Overtone scale

An overtone scale is a scale made from an octave-spanning segment of the harmonic series, repeated at the octave. Overtone scales are also called modes of the harmonic series. Their inverses, made from the subharmonic series, are undertone scales.

This article focuses on a systematic approach to building overtone scales, 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.

Modes of the harmonic series

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:

Harmonic	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-odd-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.

A mode in other musical contexts is usually a different rotation of the same intervals. In the case of different harmonic modes, that is not exactly the case. However, in some sense it is a reasonable comparison, because as you slide the subset of harmonics around, you are essentially sampling different segments of integers whose prime factorizations follow simple, constant patterns (every 2nd number has a 2, every 3rd number has a 3, every 5th number has a 5) and therefore the full internal interval set (all dyads, triads, tetrads, etc.) from one mode to the next is more alike than it is different.

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 or /5 scale. As 5 divides 10, 15, 20, etc., any scale with one of those in the denominator of every tone could be called an over-5 scale. So let us consider mode 10 – 10:11:12:13:14:15:16:17:18:19:20 –

Harmonic	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 is just a reduced form of 15/10, which has 10 in the denominator. Any subset of mode 10 – for instance 10:11:13:15:17:20, can also be considered an over-5 scale. Because 15 is divisible by 3 and 5, mode 15 is both a /3 and /5 scale, though usually the largest divisor is considered the most significant to its sound.

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-2 scales

Mode 2

2:3:4 – one tone and perfect fifth (plus octaves). Rather limited.

Mode 4 scale

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

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.

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 to the classic major. Incorporating harmonics 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-2 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 scale

Mode 3

3:4:5:6 – a major triad in 2nd inversion – that is, with the perfect fifth in the bass.

Mode 6 scale

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 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 the Scala file.

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

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 is available in all the higher Over-5 modes as well.

Mode 10 scale

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 is great for making subsets. In addition to the chords above, there is a 4:5:6:7 tetrad on 20:25:30:35. There is 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

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 scale

Mode 9

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

Ptolemy gave the 9:10:11:12 tuning of the tetrachord, calling it διάτονον ὁμᾰλόν (diatonon homalon, "equal/even diatonic") and describing it as "foreign" or "rustic"; the 12/11 neutral second is common in Southwest Asian music.

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 is 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 scale

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; 22:28:33, 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 is a fairly good choice if we want a tonic that is not 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.

Solfege systems

Andrew Heathwaite proposes a solfege system for harmonics 16–32 (Mode 16):

Harmonic	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

Kite Giedraitis proposes a solfege for harmonics 16–32 based on his uniform solfeges. This solfege is easy to remember because there is a uniform vowel sequence for each degree: -o (-i) -a. These vowels stand for down, mid and plain.

Harmonic	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	da	ro	ra	mo	ma	fo	fa	so	sa	lo	li	la	to	ti	ta	do	da

Other techniques

• Extending beyond the 24th harmonic, for example with the segments 16::32, 32::64 or 128::256.
• Using different pitches as the tonic of the scale (e.g. 12:13:14:16:18:20:22:24, which is 7::14 starting on the 12th harmonic).
• Taking subsets of larger scales, which are not strict adjacent overtone scales (e.g. 16:18:21:24:28:32). The 12-note Carlos harmonic scale, a subset of 16::32, is a well-known example; see also ringer scales.
• Inverting these scales gives the undertone scales, which are built from the subharmonic series (see e.g. subharmonics 16–32). Undertone scales have smaller steps at the bottom of the scale, which get larger as one ascends.
• Borrowing overtones and undertones from the overtones and 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 are composing in just intonation. Harry Partch's "Monophonic Fabric", which consists of 43 unequal tones per octave, is one famous example.

See also

• Harmonic series
• Carlos harmonic scale: A well-known 12-note subset of 16::32
• Ringer scale: A useful family of overtone scales that are a constant structure
• Primodality: A framework built around prime modes of the harmonic series
• AFDO: An alternative framework for describing scales which overlaps substantially with overtone scales
• 8th octave overtone tuning: A tuning system built within a large overtone scale, mode 128 of the harmonic series