Overtone scale: Difference between revisions
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=== Over-1 Scales === | === Over-1 Scales === | ||
Mode 1 | ; Mode 1: 1:2 -- only one tone. | ||
Mode 2 | ; Mode 2: 2:3:4 -- one tone and perfect fifth (plus octaves). Rather limited. | ||
Mode 4 | ; 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 | ; 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 | ; 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-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. | ||
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=== Over-3 Scales === | === Over-3 Scales === | ||
Mode 3 | ; Mode 3: 3:4:5:6 -- a major triad in 2nd inversion -- that is, with the perfect fifth in the bass. | ||
Mode 6 | ; 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 | ; 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 | ; 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 === | === Over-5 Scales === | ||
[[File:Ji-5-6-7-8-9-10-csound-foscil-220hz.mp3|right|thumb|270px|Mode 5: 5:6:7:8:9:10]] | [[File:Ji-5-6-7-8-9-10-csound-foscil-220hz.mp3|right|thumb|270px|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's available in all the higher Over-5 modes as well. This is the scale used by the Wagogo people of Tanzania. | ; 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. | ||
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=== Over-7 Scales === | === Over-7 Scales === | ||
Mode 7 | ; 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 | ; 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. | ||
=== Over-9 Scales === | === Over-9 Scales === | ||
Mode 9 | ; 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 | ; 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 === | === Over-11 Scales === | ||
Mode 11 | ; Mode 11: 11:12:13:14:15:16:17:18:19:20:21:22 | ||
Mode 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 === | === Over-13 Scales === | ||
Mode 13 | ; Mode 13: 13:14:15:16:17:18:19:20:21:22:23:24:25:26 | ||
Mode 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 === | === Over-15 Scales === | ||
Mode 15 | ; Mode 15: 15:16:17:18:19:20:21:22:23:24:25:26:27:28:29:30 | ||
Mode 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. | 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. | ||