5L 3s
| ↖ 4L 2s | ↑ 5L 2s | 6L 2s ↗ |
| ← 4L 3s | 5L 3s | 6L 3s → |
| ↙ 4L 4s | ↓ 5L 4s | 6L 4s ↘ |
┌╥╥┬╥╥┬╥┬┐ │║║│║║│║││ ││││││││││ └┴┴┴┴┴┴┴┴┘
sLsLLsLL
- For the tritave-equivalent MOS structure with the same step pattern, see 5L 3s (3/1-equivalent).
5L 3s, named oneirotonic in TAMNAMS, is a 2/1-equivalent (octave-equivalent) moment of symmetry scale containing 5 large steps and 3 small steps, repeating every octave. Generators that produce this scale range from 450¢ to 480¢, or from 720¢ to 750¢. 5L 3s can be seen as a warped diatonic scale, because it has one extra small step compared to diatonic (5L 2s).
Name
TAMNAMS suggests the temperament-agnostic name oneirotonic as the name of 5L 3s. The name was originally used as a name for the 5L 3s scale in 13edo. 'Oneiro' is sometimes used as a shortened form.
'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[8]' or 'father octatonic'. "Father" is also vague regarding the number of notes, because optimal generators for it also generate 3L 2s.
Proposed mode names
The following names have been proposed for the modes of 5L 3s, and are named after cities in the Dreamlands.
| UDP | Cyclic order |
Step pattern |
Mode names |
|---|---|---|---|
| 7|0 | 1 | LLsLLsLs | Dylathian |
| 6|1 | 4 | LLsLsLLs | Ilarnekian |
| 5|2 | 7 | LsLLsLLs | Celephaïsian |
| 4|3 | 2 | LsLLsLsL | Ultharian |
| 3|4 | 5 | LsLsLLsL | Mnarian |
| 2|5 | 8 | sLLsLLsL | Kadathian |
| 1|6 | 3 | sLLsLsLL | Hlanithian |
| 0|7 | 6 | sLsLLsLL | Sarnathian |
Scale properties
Intervals
The intervals of 5L 3s are named after the number of mossteps (L and s) they subtend. Each interval, apart from the root and octave (perfect 0-oneirostep and perfect 8-oneirostep), has two varieties, or sizes, each. Interval varieties are named major and minor for the large and small sizes, respectively, and augmented, perfect, and diminished for the scale's generators.
| Intervals | Steps subtended |
Range in cents | ||
|---|---|---|---|---|
| Generic | Specific | Abbrev. | ||
| 0-oneirostep | Perfect 0-oneirostep | P0oneis | 0 | 0.0¢ |
| 1-oneirostep | Minor 1-oneirostep | m1oneis | s | 0.0¢ to 150.0¢ |
| Major 1-oneirostep | M1oneis | L | 150.0¢ to 240.0¢ | |
| 2-oneirostep | Minor 2-oneirostep | m2oneis | L + s | 240.0¢ to 300.0¢ |
| Major 2-oneirostep | M2oneis | 2L | 300.0¢ to 480.0¢ | |
| 3-oneirostep | Diminished 3-oneirostep | d3oneis | L + 2s | 240.0¢ to 450.0¢ |
| Perfect 3-oneirostep | P3oneis | 2L + s | 450.0¢ to 480.0¢ | |
| 4-oneirostep | Minor 4-oneirostep | m4oneis | 2L + 2s | 480.0¢ to 600.0¢ |
| Major 4-oneirostep | M4oneis | 3L + s | 600.0¢ to 720.0¢ | |
| 5-oneirostep | Perfect 5-oneirostep | P5oneis | 3L + 2s | 720.0¢ to 750.0¢ |
| Augmented 5-oneirostep | A5oneis | 4L + s | 750.0¢ to 960.0¢ | |
| 6-oneirostep | Minor 6-oneirostep | m6oneis | 3L + 3s | 720.0¢ to 900.0¢ |
| Major 6-oneirostep | M6oneis | 4L + 2s | 900.0¢ to 960.0¢ | |
| 7-oneirostep | Minor 7-oneirostep | m7oneis | 4L + 3s | 960.0¢ to 1050.0¢ |
| Major 7-oneirostep | M7oneis | 5L + 2s | 1050.0¢ to 1200.0¢ | |
| 8-oneirostep | Perfect 8-oneirostep | P8oneis | 5L + 3s | 1200.0¢ |
Generator chain
A chain of bright generators, each a perfect 3-oneirostep, produces the following scale degrees. A chain of 8 bright generators contains the scale degrees of one of the modes of 5L 3s. Expanding the chain to 13 scale degrees produces the modes of either 8L 5s (for soft-of-basic tunings) or 5L 8s (for hard-of-basic tunings).
| Bright gens | Scale Degree | Abbrev. |
|---|---|---|
| 12 | Augmented 4-oneirodegree | A4oneid |
| 11 | Augmented 1-oneirodegree | A1oneid |
| 10 | Augmented 6-oneirodegree | A6oneid |
| 9 | Augmented 3-oneirodegree | A3oneid |
| 8 | Augmented 0-oneirodegree | A0oneid |
| 7 | Augmented 5-oneirodegree | A5oneid |
| 6 | Major 2-oneirodegree | M2oneid |
| 5 | Major 7-oneirodegree | M7oneid |
| 4 | Major 4-oneirodegree | M4oneid |
| 3 | Major 1-oneirodegree | M1oneid |
| 2 | Major 6-oneirodegree | M6oneid |
| 1 | Perfect 3-oneirodegree | P3oneid |
| 0 | Perfect 0-oneirodegree Perfect 8-oneirodegree |
P0oneid P8oneid |
| -1 | Perfect 5-oneirodegree | P5oneid |
| -2 | Minor 2-oneirodegree | m2oneid |
| -3 | Minor 7-oneirodegree | m7oneid |
| -4 | Minor 4-oneirodegree | m4oneid |
| -5 | Minor 1-oneirodegree | m1oneid |
| -6 | Minor 6-oneirodegree | m6oneid |
| -7 | Diminished 3-oneirodegree | d3oneid |
| -8 | Diminished 8-oneirodegree | d8oneid |
| -9 | Diminished 5-oneirodegree | d5oneid |
| -10 | Diminished 2-oneirodegree | d2oneid |
| -11 | Diminished 7-oneirodegree | d7oneid |
| -12 | Diminished 4-oneirodegree | d4oneid |
Modes
| UDP | Cyclic order |
Step pattern |
Scale degree (oneirodegree) | ||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |||
| 7|0 | 1 | LLsLLsLs | Perf. | Maj. | Maj. | Perf. | Maj. | Aug. | Maj. | Maj. | Perf. |
| 6|1 | 4 | LLsLsLLs | Perf. | Maj. | Maj. | Perf. | Maj. | Perf. | Maj. | Maj. | Perf. |
| 5|2 | 7 | LsLLsLLs | Perf. | Maj. | Min. | Perf. | Maj. | Perf. | Maj. | Maj. | Perf. |
| 4|3 | 2 | LsLLsLsL | Perf. | Maj. | Min. | Perf. | Maj. | Perf. | Maj. | Min. | Perf. |
| 3|4 | 5 | LsLsLLsL | Perf. | Maj. | Min. | Perf. | Min. | Perf. | Maj. | Min. | Perf. |
| 2|5 | 8 | sLLsLLsL | Perf. | Min. | Min. | Perf. | Min. | Perf. | Maj. | Min. | Perf. |
| 1|6 | 3 | sLLsLsLL | Perf. | Min. | Min. | Perf. | Min. | Perf. | Min. | Min. | Perf. |
| 0|7 | 6 | sLsLLsLL | Perf. | Min. | Min. | Dim. | Min. | Perf. | Min. | Min. | Perf. |
Tunings
Simple tunings
The simplest tuning for 5L 3s correspond to 13edo, 18edo, and 21edo, with step ratios 2:1, 3:1, and 3:2, respectively.
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Hypohard tunings
Hypohard oneirotonic tunings have step ratios between 2:1 and 3:1 and can be considered "meantone oneirotonic", sharing the following features with meantone diatonic tunings:
- The large step is a "meantone", around the range of 10/9 to 9/8.
- The major 2-mosstep is a meantone- to flattone-sized major third, thus is a stand-in for the classical diatonic major third.
With step ratios between 5:2 and 2:1, the minor 2-mosstep is close to 7/6.
EDOs that are in the hypohard range include 13edo, 18edo, and 31edo, and are 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.
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Hyposoft tunings
Hyposoft oneirotonic tunings have step ratios between 3:2 and 2:1, which 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.
This set of JI identifications is associated with petrtri temperament. (P1-M1ms-P3ms could be said to approximate 5:11:13 in all soft-of-basic tunings, which is what "basic" petrtri temperament is.)
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Parasoft and ultrasoft tunings
The range of oneirotonic tunings of step ratio between 6:5 and 3:2 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. The chord 10:11:13 is very well approximated in 29edo.
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Parahard tunings
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).
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Ultrahard tunings
Buzzard is a rank-2 temperament in the pseudocollapsed range. It represents the only harmonic entropy minimum of the oneirotonic spectrum.
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.
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Approaches
Samples
The Angels' Library by Inthar in the Sarnathian (23233233) mode of 21edo oneirotonic (score)
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 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
Scale tree
| Generator(edo) | Cents | Step ratio | Comments | |||||||
|---|---|---|---|---|---|---|---|---|---|---|
| Bright | Dark | L:s | Hardness | |||||||
| 3\8 | 450.000 | 750.000 | 1:1 | 1.000 | Equalized 5L 3s | |||||
| 17\45 | 453.333 | 746.667 | 6:5 | 1.200 | ||||||
| 14\37 | 454.054 | 745.946 | 5:4 | 1.250 | ||||||
| 25\66 | 454.545 | 745.455 | 9:7 | 1.286 | ||||||
| 11\29 | 455.172 | 744.828 | 4:3 | 1.333 | Supersoft 5L 3s | |||||
| 30\79 | 455.696 | 744.304 | 11:8 | 1.375 | ||||||
| 19\50 | 456.000 | 744.000 | 7:5 | 1.400 | ||||||
| 27\71 | 456.338 | 743.662 | 10:7 | 1.429 | ||||||
| 8\21 | 457.143 | 742.857 | 3:2 | 1.500 | Soft 5L 3s | |||||
| 29\76 | 457.895 | 742.105 | 11:7 | 1.571 | ||||||
| 21\55 | 458.182 | 741.818 | 8:5 | 1.600 | ||||||
| 34\89 | 458.427 | 741.573 | 13:8 | 1.625 | ||||||
| 13\34 | 458.824 | 741.176 | 5:3 | 1.667 | Semisoft 5L 3s | |||||
| 31\81 | 459.259 | 740.741 | 12:7 | 1.714 | ||||||
| 18\47 | 459.574 | 740.426 | 7:4 | 1.750 | ||||||
| 23\60 | 460.000 | 740.000 | 9:5 | 1.800 | ||||||
| 5\13 | 461.538 | 738.462 | 2:1 | 2.000 | Basic 5L 3s Scales with tunings softer than this are proper | |||||
| 22\57 | 463.158 | 736.842 | 9:4 | 2.250 | ||||||
| 17\44 | 463.636 | 736.364 | 7:3 | 2.333 | ||||||
| 29\75 | 464.000 | 736.000 | 12:5 | 2.400 | ||||||
| 12\31 | 464.516 | 735.484 | 5:2 | 2.500 | Semihard 5L 3s | |||||
| 31\80 | 465.000 | 735.000 | 13:5 | 2.600 | ||||||
| 19\49 | 465.306 | 734.694 | 8:3 | 2.667 | ||||||
| 26\67 | 465.672 | 734.328 | 11:4 | 2.750 | ||||||
| 7\18 | 466.667 | 733.333 | 3:1 | 3.000 | Hard 5L 3s | |||||
| 23\59 | 467.797 | 732.203 | 10:3 | 3.333 | ||||||
| 16\41 | 468.293 | 731.707 | 7:2 | 3.500 | ||||||
| 25\64 | 468.750 | 731.250 | 11:3 | 3.667 | ||||||
| 9\23 | 469.565 | 730.435 | 4:1 | 4.000 | Superhard 5L 3s | |||||
| 20\51 | 470.588 | 729.412 | 9:2 | 4.500 | ||||||
| 11\28 | 471.429 | 728.571 | 5:1 | 5.000 | ||||||
| 13\33 | 472.727 | 727.273 | 6:1 | 6.000 | ||||||
| 2\5 | 480.000 | 720.000 | 1:0 | → ∞ | Collapsed 5L 3s | |||||