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{{User:IlL/Template:RTT_restriction}}
{{Interwiki
:''For the tritave-equivalent MOS structure with the same step pattern, see [[5L 3s (tritave-equivalent)]].''
| en = 5L 3s
| de =
| es =
| ja =
| ko = 5L3s (Korean)
}}
{{Infobox MOS
{{Infobox MOS
| Name = oneirotonic
| Neutral = 2L 6s
| Periods = 1
| nLargeSteps = 5
| nSmallSteps = 3
| Equalized = 3
| Paucitonic = 2
| Pattern = LLsLLsLs
}}
}}
'''5L 3s''' or '''oneirotonic''' (/oʊnaɪrəˈtɒnɪk/ ''oh-ny-rə-TON-ik'' or /ənaɪrə-/ ''ə-ny-rə-'') 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). The name ''oneirotonic'' (from Greek ''oneiros'' 'dream') was coined by [[Cryptic Ruse]] after the Dreamlands in H.P. Lovecraft's Dream Cycle mythos.
: ''For the tritave-equivalent MOS structure with the same step pattern, see [[5L 3s (3/1-equivalent)]].''
{{MOS intro}}
5L 3s can be seen as a [[Warped diatonic|warped diatonic scale]], because it has one extra small step compared to diatonic ([[5L 2s]]).
Oneirotonic is a distorted diatonic, 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 oneirotonic mode LLsLLsLs.
== Name ==
{{TAMNAMS name}} 'Oneiro' is sometimes used as a shortened form.
Any edo with an interval between 450¢ and 480¢ has an oneirotonic scale. [[13edo]] is the smallest edo with a (non-degenerate) 5L3s oneirotonic scale and thus is the most commonly used oneirotonic tuning.
'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]].
5L 3s has a pentatonic MOS subset [[3L 2s]] (SLSLL), and in this context we call this the ''oneiro-pentatonic''. When viewed as a chord (with undetermined voicing) we call it the Oneiro Core Pentad. (Note: [[3L 5s]] scales also have 3L 2s subsets.)
== Scale properties ==
== Notation==
=== Intervals ===
The notation used in this article is J Ultharian (LsLLsLsL) = JKLMNOPQJ, with reference pitch N = 261.6255653 Hz, 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".) Ultharian has been chosen as the default mode because we want to carry over the diatonic idea of sharpening the second-to-last degree to get the leading tone for minor keys and the sharpened "Vmaj", and we also have the "sharp V" for the oneiromajor tonality by default.
{{MOS intervals}}
The chain of oneirofourths becomes: ... P@ K@ N@ Q L O J M P K N Q& L& O& ...
[[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]].
== Intervals ==
EDOs that are in the hypohard range include [[13edo]], [[18edo]], and [[31edo]], and are associated with [[5L 3s/Temperaments#A-Team|A-Team]] temperament.
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 of generators it takes to reach the interval and reducing mod 1200 if necessary (The % sign can be used for the modulo operation on many search engines). For example, since the major oneirothird is reached by six subfourth generators, [[18edo]]'s major oneirothird is 6*466.67 mod 1200 = 2800 mod 1200 = 400¢, same as the [[12edo]] major third.
* 13edo has characteristically small 1-mossteps of about 185{{c}}. 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.
{| class="wikitable center-all"
* 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{{c}}, a perfect 5-mosstep) and falling fifths (666.7{{c}}, 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.
! # generators up
* [[44edo]] (generator {{nowrap|17\44 {{=}} 463.64{{c}}}}), [[57edo]] (generator {{nowrap|22\57 {{=}} 463.16{{c}}}}), and [[70edo]] (generator 27\70 {{=}} 462.857{{c}}}}) 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.
! Notation (1/1 = J)
! Octatonic interval category name
! Abbrev.
! # generators up
! Notation of 2/1 inverse
! Octatonic interval category name
! Abbrev.
|-
| colspan="8" style="text-align:center" | The 8-note MOS has the following intervals (from some root):
|-
| 0
| J
| perfect unison
| P1
| 0
| J
| octave
| Po9
|-
| 1
| M
| perfect oneirofourth (aka minor fourth, falling fourth)
| Po4
| -1
| O
| perfect oneirosixth (aka major fifth, rising fifth)
| Po6
|-
| 2
| P
| major oneiroseventh
| Mo7
| -2
| L
| minor oneirothird
| mo3
|-
| 3
| K
| major oneirosecond
| Mo2
| -3
| Q
| minor oneiroeighth
| mo8
|-
| 4
| N
| major oneirofifth (aka minor fifth, falling fifth)
| Mo5
| -4
| N@
| minor oneirofifth (aka major fourth, rising fourth)
| mo4
|-
| 5
| Q&
| major oneiroeighth
| Mo8
| -5
| K@
| minor oneirosecond
| mo2
|-
| 6
| L&
| major oneirothird
| Mo3
| -6
| P@
| minor oneiroseventh
| mo7
|-
| 7
| O&
| augmented oneirosixth
| Ao6
| -7
| M@
| diminished oneirofourth
| d-o4
|-
| colspan="8" style="text-align:center" | The chromatic 13-note MOS (either [[5L 8s]], [[8L 5s]], or [[13edo]]) also has the following intervals (from some root):
{{MOS tunings|Step Ratios=Hypohard|JI Ratios=Subgroup: 2.5.9.21; Int Limit:40; Complements Only: 1|Tolerance=15}}
=== 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:
=== Hyposoft tunings ===
* The large step is a "meantone", somewhere between near-10/9 (as in [[13edo]]) and near-9/8 (as in [[18edo]]).
[[Hyposoft]] oneirotonic tunings have step ratios between 3:2 and 2:1, which remains relatively unexplored. In these tunings,
* 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 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{{c}} to 114{{c}}.
* 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{{c}}) to 4\13 (369{{c}}).
EDOs that are in the hypohard range include [[13edo]], [[18edo]], and [[31edo]].
* [[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{{c}}) and Baroque diatonic semitones (114.29{{c}}, close to quarter-comma meantone's 117.11{{c}}).
* 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.
* [[34edo]]'s 9:10:11:13 is even better.
* 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.
This set of JI identifications is associated with [[5L 3s/Temperaments#Petrtri|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.)
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.
! class="unsortable"|Degree
! Size in 13edo (basic)
! Size in 18edo (hard)
! Size in 31edo (semihard)
! class="unsortable"| Note name on J
! class="unsortable"| Approximate ratios<ref>The ratio interpretations that are not valid for 18edo are italicized.</ref>
! #Gens up
|-
| unison
| 0\13, 0.00
| 0\18, 0.00
| 0\31, 0.00
| J
| 1/1
| 0
|-
| min. o2nd
| 1\13, 92.31
| 1\18, 66.67
| 2\31, 77.42
| K@
| 21/20, ''22/21''
| -5
|-
| maj. o2nd
| 2\13, 184.62
| 3\18, 200.00
| 5\31, 193.55
| K
| 9/8, 10/9
| +3
|-
| min. o3rd
| 3\13, 276.92
| 4\18, 266.67
| 7\31, 270.97
| L
| 7/6
| -2
|-
| maj. o3rd
| 4\13, 369.23
| 6\18, 400.00
| 10\31, 387.10
| L&
| 5/4
| +6
|-
| dim. o4th
| 4\13, 369.23
| 5\18, 333.33
| 9\31, 348.39
| M@
| ''16/13, 11/9''
| -7
|-
| perf. o4th
| 5\13, 461.54
| 7\18, 466.67
| 12\31, 464.52
| M
| 21/16, ''13/10'', 17/13
| +1
|-
| min. o5th
| 6\13, 553.85
| 8\18, 533.33
| 14\31, 541.94
| N@
| ''11/8''
| -4
|-
| maj. o5th
| 7\13, 646.15
| 10\18, 666.66
| 17\31, 658.06
| N
| ''13/9'', ''16/11''
| +4
|-
| perf. o6th
| 8\13, 738.46
| 11\18, 733.33
| 19\31, 735.48
| O
| 26/17
| -1
|-
| aug. o6th
| 9\13, 830.77
| 13\18, 866.66
| 22\31, 851.61
| O&
| ''13/8'', ''18/11''
| +7
|-
| min. o7th
| 9\13, 830.77
| 12\18, 800.00
| 21\31, 812.90
| P@
| 8/5
| -6
|-
| maj. o7th
| 10\13, 923.08
| 14\18, 933.33
| 24\31, 929.03
| P
| 12/7
| +2
|-
| min. o8th
| 11\13, 1015.39
| 15\18, 1000.00
| 26\31, 1006.45
| Q
| 9/5, 16/9
| -3
|-
| maj. o8th
| 12\13, 1107.69
| 17\18, 1133.33
| 29\31, 1122.58
| Q&
|
| +5
|}
<references/>
=== 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-Mo2-Mo3-Mo5 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¢).
{{MOS tunings
* [[34edo]]'s 9:10:11:13 is even better.
| Step Ratios = 6/5; 3/2; 4/3
| JI Ratios =
1/1;
14/13;
11/10;
9/8;
15/13;
13/11;
14/11;
13/10;
4/3;
15/11;
7/5;
10/7;
22/15;
3/2;
20/13;
11/7;
22/13;
26/15;
16/9;
20/11;
13/7;
2/1
}}
The sizes of the generator, large step and small step of oneirotonic are as follows in various hyposoft oneiro tunings (13edo not shown).
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).
|-
!
! [[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 ====
{{MOS tunings
Sortable table of major and minor intervals in hyposoft tunings (13edo not shown):
| JI Ratios =
1/1;
21/17;
17/16;
14/11;
6/5;
21/16;
21/17;
34/21;
32/21;
5/3;
11/7;
32/17;
34/21;
2/1
| Step Ratios = 4/1
}}
{| class="wikitable right-2 right-3 sortable "
=== Ultrahard tunings ===
|-
{{Main|5L 3s/Temperaments#Buzzard}}
! class="unsortable"|Degree
! Size in 21edo (soft)
! Size in 34edo (semisoft)
! class="unsortable"| Note name on J
! class="unsortable"| Approximate ratios
! #Gens up
|-
| unison
| 0\21, 0.00
| 0\34, 0.00
| J
| 1/1
| 0
|-
| min. o2nd
| 2\21, 114.29
| 3\34, 105.88
| K@
| 16/15
| -5
|-
| maj. o2nd
| 3\21, 171.43
| 5\34, 176.47
| K
| 10/9, 11/10
| +3
|-
| min. o3rd
| 5\21, 285.71
| 8\34, 282.35
| L
| 13/11, 20/17
| -2
|-
| maj. o3rd
| 6\21, 342.86
| 10\34, 352.94
| L&
| 11/9
| +6
|-
| dim. o4th
| 7\21, 400.00
| 11\34, 388.24
| M@
| 5/4
| -7
|-
| perf. o4th
| 7\18, 457.14
| 12\31, 458.82
| M
| 13/10
| +1
|-
| min. o5th
| 10\21, 571.43
| 16\34, 564.72
| N@
| 18/13, 32/23
| -4
|-
| maj. o5th
| 11\21, 628.57
| 18\34, 635.29
| N
| 13/9, 23/16
| +4
|-
| perf. o6th
| 13\21, 742.86
| 21\34, 741.18
| O
| 20/13
| -1
|-
| aug. o6th
| 14\21, 800.00
| 23\34, 811.77
| O&
| 8/5
| +7
|-
| min. o7th
| 15\21, 857.14
| 24\34, 847.06
| P@
| 18/11
| -6
|-
| maj. o7th
| 16\21, 914.29
| 26\34, 917.65
| P
| 22/13, 17/10
| +2
|-
| maj. o8th
| 18\21, 1028.57
| 29\34, 1023.53
| Q
| 9/5
| -3
|-
| maj. o8th
| 19\21, 1085.71
| 31\34, 1094.12
| Q&
| 15/8
| +5
|}
=== Parasoft to ultrasoft tunings ===
[[Buzzard]] is a rank-2 temperament in the [[Step ratio|pseudocollapsed]] range. It represents the only [[harmonic entropy]] minimum of the oneirotonic spectrum.
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.
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 [[The_Archipelago|archipelago]] harmonies, as by dividing the 5th in 4 it obviously also divides it in two as well.
Buzzard is an oneirotonic rank-2 temperament in the [[Step ratio|pseudopaucitonic]] 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 it's own in terms of harmonies, providing not only an excellent [[3/2]], but also [[7/4]] and [[The_Archipelago|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.
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.
** Enharmonic with K@ Oneirominor, M@ Oneiromajor in 13edo
== Modes ==
Oneirotonic modes are named after cities in the Dreamlands. (The names are by Cryptic Ruse.)
# Dylathian (də-LA(H)TH-iən): LLSLLSLS
# Illarnekian (ill-ar-NEK-iən): LLSLSLLS
# Celephaïsian (kel-ə-FAY-zhən): LSLLSLLS
# Ultharian (ul-THA(I)R-iən): LSLLSLSL
# Mnarian (mə-NA(I)R-iən): LSLSLLSL
# Kadathian (kə-DA(H)TH-iən): SLLSLLSL
# Hlanithian (lə-NITH-iən): SLLSLSLL
# Sarnathian (sar-NA(H)TH-iən): SLSLLSLL
The modes on the white keys JKLMNOPQJ are:
* J Ultharian
* K Hlanithian
* L Illarnekian
* M Mnarian
* N Sarnathian
* O Celephaïsian
* P Kadathian
* Q Dylathian
{| class="wikitable"
|-
|+ Table of modes (based on J, from brightest to darkest)
|-
! Mode
! 1
! 2
! 3
! 4
! 5
! 6
! 7
! 8
! (9)
|-
| Dylathian
| J
| K
| L&
| M
| N
| O&
| P
| Q&
| (J)
|-
| Illarnekian
| J
| K
| L&
| M
| N
| O
| P
| Q&
| (J)
|-
| Celephaïsian
| J
| K
| L
| M
| N
| O
| P
| Q&
| (J)
|-
| Ultharian
| J
| K
| L
| M
| N
| O
| P
| Q
| (J)
|-
| Mnarian
| J
| K
| L
| M
| N@
| O
| P
| Q
| (J)
|-
| Kadathian
| J
| K@
| L
| M
| N@
| O
| P
| Q
| (J)
|-
| Hlanithian
| J
| K@
| L
| M
| N@
| O
| P@
| Q
| (J)
|-
| Sarnathian
| J
| K@
| L
| M@
| N@
| O
| P@
| Q
| (J)
|}
For classical-inspired functional harmony, we use the terms ''(Functional) Oneiromajor'' and ''(Functional) Oneirominor'': Oneiromajor for Illarnekian where the 6th degree (the rising fifth) can be sharpened, and Oneirominor for Ultharian where the 8th degree (the leading tone) can be sharpened. The respective purposes of these alterations are:
# in Oneiromajor, to have both major (requiring a sharpened 6th degree) on the flat fourth "subdominant" and the sharp fifth as "dominant"
# in Oneirominor, to have both the flat 8th degree as the dominant of the "mediant" (relative major) and the sharp 8th degree as leading tone
In key signatures, Oneirominor should be treated as Ultharian and Oneiromajor should be treated as Illarnekian. Note that Oneiromajor and Oneirominor still have the relative major-minor relationship; they are related by a major mosthird, just like diatonic major/minor.
=== Alterations ===
==== Archeodim ====
We call the LSLLLSLS pattern (independently of modal rotation) '''archeodim''', because the "LLL" resembles the [[archeotonic]] scale in 13edo and the "LSLSLS" resembles the diminished scale. Archeodim is the most important oneirotonic [[MODMOS]] pattern (a MODMOS is a MOS with one or more alterations), because it allows one to evoke certain ana or kata diatonic modes where three whole steps in a row are important (Dorian, Phrygian, Lydian or Mixo) in an octatonic context. The MOS would not always be able to do this because it has at most two consecutive large steps. Archeodim modes exist in all oneirotonic tunings, since they use the same large and small steps as the oneirotonic scale itself.
As with the MOS, archeodim has four ana and four kata rotations:
* Ana:
** LLLSLSLS: Dylathian &4, Dylydian
** LLSLSLSL: Illarnekian @8, Illarmixian
** LSLLLSLS: Celephaïsian &6, Celdorian
** SLLLSLSL: Ultharian @2, Ulphrygian
* Kata:
** LSLSLLLS: Mnarian &8, Mnionian
** SLSLLLSL: Sarnathian &7, Sardorian
** LSLSLSLL: Mnarian @7, Mnaeolian
** SLSLSLLL: Sarnathian @6, Sarlocrian
{| class="wikitable"
|-
|+ Table of archeodim modes (based on J)
|-
! Mode
! 1
! 2
! 3
! 4
! 5
! 6
! 7
! 8
! (9)
|-
| Dylydian
| J
| K
| L&
| M&
| N
| O&
| P
| Q&
| (J)
|-
| Illarmixian
| J
| K
| L&
| M
| N
| O
| P
| Q
| (J)
|-
| Celdorian
| J
| K
| L
| M
| N
| O&
| P
| Q&
| (J)
|-
| Ulphrygian
| J
| K@
| L
| M
| N
| O
| P
| Q
| (J)
|-
| Mnionian
| J
| K
| L
| M
| N@
| O
| P
| Q&
| (J)
|-
| Sardorian
| J
| K@
| L
| M@
| N@
| O
| P
| Q
| (J)
|-
| Mnaeolian
| J
| K
| L
| M
| N@
| O
| P@
| Q
| (J)
|-
| Sarlocrian
| J
| K@
| L
| M@
| N@
| O@
| P@
| Q
| (J)
|}
==== Other MODMOSes ====
Other potentially interesting oneirotonic MODMOSes (that do not use half-sharps or half-flats) are:
* the distorted harmonic minor LSLLSALS (A = aug oneiro2nd = L + chroma)
* the distorted Freygish SASLSLLS
* Celephaïsian &4 &6 LsAsLsLs
== Hypohard oneiro theory ==
Oneirotonic is often used as distorted diatonic. Because distorted diatonic modal harmony and functional harmony both benefit from a recognizable major third, the following theory essentially assumes a hypohard tuning, i.e. an oneirotonic tuning with generator between 5\13 and 7\18 (or possibly an approximation of such a tuning, such as a [[neji]]). The reader is encouraged to experiment and see what ideas work for other oneirotonic tunings.
=== Ana modes ===
We call modes with a major mos5th ''ana modes'' (from Greek for 'up'), because the sharper 5th degree functions as a flattened melodic fifth when moving from the tonic up. The ana modes of the MOS are the 4 brightest modes, namely Dylathian, Illarnekian, Celephaïsian and Ultharian.
The ana modes have squashed versions of the classical major and minor pentachords R-M2-M3-P4-P5 and R-M2-m3-P4-P5 and can be viewed as providing a distorted version of classical diatonic functional harmony. For example, in the Dylathian mode, the 4:5:9 triad on the sixth degree can sound like both "V" and "III of iv" depending on context.
In pseudo-classical functional harmony, the 6th scale degree (either an augmented mossixth or a perfect mossixth) could be treated as mutable. The perfect mossixth would be used when invoking the diatonic V-to-I trope by modulating by a perfect mosfourth from the sixth degree "dominant". The augmented mossixth would be used when a major key needs to be used on the fourth degree "subdominant".
==== Pentatonic subsets ====
The ''Oneiro Falling Suspended Pentatonic'', i.e. R-Mo2-Po4-Mo5-Mo7 (on J, J-K-M-N-P), is also an important subset in ana modes: it roughly implies the "least" tonality (In particular, it only implies ana-ness, not major or minor tonality), and it sounds floaty, and suspended, much like suspended and quartal chords do in diatonic contexts. The ''Oneiro Rising Suspended Pentatonic'' R-Mo2-Po4-Po6-Mo7 (J-K-M-O-P) can be used for similar effect.
Oneiro has at least two different types of "V-to-I" resolution because of the two fifth sizes:
# One uses the sharp fifth as the "V" and uses a true major third. The sharp "V" voiceleads naturally to the flat fifth in the resolved falling tonic triad on the I: e.g. P6-M8-P2 > M5-P1-(M/m)3.
# One uses the flat fifth as the "V" and the chord on the "V" is a "false major triad" R-P4-P6 (root-falling 4th-rising 5th).
Some suggested basic ana functional harmony progressions are listed below, outlined very roughly. Note that VI, VII and VIII are sharp 5th, 6th-like and 7th-like degrees respectively. A Roman numeral without maj or min means either major or minor. The "Natural" Roman numerals follow the Illarnekian mode.
* I-IVmin-VImaj-I
* Imaj-VIImin-IVmin-Imaj
* Imin-@IIImaj-VImaj-Imaj
* Imin-@IIImaj-Vdim-VImaj-Imin
* Imin-@VIIImin-IIImaj-VImaj-Imin
* Imin-IVmin-@VIIImin-@IIImaj-VImaj-Imin
* Imin-IVmin-IIdim-VImaj-Imin
* Imin-IVmin-IIdim-@IIImaj-Imin
* I-VIImin-IImin-VImaj-I
* Imaj-VIImin-IVmin-VImaj-Imaj
* Modulations by major mos2nd:
** I-IV-VII-II
** I-IVmaj-II
** I-VIImin-II
* Modulations by major mos3rd:
** Modulate up major mos2nd twice
** Imin-VImin-III (only in 13edo)
** Imaj-&VImin-III (only in 13edo)
* Modulations by minor mos3rd:
** I-VI-@III
** I-IVmin-VImin-@VIIImaj-@III
Another Western-classical-influenced approach to oneirotonic chord progressions is to let the harmony emerge from counterpoint. This would allow, for example, using the perfect oneirofourth and minor oneirofifth (instead of the major oneirothird and the perfect oneirofourth) as stand-ins for major thirds and fourths in neobaroque contexts (this adds some dissonance which might be what you want sometimes, e.g. in a chord that is supposed to resolve to a more consonant chord).
===== Samples =====
[[File:Oneiro Baroque Exercises 13edo.mp3]]
(A short contrapuntal 13edo keyboard exercise, meant to be played in all 13 keys. The first part is in Oneiromajor, i.e. Illarnekian with mutable 6th degree, and the second part is in Oneirominor, i.e. Celephaïsian with mutable 7th degree.)
[[File:Oneiro Baroque Exercises 18edo.mp3]]
([[18edo]])
[[File:Oneiro Baroque Exercises 31edo.mp3]]
([[31edo]])
[[File:Oneiro Classical Exercises 21edo.mp3]]
([[21edo]] for comparison)
[[File:Oneirotonic 3 part sample.mp3]]
(A rather classical-sounding 3-part harmonization of the ascending J Illarnekian scale; tuning is 13edo)
=== Kata modes ===
We call modes with a minor mos5th ''kata modes'' (from Greek for 'down'). The kata modes of the MOS are the 4 darkest modes, namely Mnarian, Kadathian, Hlanithian and Sarnathian. In kata modes, the melodically squashed fifth from the tonic downwards is the flatter 5th degree. Kata modes could be used to distort diatonic tropes that start from the tonic and work downwards or work upwards towards the tonic from below it. For example:
* Mnarian (LSLSLLSL) and Kadathian (SLLSLLSL) are kata-Mixolydians
* Hlanithian (SLLSLSLL) is a kata-melodic major (the 4th degree sounds like a major third; it's actually a perfect mosfourth.)
* Sarnathian (SLSLLSLL) is a kata-melodic minor (When starting from the octave above, the 4th degree sounds like a minor third; it's actually a diminished mosfourth.)
When used in an "ana" way, the kata modes are radically different in character than the ana modes. Particularly in 13edo and tunings close to it, the fifth and seventh scale degrees become the more concordant 11/8 and quasi-13/8 respectively, so they may sound more like stable scale functions. Hlanithian, in particular, may be like a more stable version of the Locrian mode in diatonic.
=== Chords and extended harmony ===
Chords are given in oneirotonic MOS interval notation. For example, M5 means major mosfifth (squashed fifth).
"Rising" means that a triad uses the perfect mos6th (major 5th); "falling" means that a triad uses a major mos5th (minor 5th)
* R-Mo3-Mo5: Falling Major Triad; Squashed Major Triad
* R-mo3-Mo5: Falling Minor Triad; Squashed Minor Triad
* R-mo3-mo5: Squashed Dim Triad
* R-Mo3-Ao5: Squashed Aug Triad
* R-Mo3-Mo5-Ao6: Falling Major Triad Add6
* R-mo3-Mo5-Ao6: Falling Minor Triad Add6
* R-Mo3-Mo5-Mo7: Falling Major Tetrad
* R-mo3-Mo5-Mo7: Falling Minor Tetrad
* R-mo3-mo5-Mo7: Half-Diminished Tetrad
* R-mo3-mo5-mo7: Orwell Tetrad, Diminished Tetrad
* R-Mo3-Ao6: Squashed 1st Inversion Minor Triad; Sephiroth Triad (approximates 8:10:13 in 13edo and 31edo)
* R-Mo3-Ao6-Mo8: Sephiroth Triad Add7
* R-Mo3-Ao6-(Mo2)-(Po4): Sephiroth Tetrad Add9
* R-Mo3-Ao6-(mo2)-(Po4): Sephiroth Tetrad Addm9
* R-Mo3-Ao6-(Po4): Sephiroth Tetrad
* R-mo3-Po6: Rising Minor Triad; Squashed 1st Inversion Major Triad
* R-Mo3-Po6: Rising Major Triad
* R-mo3-Mo7: Minor add6 no5
* R-mo3-mo7: Minor addm6 no5
* R-mo5-Mo7: Falling no3 add6
* R-mo5-mo7: Falling no3 add6
* R-Mo3-Mo8: Major 7th no5
* R-mo3-Mo8: Minor Major 7th no5
* R-Mo3-Mo5-Mo8: Falling Major Seventh Tetrad
* R-mo3-Mo5-Mo8: Falling Minor Major Seventh Tetrad
* R-Mo3-Mo7-Mo8: no5 Major Seventh Add6
* R-mo3-Mo7-Mo8: no5 Minor Major Seventh Add6
* R-Mo3-Po6-Mo8: Rising Major Seventh
* R-mo3-Po6-Mo8: Rising Oneiro Minor Major Seventh
21edo has the [[Step ratio|soft]] [[oneirotonic]] (5L 3s) MOS with generator 8\21; in addition to the [[naiadic]]s (457.14¢) and extremely sharp fifths (742.85¢) that generate it, it has neutral thirds (instead of major thirds as in [[13edo]] oneirotonic), neogothic minor thirds, and Baroque diatonic semitones. The oneirofifths (4-step intervals) are more tritone-like than fifth-like, unlike in 13edo, although they do have a consonant, even JI-like quality to them. In terms of JI, it mainly approximates 5:9:11:13 (R-mo8-Mo2-Po4) and 16:23:30 (R-Mo5-Mo8). Importantly, the sharp fifth is now harmonically much more fifth-like than the flat fifth, unlike in [[13edo]] and harder tunings. Rather than squashed tertian triads, it may be preferable to use triads with sharp fifths, quartal harmony, stacks of seconds and thirds, third+sixth and third+seventh chords, and using the JI approximations (subsets of 5:9:11:13 (R-mo8-Mo2-Po4), 9:10:11:13 (R-Mo2-Mo3-Mo5), and 8:15:23 (R-Mo7-Mo5)).
== Primodal theory ==
:''Main article: [[5L 3s/Primodal theory]]''
== Temperaments ==
:''Main article: [[5L 3s/Temperaments]]''
== Samples ==
[[File:13edo Prelude in J Oneirominor.mp3]]
[[File:13edo Prelude in J Oneirominor.mp3]]
Line 1,141:
Line 188:
[[File:A Moment of Respite.mp3]]
[[File:A Moment of Respite.mp3]]
(13edo, L Illarnekian)
(13edo, L Ilarnekian)
[[File:Lunar Approach.mp3]]
[[File:Lunar Approach.mp3]]
Line 1,147:
Line 194:
(by [[Igliashon Jones]], 13edo, J Celephaïsian)
(by [[Igliashon Jones]], 13edo, J Celephaïsian)
== See also ==
=== 13edo Oneirotonic Modal Studies ===
* [[Well-Tempered 13-Tone Clavier]] (collab project to create 13edo oneirotonic keyboard pieces in a variety of keys and modes)
* [[File:Inthar-13edo Oneirotonic Studies 1 Dylathian.mp3]]: Tonal Study in Dylathian
* [[File:Inthar-13edo Oneirotonic Studies 2 Ultharian.mp3]]: Tonal Study in Ultharian
* [[File:Inthar-13edo Oneirotonic Studies 3 Hlanithian.mp3]]: Tonal Study in Hlanithian
* [[File:Inthar-13edo Oneirotonic Studies 4 Illarnekian.mp3]]: Tonal Study in Ilarnekian
* [[File:Inthar-13edo Oneirotonic Studies 5 Mnarian.mp3]]: Tonal Study in Mnarian
* [[File:Inthar-13edo Oneirotonic Studies 6 Sarnathian.mp3]]: Tonal Study in Sarnathian
* [[File:Inthar-13edo Oneirotonic Studies 7 Celephaisian.mp3]]: Tonal Study in Celephaïsian
* [[File:Inthar-13edo Oneirotonic Studies 8 Kadathian.mp3]]: Tonal Study in Kadathian
== Scale tree ==
== Scale tree ==
{| class="wikitable" style="text-align:center;"
{{MOS tuning spectrum
|-
| 13/8 = Golden oneirotonic (458.3592{{c}})
! colspan="5" | generator
| 13/5 = Golden A-Team (465.0841{{c}})
! | tetrachord
}}
! | g in cents
! | 2g
! | 3g
! | 4g
! | Comments
|-
| | 2\5
| |
| |
| |
| |
| | 1 0 1
| | 480.000
| | 960.000
| | 240.00
| | 720.000
| |
|-
| | 21\53
| |
| |
| |
| |
| | 10 1 10
| | 475.472
| | 950.943
| | 226.415
| | 701.887
| |
|-
| | 19\48
| |
| |
| |
| |
| | 9 1 9
| | 475
| | 950
| | 225
| | 700
| |
|-
| | 17\43
| |
| |
| |
| |
| | 8 1 8
| | 474.419
| | 948.837
| | 223.256
| | 697.674
| |
|-
| | 15\38
| |
| |
| |
| |
| | 7 1 7
| | 473.684
| | 947.368
| | 221.053
| | 694.737
| |
|-
| | 13\33
| |
| |
| |
| |
| | 6 1 6
| | 472.727
| | 945.455
| | 218.181
| | 690.909
| |
|-
| | 11\28
| |
| |
| |
| |
| | 5 1 5
| | 471.429
| | 942.857
| | 214.286
| | 685.714
| |
|-
| | 9\23
| |
| |
| |
| |
| | 4 1 4
| | 469.565
| | 939.130
| | 208.696
| | 678.261
| | L/s = 4
|-
|
|16\41
|
|
|
|7 2 7
|468.293
|936.585
|204.878
|673.171
|
|-
| | 7\18
| |
| |
| |
| |
| | 3 1 3
| | 466.667
| | 933.333
| | 200.000
| | 666.667
| | L/s = 3
|-
| |
| | 19\49
| |
| |
| |
| | 8 3 8
| | 465.306
| | 930.612
| | 195.918
| | 661.2245
| |
|-
| |
| |
| | 50\129
| |
| |
| | 21 8 21
| | 465.116
| | 930.233
| | 195.349
| | 660.465
| |
|-
| |
| |
| |
| | 131\338
| |
| | 55 21 55
| | 465.089
| | 930.1775
| | 195.266
| | 660.335
| |
|-
| |
| |
| |
| |
| | 212\547
| | 89 34 89
| | 465.082
| | 930.1645
| | 195.247
| | 660.329
| |
|-
| |
| |
| |
| | 81\209
| |
| | 34 13 34
| | 465.072
| | 930.1435
| | 195.215
| | 660.287
| |
|-
| |
| |
| | 31\80
| |
| |
| | 13 5 13
| | 465
| | 930
| | 195
| | 660
| |
|-
| |
| | 12\31
| |
| |
| |
| | 5 2 5
| | 464.516
| | 929.032
| | 193.549
| | 658.065
| |
|-
| | 5\13
| |
| |
| |
| |
| | 2 1 2
| | 461.538
| | 923.077
| | 184.615
| | 646.154
| | Boundary of propriety (generators smaller than this are proper)
|-
| |
| | 13\34
| |
| |
| |
| | 5 3 5
| | 458.824
| | 917.647
| | 176.471
| | 635.294
| |
|-
| |
| |
| | 34\89
| |
| |
| | 13 8 13
| | 458.427
| | 916.854
| | 175.281
| | 633.708
| |
|-
| |
| |
| |
| | 89\233
| |
| | 34 21 34
| | 458.369
| | 916.738
| | 175.107
| | 633.473
| |
|-
| |
| |
| |
| |
| | 233\610
| | 89 55 89
| | 458.361
| | 916.721
| | 175.082
| | 633.443
| | Golden oneirotonic; generator is 2 octaves minus logarithmic [[phi]]
|-
| |
| |
| |
| | 144\377
| |
| | 55 34 55
| | 458.355
| | 916.711
| | 175.066
| | 633.422
| |
|-
| |
| |
| | 55\144
| |
| |
| | 21 13 21
| | 458.333
| | 916.666
| | 175
| | 633.333
| |
|-
| |
| | 21\55
| |
| |
| |
| | 8 5 8
| | 458.182
| | 916.364
| | 174.545
| | 632.727
| |
|-
| | 8\21
| |
| |
| |
| |
| | 3 2 3
| | 457.143
| | 914.286
| | 171.429
| | 628.571
| | Optimum rank range (L/s=3/2) oneirotonic
|-
| | 11\29
| |
| |
| |
| |
| | 4 3 4
| | 455.172
| | 910.345
| | 165.517
| | 620.690
| |
|-
| | 14\37
| |
| |
| |
| |
| | 5 4 5
| | 454.054
| | 908.108
| | 162.162
| | 616.216
| |
|-
| | 17\45
| |
| |
| |
| |
| | 6 5 6
| | 453.333
| | 906.667
| | 160
| | 613.333
| |
|-
| | 20\53
| |
| |
| |
| |
| | 7 6 7
| | 452.83
| | 905.66
| | 158.491
| | 611.321
| |
|-
| | 23\61
| |
| |
| |
| |
| | 8 7 8
| | 452.459
| | 904.918
| | 157.377
| | 609.836
| |
|-
| | 26\69
| |
| |
| |
| |
| | 9 8 9
| | 452.174
| | 904.348
| | 156.522
| | 608.696
| |
|-
| | 29\77
| |
| |
| |
| |
| | 10 9 10
| | 451.948
| | 903.896
| | 155.844
| | 607.792
| |
|-
| | 3\8
| |
| |
| |
| |
| | 1 1 1
| | 450.000
| | 900.000
| | 150.000
| | 600.000
| |
|}
[[Category:Scales]]
[[Category:Oneirotonic| ]] <!-- sort order in category: this page shows above A -->
[[Category:Oneirotonic| ]] <!-- sort order in category: this page shows above A -->
[[Category:Mos]]
[[Category:Pages with internal sound examples]]
[[Category:MOS scales]]
[[Category:Abstract MOS patterns]][[Category:Oneirotonic]]
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).
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.
Scale properties
Intervals
Intervals of 5L 3s
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
Generator chain of 5L 3s
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
Scale degrees of the modes of 5L 3s
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.
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Maj.
Maj.
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5|2
7
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Maj.
Min.
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Maj.
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Maj.
Maj.
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4|3
2
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Perf.
Maj.
Min.
Perf.
Maj.
Perf.
Maj.
Min.
Perf.
3|4
5
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Maj.
Min.
Perf.
Min.
Perf.
Maj.
Min.
Perf.
2|5
8
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Perf.
Min.
Min.
Perf.
Min.
Perf.
Maj.
Min.
Perf.
1|6
3
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Min.
Min.
Perf.
Min.
Perf.
Min.
Min.
Perf.
0|7
6
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Dim.
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Proposed mode names
The following names have been proposed for the modes of 5L 3s, and are named after cities in the Dreamlands.
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.
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 ¢).
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.)
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.
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 22edosuperpythdiatonic and a warped 24edosemaphoresemiquartal (and both nearby scales are superhard MOSes).
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.