- For the tritave-equivalent MOS structure with the same step pattern, see 5L 3s (tritave-equivalent).
5L 3s refers to the structure of octave-equivalent MOS scales with generators ranging from 2\5 (two degrees of 5edo = 480¢) to 3\8 (three degrees of 8edo = 450¢). In the case of 8edo, L and s are the same size; in the case of 5edo, s becomes so small it disappears (and all that remains are the five equal L's).
Any edo with an interval between 450¢ and 480¢ has a 5L 3s scale. 13edo is the smallest edo with a (non-degenerate) 5L 3s scale and thus is the most commonly used 5L 3s tuning.
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 mosfourths becomes: ... P@ K@ N@ Q L O J M P K N Q& L& O& ...
Thus the 13edo gamut is as follows:
J/Q& J&/K@ K/L@ L/K& L&/M@ M M&/N@ N/O@ O/N& O&/P@ P Q Q&/J@ J
The 18edo gamut is notated as follows:
J Q&/K@ J&/L@ K L K&/M@ L& M N@ M&/O@ N O P@ O& 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. A more correct way to say it 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 mosthird is reached by six subfourth generators, 18edo's major mosthird is 6*466.67 mod 1200 = 2800 mod 1200 = 400¢, same as the 12edo major third.
|# generators up||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 mosfourth||2L + 1s||-1||O||perfect mossixth||3L + 2s|
|2||P||major mosseventh||4L + 2s||-2||L||minor mosthird||1L + 1s|
|3||K||major mossecond||1L + 0s||-3||Q||minor moseighth||4L + 3s|
|4||N||major mosfifth||3L + 1s||-4||N@||minor mosfifth||2L + 2s|
|5||Q&||major moseighth||5L + 2s||-5||K@||minor mossecond||0L + 1s|
|6||L&||major mosthird||2L + 0s||-6||P@||minor mosseventh||3L + 3s|
|7||O&||augmented mossixth||4L + 1s||-7||M@||diminished mosfourth||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 mosunison (aka moschroma)||1L - 1s||-8||J@||diminished mosoctave (aka diminished mosninth)||4L + 4s|
|9||M&||augmented mosfourth||3L + 0s||-9||O@||diminished mossixth||2L + 3s|
|10||P&||augmented mosseventh||5L + 1s||-10||L@||diminished mosthird||0L + 2s|
|11||K&||augmented mossecond||2L - 1s||-11||Q@||diminished moseighth||3L + 4s|
|12||N&||augmented mosfifth||4L + 0s||-12||N@@||diminished mosfifth||1L + 3s|
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 mosthird (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 major mosseconds of about 185c. 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 (733.3c, a perfect mossixth) and falling fifths (666.7c, a major mosfifth) 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 mos3rd 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 mos2nd||1\13, 92.31||1\18, 66.67||2\31, 77.42||K@||21/20, 22/21||-5|
|major mos2nd||2\13, 184.62||3\18, 200.00||5\31, 193.55||K||9/8, 10/9||+3|
|minor mos3rd||3\13, 276.92||4\18, 266.67||7\31, 270.97||L||7/6||-2|
|major mos3rd||4\13, 369.23||6\18, 400.00||10\31, 387.10||L&||5/4||+6|
|dim. mos4th||4\13, 369.23||5\18, 333.33||9\31, 348.39||M@||16/13, 11/9||-7|
|perf. mos4th||5\13, 461.54||7\18, 466.67||12\31, 464.52||M||21/16, 13/10, 17/13||+1|
|minor mos5th||6\13, 553.85||8\18, 533.33||14\31, 541.94||N@||11/8||-4|
|major mos5th||7\13, 646.15||10\18, 666.66||17\31, 658.06||N||13/9, 16/11||+4|
|perf. mos6th||8\13, 738.46||11\18, 733.33||19\31, 735.48||O||26/17||-1|
|aug. mos6th||9\13, 830.77||13\18, 866.66||22\31, 851.61||O&||13/8, 18/11||+7|
|minor mos7th||9\13, 830.77||12\18, 800.00||21\31, 812.90||P@||8/5||-6|
|major mos7th||10\13, 923.08||14\18, 933.33||24\31, 929.03||P||12/7||+2|
|minor mos8th||11\13, 1015.39||15\18, 1000.00||26\31, 1006.45||Q||9/5, 16/9||-3|
|major mos8th||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 mosthird (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-Lmos2-Lmos3-Lmos5 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.
This set of JI identifications is associated with petrtri temperament. (P1-Mmos2-Pmos4 could be said to approximate 5:11:13 in all soft-of-basic tunings, which is what "basic" petrtri temperament is.)
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 mos2nd||2\21, 114.29||3\34, 105.88||K@||16/15||-5|
|major mos2nd||3\21, 171.43||5\34, 176.47||K||10/9, 11/10||+3|
|minor mos3rd||5\21, 285.71||8\34, 282.35||L||13/11, 20/17||-2|
|major mos3rd||6\21, 342.86||10\34, 352.94||L&||11/9||+6|
|dim. mos4th||7\21, 400.00||11\34, 388.24||M@||5/4||-7|
|perf. mos4th||7\18, 457.14||12\31, 458.82||M||13/10||+1|
|minor mos5th||10\21, 571.43||16\34, 564.72||N@||18/13, 32/23||-4|
|major mos5th||11\21, 628.57||18\34, 635.29||N||13/9, 23/16||+4|
|perf. mos6th||13\21, 742.86||21\34, 741.18||O||20/13||-1|
|aug. mos6th||14\21, 800.00||23\34, 811.77||O&||8/5||+7|
|minor mos7th||15\21, 857.14||24\34, 847.06||P@||18/11||-6|
|major mos7th||16\21, 914.29||26\34, 917.65||P||22/13, 17/10||+2|
|minor mos8th||18\21, 1028.57||29\34, 1023.53||Q||9/5||-3|
|major mos8th||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 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. mos2nd||2\29, 82.8||K@@||-13|
|minor mos2nd||3\29, 124.1||K@||14/13||-5|
|major mos2nd||4\29, 165.5||K||11/10||+3|
|aug. mos2nd||5\29, 206.9||K&||9/8||+11|
|dim. mos3rd||6\29, 248.3||L@||15/13||-10|
|minor mos3rd||7\29, 289.7||L||13/11||-2|
|major mos3rd||8\29, 331.0||L&||+6|
|aug. mos3rd||9\29, 372.4||L&&||+14|
|doubly dim. mos4th||9\29, 372.4||M@@||-15|
|dim. mos4th||10\29, 413.8||M@||14/11||-7|
|perf. mos4th||11\29, 455.2||M||13/10||+1|
|aug. mos4th||12\29, 496.6||M&||4/3||+9|
|dim. mos5th||13\29, 537.9||N@@||15/11||-12|
|minor mos5th||14\29, 579.3||N@||7/5||-4|
|major mos5th||15\29 620.7||N||10/7||+4|
|aug. mos5th||16\29 662.1||N&||22/15||+12|
|dim. mos6th||17\29, 703.4||O@||3/2||-9|
|perf. mos6th||18\29, 755.2||O||20/13||-1|
|aug. mos6th||19\29, 786.2||O&||11/7||+7|
|doubly aug. mos6th||20\29 827.6||O&&||+15|
|dim. mos7th||20\29 827.6||P@@||-14|
|minor mos7th||21\29 869.0||P@||-6|
|major mos7th||22\29, 910.3||P||22/13||+2|
|aug. mos7th||23\29, 951.7||P&||26/15||+10|
|dim. mos8th||24\29, 993.1||Q@||16/9||-11|
|minor mos8th||25\29, 1034.5||Q||20/11||-3|
|major mos8th||26\29, 1075.9||Q&||13/7||+5|
|aug. mos8th||27\29, 1117.2||Q&&||+13|
|dim. mos9th||28\29, 1158.6||J@||-8|
Oneirotonic modes are named after cities in the Dreamlands.
|sLsLLsLL||0|7||Sarnathian (sar-NA(H)TH-iən), can be shortened to "Sarn"|
(13edo, first 30 seconds is in J Celephaïsian)
(13edo, L Illarnekian)
(by Igliashon Jones, 13edo, J Celephaïsian)
- Well-Tempered 13-Tone Clavier (collab project to create 13edo oneirotonic keyboard pieces in a variety of keys and modes)
|8\21||457.143||3||2||1.500||L/s = 3/2|
(generators smaller than this are proper)
|7\18||466.667||3||1||3.000||L/s = 3/1|