! = Fokblock([55/54, 40/39, 27/26, 22/21, 15/14], [6, 0, 5, 1, 4])&lt;br /&gt;
! = Fokblock([55/54, 40/39, 36/35, 55/52, 15/14], [2, 0, 1, 5, 4])&lt;br /&gt;
! = Fokblock([55/54, 40/39, 65/63, 22/21, 15/14], [6, 0, 1, 1, 6])&lt;br /&gt;
! = Fokblock([55/54, 40/39, 65/63, 27/26, 22/21], [6, 0, 4, 6, 1])&lt;br /&gt;
! = Fokblock([66/65, 36/35, 22/21, 96/91, 15/14], [5, 3, 2, 0, 6])&lt;br /&gt;
! = Fokblock([66/65, 36/35, 22/21, 96/91, 99/91], [2, 3, 2, 0, 6])&lt;br /&gt;
! = Fokblock([66/65, 40/39, 22/21, 96/91, 15/14], [5, 3, 2, 0, 6])&lt;br /&gt;
! = Fokblock([66/65, 40/39, 22/21, 96/91, 99/91], [2, 3, 2, 0, 6])&lt;br /&gt;
! = Fokblock([66/65, 40/39, 36/35, 22/21, 15/14], [5, 0, 0, 2, 6])&lt;br /&gt;
! = Fokblock([66/65, 40/39, 36/35, 22/21, 99/91], [2, 0, 0, 2, 6])&lt;br /&gt;
! = Fokblock([66/65, 40/39, 65/63, 80/77, 15/14], [1, 3, 2, 0, 6])&lt;br /&gt;
! = Fokblock([66/65, 40/39, 80/77, 22/21, 15/14], [2, 3, 0, 2, 6])&lt;br /&gt;
! = Fokblock([78/77, 40/39, 36/35, 22/21, 15/14], [1, 0, 2, 2, 6])&lt;br /&gt;
! = Fokblock([78/77, 40/39, 36/35, 22/21, 55/52], [4, 0, 2, 2, 6])&lt;br /&gt;
! = Fokblock([78/77, 40/39, 65/63, 22/21, 15/14], [0, 0, 4, 1, 6])&lt;br /&gt;
! = Fokblock([78/77, 55/54, 40/39, 22/21, 15/14], [1, 4, 0, 1, 6])&lt;br /&gt;
! = Fokblock([78/77, 55/54, 40/39, 22/21, 55/52], [4, 4, 0, 1, 6])&lt;br /&gt;
! = Fokblock([78/77, 55/54, 40/39, 36/35, 15/14], [1, 2, 0, 1, 6])&lt;br /&gt;
! = Fokblock([78/77, 55/54, 40/39, 36/35, 55/52], [4, 2, 0, 1, 6])&lt;br /&gt;
! = Fokblock([78/77, 65/63, 80/77, 22/21, 15/14], [3, 4, 0, 1, 6])&lt;br /&gt;
! = Fokblock([78/77, 66/65, 40/39, 22/21, 15/14], [0, 2, 0, 2, 6])&lt;br /&gt;
! = Fokblock([78/77, 66/65, 40/39, 65/63, 15/14], [0, 1, 0, 2, 6])&lt;br /&gt;
! = Fokblock([78/77, 66/65, 65/63, 80/77, 15/14], [3, 1, 2, 0, 6])&lt;br /&gt;
! = Fokblock([78/77, 66/65, 80/77, 22/21, 15/14], [3, 2, 0, 2, 6])&lt;br /&gt;
! = Fokblock([144/143, 40/39, 27/26, 22/21, 15/14], [0, 3, 5, 1, 4])&lt;br /&gt;
! = Fokblock([144/143, 40/39, 33/32, 22/21, 15/14], [2, 3, 5, 1, 4])&lt;br /&gt;
! = Fokblock([144/143, 40/39, 33/32, 80/77, 22/21], [2, 3, 6, 4, 1])&lt;br /&gt;
! = Fokblock([144/143, 40/39, 33/32, 80/77, 96/91], [4, 3, 6, 4, 1])&lt;br /&gt;
! = Fokblock([144/143, 40/39, 33/32, 96/91, 15/14], [4, 3, 5, 1, 4])&lt;br /&gt;
! = Fokblock([144/143, 40/39, 65/63, 22/21, 15/14], [0, 3, 1, 1, 6])&lt;br /&gt;
! = Fokblock([144/143, 40/39, 65/63, 27/26, 22/21], [0, 3, 4, 6, 1])&lt;br /&gt;
! = Fokblock([144/143, 40/39, 80/77, 22/21, 15/14], [2, 3, 1, 1, 6])&lt;br /&gt;
! = Fokblock([144/143, 40/39, 80/77, 96/91, 15/14], [4, 3, 1, 1, 6])&lt;br /&gt;
! = Fokblock([144/143, 55/54, 27/26, 22/21, 15/14], [0, 3, 5, 1, 4])&lt;br /&gt;
! = Fokblock([144/143, 55/54, 65/63, 22/21, 15/14], [0, 3, 1, 1, 6])&lt;br /&gt;
! = Fokblock([144/143, 55/54, 65/63, 27/26, 22/21], [0, 3, 4, 6, 1])&lt;/body&gt;&lt;/html&gt;</pre></div>
Retrieved from "https://en.xen.wiki/w/Otonal_heptad"