Overtone scale: Difference between revisions

This page was originally developed by [[Andrew Heathwaite]], but others are welcome to add to it. For another take on the subject, see [[Mike Sheiman's Very Easy Scale Building From The Harmonic Series Page]]. This article focuses on a systematic approach to building modes of the harmonic series and taking subsets of it, with attention paid to the different kinds of relationships available depending on the starting pitch, or tonic notes. It is not concerned with "purity", "consonance", "naturalness" or avoidance of "dissonance." Here, what might be called dissonant is instead called complex, and the reader is encouraged to explore the sounds of harmonic ratios ranging from the simplest to the most complex. This does not mean that the more complex intervals can be treated exactly the same way as the simpler ones, but that different levels of complexity can be valuable to explore in a tuning system. The usefulness of all this is left to each composer to determine through experimentation.

One way of using the [[harmonic series]] to generate scalar material is to take an octave-long subset of the series and make it repeat at the octave. So for instance, starting at the fifth harmonic and continuing up the sequence to the tenth harmonic (which is a doubling of five, and thus an octave higher) produces a pentatonic scale:

|}

Another way to describe Mode 5 is that it is an example of an "Over-5 Scale." As 5 is octave-redundant with 10, 20, 40, 80 etc, any scale with one of those (the form is technically 2<span style="vertical-align: super;">n</span>*5, where n is any integer greater than or equal to zero) in the denominator of every tone could be called an Over-5 Scale. So let's consider Mode 10 -- 10:11:12:13:14:15:16:17:18:19:20 --

Notice that the 15th harmonic is a 3/2 above 10. Although this may look like it breaks the Over-5 rule, it's just a reduced form of 15/10, which has a number of the form 2ⁿ*5 in the denominator. 10 may be too many notes for a particular purpose; we could take a subset of Mode 10 -- for instance 10:11:13:15:17:20, and it would also be an Over-5 scale. Below are some of the simplest Over-n scales as Modes of the Harmonic Series. All of them are ripe for the taking of subsets.

; Mode 1: 1:2 -- only one tone.

Over-1 scales have a very strong attraction to their tonic, which is the fundamental of the series. Other Over-n scales may have more complex relationships to their tonics, which are not fundamentals. Indeed, when taking subsets, the fundamental may not even be present.

[[File:Ji-3-4-5-6-csound-foscil-220hz.mp3|right|thumb|270px|Mode 3 scale]]

; Mode 24: 24:25:26:27:28:29:30:31:32:33:34:35:36:37:38:39:40:41:42:43:44:45:46:47:48 -- a great variety available here, with 47 as the highest prime. Added to the classic major and septimal subminor triads, we have a 29-limit supraminor triad -- 24:29:36 and 31-limit supermajor triad -- 24:31:36. Andrew Heathwaite has refretted a mountain dulcimer to this scale (and has plans to refret more instruments to match). There are 3/2 perfect fifths available from 1, 3, 5, 7, 9, 11, and 13, allowing the possibility of making Over-n scales that start on any of those pitches.

[[File:Ji-5-6-7-8-9-10-csound-foscil-220hz.mp3|right|thumb|270px|Mode 5: 5:6:7:8:9:10]]

; Mode 20: 20:21:22:23:24:25:26:27:28:29:30:31:32:33:34:35:36:37:38:39:40 — this has a lot of variety as it great for making subsets. In addition to the chords above, there's a 4:5:6:7 tetrad on 20:25:30:35. There's also a 23-limit inframinor triad on 20:23:30 and a variety of sevenths.

[[File:Ji-7-8-9-10-11-12-13-14-csound-foscil-220hz.mp3|right|thumb|270px|Mode 7: 7:8:9:10:11:12:13:14]]

; Mode 14: 14:15:16:17:18:19:20:21:22:23:24:25:26:27:28 -- 21 is 3/2 above 14, so we can get some [[List_of_root-3rd-P5_triads_in_JI|root-3rd-P5 triads]], such as 14:18:21, a septimal supermajor triad, which also sounds good with 27/14 -- a supermajor seventh; 14:17:21, a septendecimal ([[17-limit]]) supraminor triad, which works well with a 13/7 low major seventh. [[19/14]] is notable here as a wide and complex perfect fourth.

[[File:Ji-9-10-11-12-13-14-15-16-17-18-csound-foscil-220hz.mp3|right|thumb|270px|Mode 9 scale]]

; Mode 18: 18:19:20:21:22:23:24:25:26:27:28:29:30:31:32:33:34:35:36 -- now we have 27, a 3/2 above with bass, which allows 18:22:27:33, an undecimal neutral seventh chord; and 18:23:27, a [[23-limit]] supermajor triad (close to [[17edo]]). It's also worth noting that the entirety of Mode 6 is available here starting on 18 -- 18:21:24:27:30:33:36.

[[File:Ji-11-12-13-14-15-16-17-18-19-20-21-22-csound-foscil-220hz.mp3|right|thumb|270px|Mode 11 scale]]

; Mode 22: 22:23:24:25:26:27:28:29:30:31:32:33:34:35:36:37:38:39:40:41:42:43:44 -- with 33, we have a perfect fifth above the bass and can make such root-3rd-P5 triads as 22:26:33, a middle "Gothic" tridecimal minor triad; 22:27:33, an undecimal neutral triad; 22:28:23, a "Gothic" undecimal supermajor triad. The sevenths are all complex, ranging from an [[interseptimal]] [[19/11]]; to a neutral seventh [[20/11]] (close to that of [[22edo]]); to a wide major seventh at [[21/11]].

; Mode 13: 13:14:15:16:17:18:19:20:21:22:23:24:25:26

; Mode 26: 26:27:28:29:30:31:32:33:34:35:36:37:38:39:40:41:42:43:44:45:46:47:48:49:50:51:52 -- 39/26 is a 3/2 perfect fifth. Root-3rd-P5 chords include the tridecimal inframinor 26:30:39; a 31-limit minor triad at 26:31:39 (oddly normal-sounding on its own); a tridecimal neutral triad at 26:32:39; and a wide tridecimal major at 26:33:39. As odd harmonics go up to 51, a great variety is possible here.

; Mode 15: 15:16:17:18:19:20:21:22:23:24:25:26:27:28:29:30

[[Primodality]] involves the use of large prime modes of the harmonic series.  

[[Andrew Heathwaite]] proposes a solfege system for harmonics 16-32 (Mode 16):

Thus, the pentatonic scale in the example at the top (Mode 5) could be sung: '''mi sol ta do re mi'''

For those interested in learning to sing and hear just intervals, here are twelve of the simplest otonal scales to try. I leave it up to the curious learner to decide the value, beauty, or usefulness of these particular scales for their compositional purposes.

|}

Here are some next steps:

* Borrow overtones &amp; undertones from the overtones &amp; undertones of the fundamental -- this process can produce rich fields of interlocking harmonic series, and is often the sort of thing that composers do when they're composing in just intonation. Harry Partch's "Monophonic Fabric," which consists of 43 unequal tones per octave, is one famous example.

* [[8th Octave Overtone Tuning]]