- For the tritave-equivalent MOS structure with the same step pattern, see 5L 3s (3/1-equivalent).
5L 3s, also called oneirotonic, is a moment of symmetry scale consisting of 5 large steps and 3 small steps, repeating every octave. This scale is made using a generator ranging from 450¢ to 480¢, or from 720¢ to 750¢.
The TAMNAMS system is used in this article to name 5L 3s intervals and step size ratios and step ratio ranges.
The notation used in this article is J Ultharian (LsLLsLsL) = JKLMNOPQJ, unless specified otherwise. We denote raising and lowering by a chroma (L − s) by & "amp" and @ "at". (Mnemonics: & "and" means additional pitch. @ "at" rhymes with "flat".)
The chain of perfect 3-mossteps becomes: ... P@ K@ N@ Q L O J M P K N Q& L& O& ...
Thus the 13edo gamut is as follows:
J J&/K@ K/L@ L/K& L&/M@ M M&/N@ N/O@ O/N& O&/P@ P/Q@ Q/P& Q&/J@ J
The 18edo gamut is notated as follows:
J K@ J&/L@ K L K&/M@ L& M N@ M&/O@ N O N&/P@ O&/Q@ P Q P&/J@ Q& J
The 21edo gamut:
J J& K@ K K&/L@ L L& M@ M M& N@ N N&/O@ O O& P@ P P&/Q@ Q Q& J@ J
The TAMNAMS system suggests the name oneirotonic (/oʊnaɪrəˈtɒnɪk/ oh-ny-rə-TON-ik or /ənaɪrə-/ ə-ny-rə-) or 'oneiro' for short. The name oneirotonic (from Greek oneiros 'dream') is coined after the Dreamlands in H.P. Lovecraft's Dream Cycle mythos.
'Father' is sometimes also used to denote 5L 3s, but it's a misnomer, as father is technically an abstract regular temperament (although a very inaccurate one), not a generator range. There are father tunings which generate 3L 5s. A more correct but still not quite correct name would be 'father' or 'father octatonic'. "Father" is also vague regarding the number of notes, because optimal generators for it also generate 3L 2s.
The table of oneirotonic intervals below takes the flat fourth as the generator. Given the size of the subfourth generator g, any oneirotonic interval can easily be found by noting what multiple of g it is, and multiplying the size by the number k of generators it takes to reach the interval and reducing mod 1200 if necessary (so you can use "k*g % 1200" for search engines, for plugged-in values of k and g). For example, since the major 2-step is reached by six subfourth generators, 18edo's major 2-step is 6*466.67 mod 1200 = 2800 mod 1200 = 400¢, same as the 12edo major third.
Note: In TAMNAMS, a k-step interval class in oneirotonic may be called a "k-step", "k-mosstep", or "k-oneirostep". We discourage using 1-indexed terms such as "mos(k+1)th" for non-diatonic mosses in TAMNAMS.
|Notation (1/1 = J)||TAMNAMS name||In L's and s's||# generators up||Notation of 2/1 inverse||TAMNAMS name||In L's and s's|
|The 8-note MOS has the following intervals (from some root):|
|0||J||perfect unison||0L + 0s||0||J||octave||5L + 3s|
|1||M||perfect 3-step||2L + 1s||-1||O||perfect 5-step||3L + 2s|
|2||P||major 6-step||4L + 2s||-2||L||minor 2-step||1L + 1s|
|3||K||major (1-)step||1L + 0s||-3||Q||minor 7-step||4L + 3s|
|4||N||major 4-step||3L + 1s||-4||N@||minor 4-step||2L + 2s|
|5||Q&||major 7-step||5L + 2s||-5||K@||minor (1-)step||0L + 1s|
|6||L&||major 2-step||2L + 0s||-6||P@||minor 6-step||3L + 3s|
|7||O&||augmented 5-step||4L + 1s||-7||M@||diminished 3-step||1L + 2s|
|The chromatic 13-note MOS (either 5L 8s, 8L 5s, or 13edo) also has the following intervals (from some root):|
|8||J&||augmented 0-step (aka moschroma)||1L - 1s||-8||J@||diminished 8-step (aka diminished mosoctave)||4L + 4s|
|9||M&||augmented 3-step||3L + 0s||-9||O@||diminished 5-step||2L + 3s|
|10||P&||augmented 6-step||5L + 1s||-10||L@||diminished 2-step||0L + 2s|
|11||K&||augmented 1-step||2L - 1s||-11||Q@||diminished 7-step||3L + 4s|
|12||N&||augmented 4-step||4L + 0s||-12||N@@||diminished 4-step||1L + 3s|
Table of intervals in the simplest oneirotonic tunings:
|Degree||Size in 13edo (basic)||Size in 18edo (hard)||Size in 21edo (soft)||Note name on J||#Gens up|
|unison||0\13, 0.00||0\18, 0.00||0\21, 0.00||J||0|
|minor step||1\13, 92.31||1\18, 66.67||2\21, 114.29||K@||-5|
|major step||2\13, 184.62||3\18, 200.00||3\21, 171.43||K||+3|
|minor 2-step||3\13, 276.92||4\18, 266.67||5\21, 285.71||L||-2|
|major 2-step||4\13, 369.23||6\18, 400.00||6\21, 342.86||L&||+6|
|dim. 3-step||4\13, 369.23||5\18, 333.33||7\21, 400.00||M@||-7|
|perf. 3-step||5\13, 461.54||7\18, 466.67||8\21, 457.14||M||+1|
|minor 4-step||6\13, 553.85||8\18, 533.33||10\21, 571.43||N@||-4|
|major 4-step||7\13, 646.15||10\18, 666.66||11\31, 628.57||N||+4|
|perf. 5-step||8\13, 738.46||11\18, 733.33||13\21, 742.86||O||-1|
|aug. 5-step||9\13, 830.77||13\18, 866.66||14\21, 800.00||O&||+7|
|minor 6-step||9\13, 830.77||12\18, 800.00||15\21, 857.14||P@||-6|
|major 6-step||10\13, 923.08||14\18, 933.33||16\21, 914.29||P||+2|
|minor 7-step||11\13, 1015.39||15\18, 1000.00||18\21, 1028.57||Q||-3|
|major 7-step||12\13, 1107.69||17\18, 1133.33||19\21, 1085.71||Q&||+5|
Hypohard oneirotonic tunings (with generator between 5\13 and 7\18) have step ratios between 2/1 and 3/1.
Hypohard oneirotonic can be considered "meantone oneirotonic". This is because these tunings share the following features with meantone diatonic tunings:
- The large step is a "meantone", somewhere between near-10/9 (as in 13edo) and near-9/8 (as in 18edo).
- The major 2-mosstep (made of two large steps) is a meantone- to flattone-sized major third, thus is a stand-in for the classical diatonic major third.
The set of identifications above is associated with A-Team temperament.
- 13edo has characteristically small 1-mossteps of about 185¢. It is uniformly compressed 12edo, so it has distorted versions of non-diatonic 12edo scales. It essentially has the best 11/8 out of all hypohard tunings.
- 18edo can be used for a large step ratio of 3, (thus 18edo oneirotonic is distorted 17edo diatonic, or for its nearly pure 9/8 and 7/6. It also makes rising fifths (733.3¢, a perfect 5-mosstep) and falling fifths (666.7¢, a major 4-mosstep) almost equally off from a just perfect fifth. 18edo is also more suited for conventionally jazz styles due to its 6-fold symmetry.
- 31edo can be used to make the major 2-mosstep a near-just 5/4.
- 44edo (generator 17\44 = 463.64¢), 57edo (generator 22\57 = 463.16¢), and 70edo (generator 27\70 = 462.857¢) offer a compromise between 31edo's major third and 13edo's 11/8 and 13/8. In particular, 70edo has an essentially pure 13/8.
The sizes of the generator, large step and small step of oneirotonic are as follows in various hypohard oneiro tunings.
|13edo (basic)||18edo (hard)||31edo (semihard)|
|generator (g)||5\13, 461.54||7\18, 466.67||12\31, 464.52|
|L (3g - octave)||2\13, 184.62||3\18, 200.00||5\31, 193.55|
|s (-5g + 2 octaves)||1\13, 92.31||1\18, 66.67||2\31, 77.42|
Sortable table of major and minor intervals in hypohard oneiro tunings:
|Degree||Size in 13edo (basic)||Size in 18edo (hard)||Size in 31edo (semihard)||Note name on J||Approximate ratios||#Gens up|
|unison||0\13, 0.00||0\18, 0.00||0\31, 0.00||J||1/1||0|
|minor step||1\13, 92.31||1\18, 66.67||2\31, 77.42||K@||21/20, 22/21||-5|
|major step||2\13, 184.62||3\18, 200.00||5\31, 193.55||K||9/8, 10/9||+3|
|minor 2-step||3\13, 276.92||4\18, 266.67||7\31, 270.97||L||7/6||-2|
|major 2-step||4\13, 369.23||6\18, 400.00||10\31, 387.10||L&||5/4||+6|
|dim. 3-step||4\13, 369.23||5\18, 333.33||9\31, 348.39||M@||16/13, 11/9||-7|
|perf. 3-step||5\13, 461.54||7\18, 466.67||12\31, 464.52||M||21/16, 13/10, 17/13||+1|
|minor 4-step||6\13, 553.85||8\18, 533.33||14\31, 541.94||N@||11/8||-4|
|major 4-step||7\13, 646.15||10\18, 666.66||17\31, 658.06||N||13/9, 16/11||+4|
|perf. 5-step||8\13, 738.46||11\18, 733.33||19\31, 735.48||O||26/17||-1|
|aug. 5-step||9\13, 830.77||13\18, 866.66||22\31, 851.61||O&||13/8, 18/11||+7|
|minor 6-step||9\13, 830.77||12\18, 800.00||21\31, 812.90||P@||8/5||-6|
|major 6-step||10\13, 923.08||14\18, 933.33||24\31, 929.03||P||12/7||+2|
|minor 7-step||11\13, 1015.39||15\18, 1000.00||26\31, 1006.45||Q||9/5, 16/9||-3|
|major 7-step||12\13, 1107.69||17\18, 1133.33||29\31, 1122.58||Q&||+5|
- The ratio interpretations that are not valid for 18edo are italicized.
Hyposoft oneirotonic tunings (with generator between 8\21 and 5\13) have step ratios between 3/2 and 2/1. The 8\21-to-5\13 range of oneirotonic tunings remains relatively unexplored. In these tunings,
- the large step of oneirotonic tends to be intermediate in size between 10/9 and 11/10; the small step size is a semitone close to 17/16, about 92¢ to 114¢.
- The major 2-mosstep (made of two large steps) in these tunings tends to be more of a neutral third, ranging from 6\21 (342¢) to 4\13 (369¢).
- 21edo's P1-L1ms-L2ms-L4ms approximates 9:10:11:13 better than the corresponding 13edo chord does. 21edo will serve those who like the combination of neogothic minor thirds (285.71¢) and Baroque diatonic semitones (114.29¢, close to quarter-comma meantone's 117.11¢).
- 34edo's 9:10:11:13 is even better.
The sizes of the generator, large step and small step of oneirotonic are as follows in various hyposoft oneiro tunings (13edo not shown).
|21edo (soft)||34edo (semisoft)|
|generator (g)||8\21, 457.14||13\34, 458.82|
|L (3g - octave)||3\21, 171.43||5\34, 176.47|
|s (-5g + 2 octaves)||2\21, 114.29||3\34, 105.88|
Sortable table of major and minor intervals in hyposoft tunings (13edo not shown):
|Degree||Size in 21edo (soft)||Size in 34edo (semisoft)||Note name on J||Approximate ratios||#Gens up|
|unison||0\21, 0.00||0\34, 0.00||J||1/1||0|
|minor step||2\21, 114.29||3\34, 105.88||K@||16/15||-5|
|major step||3\21, 171.43||5\34, 176.47||K||10/9, 11/10||+3|
|minor 2-step||5\21, 285.71||8\34, 282.35||L||13/11, 20/17||-2|
|major 2-step||6\21, 342.86||10\34, 352.94||L&||11/9||+6|
|dim. 3-step||7\21, 400.00||11\34, 388.24||M@||5/4||-7|
|perf. 3-step||8\21, 457.14||12\31, 458.82||M||13/10||+1|
|minor 4-step||10\21, 571.43||16\34, 564.72||N@||18/13, 32/23||-4|
|major 4-step||11\21, 628.57||18\34, 635.29||N||13/9, 23/16||+4|
|perf. 5-step||13\21, 742.86||21\34, 741.18||O||20/13||-1|
|aug. 5-step||14\21, 800.00||23\34, 811.77||O&||8/5||+7|
|minor 6-step||15\21, 857.14||24\34, 847.06||P@||18/11||-6|
|major 6-step||16\21, 914.29||26\34, 917.65||P||22/13, 17/10||+2|
|minor 7-step||18\21, 1028.57||29\34, 1023.53||Q||9/5||-3|
|major 7-step||19\21, 1085.71||31\34, 1094.12||Q&||15/8||+5|
Parasoft to ultrasoft tunings
The range of oneirotonic tunings of step ratio between 6/5 and 3/2 (thus in the parasoft to ultrasoft range) may be of interest because it is closely related to porcupine temperament: these tunings equate three oneirotonic large steps to a diatonic perfect fourth, i.e. they equate the oneirotonic large step to a porcupine generator. [This identification may come in handy since many altered oneirotonic modes have three consecutive large steps.] The chord 10:11:13 is very well approximated in 29edo.
The sizes of the generator, large step and small step of oneirotonic are as follows in various tunings in this range.
|generator (g)||11\29, 455.17||14\37, 454.05|
|L (3g - octave)||4\29, 165.52||5\37, 162.16|
|s (-5g + 2 octaves)||3\29, 124.14||4\37, 129.73|
The intervals of the extended generator chain (-15 to +15 generators) are as follows in various softer-than-soft oneirotonic tunings.
|Degree||Size in 29edo (supersoft)||Note name on J||Approximate ratios (29edo)||#Gens up|
|dim. step||2\29, 82.8||K@@||-13|
|minor step||3\29, 124.1||K@||14/13||-5|
|major step||4\29, 165.5||K||11/10||+3|
|aug. step||5\29, 206.9||K&||9/8||+11|
|dim. 2-step||6\29, 248.3||L@||15/13||-10|
|minor 2-step||7\29, 289.7||L||13/11||-2|
|major 2-step||8\29, 331.0||L&||+6|
|aug. 2-step||9\29, 372.4||L&&||+14|
|doubly dim. 3-step||9\29, 372.4||M@@||-15|
|dim. 3-step||10\29, 413.8||M@||14/11||-7|
|perf. 3-step||11\29, 455.2||M||13/10||+1|
|aug. 3-step||12\29, 496.6||M&||4/3||+9|
|dim. 4-step||13\29, 537.9||N@@||15/11||-12|
|minor 4-step||14\29, 579.3||N@||7/5||-4|
|major 4-step||15\29 620.7||N||10/7||+4|
|aug. 4-step||16\29 662.1||N&||22/15||+12|
|dim. 5-step||17\29, 703.4||O@||3/2||-9|
|perf. 5-step||18\29, 755.2||O||20/13||-1|
|aug. 5-step||19\29, 786.2||O&||11/7||+7|
|doubly aug. 5-step||20\29 827.6||O&&||+15|
|dim. 6-step||20\29 827.6||P@@||-14|
|minor 6-step||21\29 869.0||P@||-6|
|major 6-step||22\29, 910.3||P||22/13||+2|
|aug. 6-step||23\29, 951.7||P&||26/15||+10|
|dim. 7-step||24\29, 993.1||Q@||16/9||-11|
|minor 7-step||25\29, 1034.5||Q||20/11||-3|
|major 7-step||26\29, 1075.9||Q&||13/7||+5|
|aug. 7-step||27\29, 1117.2||Q&&||+13|
|dim. 8-step||28\29, 1158.6||J@||-8|
23edo oneiro combines the sound of neogothic tunings like 46edo and the sounds of "superpyth" and "semaphore" scales. This is because 23edo oneirotonic has a large step of 208.7¢, same as 46edo's neogothic major second, and is both a warped 22edo superpyth diatonic and a warped 24edo semaphore semiquartal (and both nearby scales are superhard MOSes).
The intervals of the extended generator chain (-12 to +12 generators) are as follows in various oneirotonic tunings close to 23edo.
|Degree||Size in 23edo (superhard)||Note name on J||Approximate ratios (23edo)||#Gens up|
|minor step||1\23, 52.2||K@||-5|
|major step||4\23, 208.7||K||+3|
|aug. step||7\23, 365.2||K&||21/17, inverse φ||+11|
|dim. 2-step||2\23, 104.3||L@||17/16||-10|
|minor 2-step||5\23, 260.9||L||-2|
|major 2-step||8\23, 417.4||L&||14/11||+6|
|dim. 3-step||6\23, 313.0||M@||6/5||-7|
|perf. 3-step||9\23, 469.6||M||21/16||+1|
|aug. 3-step||12\23, 626.1||M&||+9|
|dim. 4-step||7\23, 365.2||N@@||21/17, inverse φ||-12|
|minor 4-step||10\23, 521.7||N@||-4|
|major 4-step||13\23, 678.3||N||+4|
|aug. 4-step||16\23, 834.8||N&||34/21, φ||+12|
|dim. 5-step||11\23, 573.9||O@||-9|
|perf. 5-step||14\23, 730.4||O||32/21||-1|
|aug. 5-step||17\23, 887.0||O&||5/3||+7|
|minor 6-step||15\23 782.6||P@||11/7||-6|
|major 6-step||18\23, 939.1||P||+2|
|aug. 6-step||21\23, 1095.7||P&||32/17||+10|
|dim. 7-step||16\23, 834.8||Q@||34/21, φ||-11|
|minor 7-step||19\23, 991.3||Q||-3|
|major 7-step||22\23, 1147.8||Q&||+5|
|dim. 8-step||20\23, 1043.5||J@||-8|
In the broad sense, Buzzard can be viewed as any tuning that divides the 3rd harmonic into 4 equal parts. 23edo, 28edo and 33edo can nominally be viewed as supporting it, but are still very flat and in an ambiguous zone between 18edo and true Buzzard in terms of harmonies. 38edo & 43edo are good compromises between melodic utility and harmonic accuracy, as the small step is still large enough to be obvious to the untrained ear, but 48edo is where it really comes into its own in terms of harmonies, providing not only an excellent 3/2, but also 7/4 and archipelago harmonies, as by dividing the 5th in 4 it obviously also divides it in two as well.
Beyond that, it's a question of which intervals you want to favor. 53edo has an essentially perfect 3/2, 58edo gives the lowest overall error for the Barbados triads 10:13:15 and 26:30:39, while 63edo does the same for the basic 4:6:7 triad. You could in theory go up to 83edo if you want to favor the 7/4 above everything else, but beyond that, general accuracy drops off rapidly and you might as well be playing equal pentatonic.
The sizes of the generator, large step and small step of oneirotonic are as follows in various buzzard tunings.
|38edo||53edo||63edo||Optimal (POTE) Buzzard tuning||JI intervals represented (22.214.171.124.13 subgroup)|
|generator (g)||15\38, 473.68||21\53, 475.47||25\63, 476.19||475.69||4/3 21/16|
|L (3g - octave)||7/38, 221.04||10/53, 226.41||12/63, 228.57||227.07||8/7|
|s (-5g + 2 octaves)||1/38, 31.57||1/53 22.64||1/63 19.05||21.55||50/49 81/80 91/90|
Sortable table of intervals in the Dylathian mode and their Buzzard interpretations:
|Degree||Size in 38edo||Size in 53edo||Size in 63edo||Size in POTE tuning||Note name on Q||Approximate ratios||#Gens up|
|1||0\38, 0.00||0\53, 0.00||0\63, 0.00||0.00||Q||1/1||0|
|2||7\38, 221.05||10\53, 226.42||12\63, 228.57||227.07||J||8/7||+3|
|3||14\38, 442.10||20\53, 452.83||24\63, 457.14||453.81||K||13/10||+6|
|4||15\38, 473.68||21\53, 475.47||25\63, 476.19||475.63||L||21/16||+1|
|5||22\38, 694.73||31\53, 701.89||37\63, 704.76||702.54||M||3/2||+4|
|6||29\38, 915.78||41\53, 928.30||49\63, 933.33||929.45||N||12/7, 22/13||+7|
|7||30\38, 947.36||42\53, 950.94||50\63, 952.38||951.27||O||26/15||+2|
|8||37\38, 1168.42||52\53, 1177.36||62\63, 1180.95||1178.18||P||49/25, 160/81||+5|
Oneirotonic modes are named after cities in the Dreamlands.
(13edo, first 30 seconds is in J Celephaïsian)
(13edo, L Ilarnekian)
(by Igliashon Jones, 13edo, J Celephaïsian)
13edo Oneirotonic Modal Studies
- : Tonal Study in Dylathian
- : Tonal Study in Ultharian
- : Tonal Study in Hlanithian
- : Tonal Study in Ilarnekian
- : Tonal Study in Mnarian
- : Tonal Study in Sarnathian
- : Tonal Study in Celephaïsian
- : Tonal Study in Kadathian
- Well-Tempered 13-Tone Clavier (collab project to create 13edo oneirotonic keyboard pieces in a variety of keys and modes)
- Bright generator: 450 cents (3\8) to 480 cents (2\5)
- Dark generator: 720 cents (3\5) to 750 cents (5\8)
|34\89||458.427||13||8||1.625||Golden oneirotonic (458.3592¢)|
|5\13||461.538||2||1||2.000||Basic oneirotonic |
(generators smaller than this are proper)
|31\80||465.000||13||5||2.600||Golden A-Team (465.0841¢)|