Overtone scale: Difference between revisions

This page was originally developed by [[Andrew_Heathwaite|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|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 intead 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.

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|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|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:

| | 6:5

| | [[11/10|11/10]]

| | 6/5

| | [[13/10|13/10]]

| | 7/5

| | [[3/2|3/2]]

| | 8/5

| | [[17/10|17/10]]

| | 9/5

| | [[19/10|19/10]]

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 2<span style="vertical-align: super;">n</span>*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 4 -- 4:5:6:7:8 -- this is the classic [[7-limit|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|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|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|First Five Octaves of the Harmonic Series]] and [[otones8-16|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|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.

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|Jacob Barton]] retuned an electric organ to this scale. See [[otones_12-24|otones 12-24]].

Mode 5 -- 5:6:7:8:9:10 -- This is essentially a [[7-limit|7-limit]] fully-diminished seventh chord. [[7/5|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.

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|17-limit]]) supraminor triad, which works well with a 13/7 low major seventh. [[19/14|19/14]] is notable here as a wide and complex perfect fourth.

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|23-limit]] supermajor triad (close to [[17edo|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.

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|interseptimal]] [[19/11|19/11]]; to a neutral seventh [[20/11|20/11]] (close to that of [[22edo|22edo]]); to a wide major seventh at [[21/11|21/11]].

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

<ul><li>Go beyond the 24th overtone (eg. overtones 16-32 or higher).</li><li>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''').</li><li>Take subsets of larger scales, which are not strict adjacent overtone scales (eg. '''do re fe sol ta do''').</li><li>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.)</li><li>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.</li></ul>      [[Category:ji]]