Overtone scales

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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.

Introduction - Modes of the Harmonic Series

One way of using the overtone 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 overtone and continuing up the sequence to the tenth overtone (which is a doubling of five, and thus an octave higher) produces a pentatonic scale:

overtone 5 6 7 8 9 10
JI ratio 1/1 6/5 7/5 8/5 9/5 2/1

Another way to write this would be 5:6:7:8:9:10, which shows that the tones form both a scale and a chord; indeed, it is a 9-limit pentad with 5 in the bass. Denny Genovese would call the above scale "Mode 5 of the Harmonic Series," or "Mode 5" for short. Further examples will be given with a mode number indicated.

Any Mode of the Harmonic Series has the characteristic of containing all superparticular steps ("superparticular" refers to ratios of the form n/(n-1)) that are decreasing in pitch size as one ascends the scale). So for Mode 5 above we have:

steps 6:5 7:6 8:7 9:8 10:9
common name just minor third septimal subminor third septimal supermajor second large major second small major second

Over-n Scales

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 2n*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 --

overtone 10 11 12 13 14 15 16 17 18 19 20
JI ratio 1/1 11/10 6/5 13/10 7/5 3/2 8/5 17/10 9/5 19/10 2/1

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.

Over-1 Scales

Mode 1
1:2 -- only one tone.
Mode 2
2:3:4 -- one tone and perfect fifth (plus octaves). Rather limited.
Mode 4
4:5:6:7:8 -- this is the classic 7-limit tetrad. In music, it appears as a chord more often than a scale, but it could be used either way. It includes the classic major triad, 4:5:6, with a harmonic seventh. Another triad available here over the bass is 4:6:7, which includes a perfect fifth and a harmonic seventh, but no major third.
Mode 8
8:9:10:11:12:13:14:15:16 -- an eight-tone scale, or a 13-limit octad. This is a very effective scale, with complexity ranging from the simple 4:5:6 major triad above (or even a 2:3:4 open fifth chord) to chords involving 13 and 11 such as the wild 9:11:13:15 tetrad. Dante Rosati calls it the "Diatonic Harmonic Series Scale" and has refretted a guitar to play it. See: First Five Octaves of the Harmonic Series and otones8-16.
Mode 16
16:17:18:19:20:21:22:23:24:25:26:27:28:29:30:31:32 -- Dante calls this the "Chromatic Harmonic Series Scale." It includes a 19-limit minor chord, 16:19:24, in addition the the classic major. Incorporating overtones through the 31st, a great variety in complexity is possible. As 16 is a lot of tones to use at once, this is a good scale for making modal subsets of. Andrew Heathwaite recommends his heptatonic "remem" scale -- 16:17:18:21:24:26:28:32 -- or his extended nonatonic "remem" scale which adds 19 and 23 -- 16:17:18:19:21:23:24:26:28:32.

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.

Over-3 Scales

Mode 3
3:4:5:6 -- a major triad in 2nd inversion -- that is, with the perfect fifth in the bass.
Mode 6
6:7:8:9:10:11:12 -- an effective 6-tone scale. 9 is 3/2 above 3, so there is a perfect fifth above the bass. A septimal subminor triad -- 6:7:9 -- is available, as well as an undecimal 6:7:9:11 tetrad, which adds a neutral seventh of 11/6 to the septimal subminor triad. Try also 6:9:11, which contains a perfect fifth and 11/6 neutral seventh, but no third above the bass.
Mode 12
12:13:14:15:16:17:18:19:20:21:22:23:24 -- as this scale has 12 tones, it fits nicely onto a traditional keyboard instrument, such as piano, melodica, organ, accordion, etc. It allows a 4:5:6:7 septimal tetrad above the bass (a reduced form of 12:15:18:21) as well as the subminor triad and undecimal tetrad given available in Mode 6. The fundamental is 4/3 above the bass, making 4/3 a strong attractor in the system. Andrew Heathwaite has composed with a 12:13:14:16:18:20:22:24 subset, and Jacob Barton retuned an electric organ to this scale. See otones12-24.
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.

Over-5 Scales

Mode 5
5:6:7:8:9:10 — This is essentially a 7-limit fully-diminished seventh chord. 7/5 makes a very nice tritone above the bass -- the simplest one available in JI -- and it's available in all the higher Over-5 modes as well. This is the scale used by the Wagogo people of Tanzania.
Mode 10
10:11:12:13:14:15:16:17:18:19:20 — from 10 to 15 is a 3/2 perfect fifth. We have access to a 10:12:15 classic minor triad, as well as a number of other nice chords like the 10:13:15 barbados triad. 11/10 makes a strange second (or ninth), while 9/5 makes a very nice minor seventh (and an alternative to the 7/4 bluesy seventh of Over-1 scales and 11/6 neutral seventh of Over-3 scales.
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.

Over-7 Scales

Mode 7
7:8:9:10:11:12:13:14 -- with no 3/2 perfect fifth, it may be difficult to make 7 sound like tonic here.
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 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.

Over-9 Scales

Mode 9
9:10:11:12:13:14:15:16:17:18 -- again, lacking a 3/2 above the bass, it's hard to make 9 sound like tonic.
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.

Over-11 Scales

Mode 11
11:12:13:14:15:16:17:18:19:20:21:22
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.

Over-13 Scales

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.

Over-15 Scales

Mode 15
15:16:17:18:19:20:21:22:23:24:25:26:27:28:29:30
Mode 30
30:31:32:33:34:35:36:37:38:39:40:41:42:43:44:45:46:47:48:49:50:51:52:53:54:55:56:57:58:59:60

Mode 30 in particular is interesting because 30 is the product of the first three primes, so it's a fairly good choice if we want a tonic that isn't a power of two. It contains modes 6 and 10 as subsets. We have the classic minor triad (from 10), the subminor triad (from 6), two major triads in 30:37:45 and 30:38:45, and the Barbados triad of 30:39:45. Chords not based on the tonic include the harmonic seventh chord (32:40:48:56). A good 13-limit subset with 16 notes in it is 30:32:33:35:36:39:40:42:44:45:48:49:50:54:55:56:60.

Primodality

Zhea Erose has considered over-p scales and chords where p is a prime, which she calls primodal scales. Primodality is an approach to JI designed to emphasize the identity of the "tonic" as the pth harmonic. Scales and chords having the identity of the prime p as the tonic are collectively called a prime family, and can be denoted simply by /p. Zhea also uses various adjectives for specific primodalities, such as septimal, undecimal, tridecimal, septendecimal, novem(decimal) for /7, /11, /13, /17, /19, which are not to be confused with the use of these adjectives to denote prime limits.

Most importantly, primodality sees any overtone as valuable on its own, rather than relative to some fundamental. Taking a specific overtone as a tonic we can get its particular scales and colors and even versions of "non-xenharmonic" scales, even when the corresponding fundamental is too low to be audible. In particular, primodality discards the concept of harmonic limit, which Zhea considers an artificial way to look at JI harmony. Zhea argues that prime families are a more natural way to categorize intervals; intervals from the same prime family (intervals with a common denominator for example, all /2, all /11 or all /13) tend to blend better together. For example, it is preferable to add 21/16 to 4:5:6:7, rather than 4/3.

To construct a primodal scale, we fix a prime p to be the denominator and take intervals of the form n/p, where n ≥ p. Zhea often takes n to range over a certain "lineal segment" (Mode mp of the harmonic series where m is a positive integer) or a subset thereof.. For example, if we use p = 13 and take all n between 13 and 26 (inclusive), this would result in the scale 13:14:15:16:17:18:19:20:21:22:23:24:25:26. We may add a 3/2 to the scale root, which corresponds to adding 3p/p. (3/2 is a natural "halfway point" for harmonic scales, since if N is even, Mode N has a 3/2 as its N/2-th note.)

Primodality emphasizes unique characters of primes, as opposed to lower-complexity JI identities such as 4:5:6:7:9 that are more common in composite modes. Mode p and Mode 2p (called respectively the first and second octaves of /p) are considered the most important for the prime's identity: for any prime p, the set of harmonics from p to 2p is unique in the sense that the sets {p/p, ..., 2p/p} and {n/n, ..., 2n/n} only intersect at {1/1, 2/1} for any positive integer n < p. Similarly, the second octaves of p and the second octave of any n < p only intersect at {1/1, 3/2, 2/1}.

Approximations of edos in certain primodalities, called nejis ("near-equal JI"), can be used to explore a prime family, while keeping the transposability, scale structures, rank-2 harmonic theory, notation, etc. associated with that edo.

Zhea Erose's theory also deals with modulations between different prime families, and combining different prime families into one scale.

Music

See also

A Solfege System

Andrew Heathwaite proposes a solfege system for overtones 16-32 (Mode 16):

overtone 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
JI ratio 1/1 17/16 9/8 19/16 5/4 21/16 11/8 23/16 3/2 25/16 13/8 27/16 7/4 29/16 15/8 31/16 2/1
solfege do ra re me mi fe fu su sol le lu la ta tu ti da do

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

Twelve Scales

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.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Mode 1 1-note do do
Mode 2 2-note do sol do
Mode 3 3-note sol do mi sol
Mode 4 4-note do mi sol ta do
Mode 5 5-note mi sol ta do re mi
Mode 6 6-note sol ta do re mi fu sol
Mode 7 7-note ta do re mi fu sol lu ta
Mode 8 8-note do re mi fu sol lu ta ti do
Mode 9 9-note re mi fu sol lu ta ti do ra re
Mode 10 10-note mi fu sol lu ta ti do ra re me mi
Mode 11 11-note fu sol lu ta ti do ra re me mi fe fu
Mode 12 12-note sol lu ta ti do ra re me mi fe fu su sol

Next Steps

Here are some next steps:

  • Go beyond the 24th overtone (eg. overtones 16-32 or higher).
  • Experiment with using different pitches as the "tonic" of the scale (eg. sol lu ta do re mi fu sol, which could be taken as the 7-note scale starting on sol).
  • Take subsets of larger scales, which are not strict adjacent overtone scales (eg. do re fe sol ta do).
  • Learn the inversions of these scales, which would be undertone scales. (Undertone scales would have smaller steps at the bottom of the scale, which would get larger as one ascends.)
  • Borrow overtones & undertones from the overtones & 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.

See Also