5L 3s

From Xenharmonic Wiki
(Redirected from Oneirotonic)
Jump to navigation Jump to search
For the tritave-equivalent MOS structure with the same step pattern, see 5L 3s (tritave-equivalent).
5L 3s
Pattern LLsLLsLs
Period 2/1
Generator range 3\8 (450.0¢) to 2\5 (480.0¢)
Parent MOS 3L 2s
Daughter MOSes 8L 5s, 5L 8s
Sister MOS 3L 5s
Neutralized MOS 2L 6s
TAMNAMS name oneirotonic
Equal temperament tunings
Supersoft (L:s = 4:3) 11\29 (455.2¢)
Soft (L:s = 3:2) 8\21 (457.1¢)
Semisoft (L:s = 5:3) 13\34 (458.8¢)
Basic (L:s = 2:1) 5\13 (461.5¢)
Semihard (L:s = 5:2) 12\31 (464.5¢)
Hard (L:s = 3:1) 7\18 (466.7¢)
Superhard (L:s = 4:1) 9\23 (469.6¢)

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).

5L 3s is a warped diatonic scale, because it has one extra small step compared to diatonic (5L 2s): for example, the Ionian diatonic mode LLsLLLs can be distorted to the Dylathian mode LLsLLsLs.

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.

5L 3s has a pentatonic MOS subset 3L 2s (SLSLL). (Note: 3L 5s scales also have 3L 2s subsets.)

Standing assumptions

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

Names

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[8]' or 'father octatonic'. "Father" is also vague regarding the number of notes, because optimal generators for it also generate 3L 2s.

Intervals

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

Tuning ranges

Hypohard

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.

Also, in 18edo and 31edo, the minor mosthird is close to 7/6.

The set of identifications above is associated with A-Team temperament.

EDOs that are in the hypohard range include 13edo, 18edo, and 31edo.

  • 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

Intervals

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[1] #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
  1. The ratio interpretations that are not valid for 18edo are italicized.

Hyposoft

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

Intervals

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.

29edo (supersoft) 37edo
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

Intervals

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
unison 0\29, 0.00 J 1/1 0
oneirochroma 1\29, 41.3 J& +8
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


Modes

Oneirotonic modes are named after cities in the Dreamlands.

Mode UDP Name
LLsLLsLs 7|0 Dylathian (də-LA(H)TH-iən)
LLsLsLLs 6|1 Illarnekian (ill-ar-NEK-iən)
LsLLsLLs 5|2 Celephaïsian (kel-ə-FAY-zhən)
LsLLsLsL 4|3 Ultharian (ul-THA(I)R-iən)
LsLsLLsL 3|4 Mnarian (mə-NA(I)R-iən)
sLLsLLsL 2|5 Kadathian (kə-DA(H)TH-iən)
sLLsLsLL 1|6 Hlanithian (lə-NITH-iən)
sLsLLsLL 0|7 Sarnathian (sar-NA(H)TH-iən), can be shortened to "Sarn"

Approaches

Samples

WT13C Prelude II (J Oneirominor) (score) – Simple two-part Baroque piece. It stays in oneirotonic even though it modulates to other keys a little.

(13edo, first 30 seconds is in J Celephaïsian)

(13edo, L Illarnekian)

(by Igliashon Jones, 13edo, J Celephaïsian)

See also

Scale tree

Generator Cents L s L/s Comments
3\8 450.000 1 1 1.000
17\45 453.333 6 5 1.200
14\37 454.054 5 4 1.250
34\59 454.545 9 7 1.286
11\29 455.172 4 3 1.333
30\79 455.696 11 8 1.375
19\50 456.000 7 5 1.400
27\71 456.338 10 7 1.429
8\21 457.143 3 2 1.500 L/s = 3/2
29\76 457.895 11 7 1.571
21\55 458.182 8 5 1.600
34\89 458.427 13 8 1.625 Golden oneirotonic
13\34 458.824 5 3 1.667
31\81 459.259 12 7 1.714
18\47 459.574 7 4 1.750
23\60 460.000 9 5 1.800
5\13 461.538 2 1 2.000 Basic oneirotonic
(generators smaller than this are proper)
22\57 463.158 9 4 2.250
17\44 463.636 7 3 2.333
29\75 464.000 12 5 2.400
12\31 464.516 5 2 2.500
31\80 465.000 13 5 2.600
19\49 465.306 8 3 2.667
26\67 465.672 11 4 2.750
7\18 466.667 3 1 3.000 L/s = 3/1
23\59 467.797 10 3 3.333
16\41 468.293 7 2 3.500
25\64 468.750 11 3 3.667
9\23 469.565 4 1 4.000
20\51 470.588 9 2 4.500
11\28 471.429 5 1 5.000
13\33 472.727 6 1 6.000
2\5 480.000 1 0 → inf