View source for 5L 3s

{{Infobox MOS
| Neutral = 2L 6s
}}
:''For the tritave-equivalent MOS structure with the same step pattern, see [[5L 3s (3/1-equivalent)]].''
{{MOS intro}}
5L 3s is a [[Warped diatonic|warped diatonic scale]], because it has one extra small step compared to diatonic ([[5L 2s]]): for example, the Ionian diatonic mode LLsLLLs can be warped to the Dylathian mode LLsLLsLs.

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 &minus; 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'''

== 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. 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]].

== 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 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.
{| class="wikitable center-all"
|-
! 
!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
|-
| colspan="8" style="text-align:center" |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
|-
| 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):
|-
| 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
|}

== Tuning ranges==
===Simple tunings ===
Table of intervals in the simplest oneirotonic tunings:
{| class="wikitable right-2 right-3 right-4 sortable "
|-
! class="unsortable" |Degree
! Size in 13edo (basic)
! Size in 18edo (hard)
! Size in 21edo (soft)
! class="unsortable" |Note name on J
! #Gens up
|- bgcolor="#eaeaff"
| 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
|- bgcolor="#eaeaff"
| minor 2-step
| 3\13, 276.92
| 4\18, 266.67
| 5\21, 285.71
| L
| -2
|- bgcolor="#eaeaff"
| 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
|- bgcolor="#eaeaff"
| minor 4-step
| 6\13, 553.85
| 8\18, 533.33
| 10\21, 571.43
| N@
| -4
|- bgcolor="#eaeaff"
| 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
|- bgcolor="#eaeaff"
| minor 6-step
| 9\13, 830.77
| 12\18, 800.00
| 15\21, 857.14
| P@
|  -6
|- bgcolor="#eaeaff"
| 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===
[[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.

Also, in [[18edo]] and [[31edo]], the minor 2-mosstep is close to [[7/6]].

The set of identifications above is associated with [[5L 3s/Temperaments#A-Team|A-Team]] temperament.

EDOs that are in the hypohard range include [[13edo]], [[18edo]], and [[31edo]].
* 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. 
{| class="wikitable right-2 right-3 right-4 right-5"
|-
!
![[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:

{| class="wikitable right-2 right-3 right-4 sortable "
|-
! 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
|- bgcolor="#eaeaff"
| 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
|- bgcolor="#eaeaff"
| minor 2-step
| 3\13, 276.92
| 4\18, 266.67
| 7\31, 270.97
| L
| 7/6
| -2
|- bgcolor="#eaeaff"
| 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
|- bgcolor="#eaeaff"
| minor 4-step
| 6\13, 553.85
| 8\18, 533.33
| 14\31, 541.94
| N@
| ''11/8''
| -4
|- bgcolor="#eaeaff"
| 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
|- bgcolor="#eaeaff"
| minor 6-step
| 9\13, 830.77
| 12\18, 800.00
| 21\31, 812.90
| P@
| 8/5
| -6
|- bgcolor="#eaeaff"
| 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
|}
<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 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 [[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 sizes of the generator, large step and small step of oneirotonic are as follows in various hyposoft oneiro tunings (13edo not shown).
{| class="wikitable right-2 right-3 right-4 right-5"
|-
!
! [[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):

{| class="wikitable right-2 right-3 sortable "
|-
! class="unsortable" |Degree
! Size in 21edo (soft)
! Size in 34edo (semisoft)
! class="unsortable" |Note name on J
! class="unsortable" |Approximate ratios
! #Gens up
|- bgcolor="#eaeaff"
| 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
|- bgcolor="#eaeaff"
| minor 2-step
| 5\21, 285.71
| 8\34, 282.35
| L
| 13/11, 20/17
| -2
|- bgcolor="#eaeaff"
| 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
|- bgcolor="#eaeaff"
| minor 4-step
| 10\21, 571.43
| 16\34, 564.72
| N@
| 18/13, 32/23
| -4
|- bgcolor="#eaeaff"
| 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
|- bgcolor="#eaeaff"
| minor 6-step
| 15\21, 857.14
| 24\34, 847.06
| P@
| 18/11
| -6
|- bgcolor="#eaeaff"
| 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.
{| class="wikitable right-2 right-3 right-4 right-5"
|-
!
! [[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.
{| class="wikitable right-2 right-3 sortable "
|-
! class="unsortable" |Degree
! Size in 29edo (supersoft)
! class="unsortable" |Note name on J
! class="unsortable" |Approximate ratios (29edo)
!#Gens up
|- bgcolor="#eaeaff"
| unison
| 0\29, 0.00
| J
| 1/1
| 0
|- bgcolor="#eaeaff"
| oneirochroma
| 1\29, 41.4
| J&
|
| +8
|-
| 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
|- bgcolor="#eaeaff"
| dim. 2-step
| 6\29, 248.3
| L@
| 15/13
| -10
|- bgcolor="#eaeaff"
| minor 2-step
| 7\29, 289.7
| L
| 13/11
| -2
|- bgcolor="#eaeaff"
| major 2-step
| 8\29, 331.0
| L&
|
| +6
|- bgcolor="#eaeaff"
| 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
|- bgcolor="#eaeaff"
| dim. 4-step
| 13\29, 537.9
| N@@
| 15/11
| -12
|- bgcolor="#eaeaff"
| minor 4-step
| 14\29, 579.3
| N@
| 7/5
| -4
|- bgcolor="#eaeaff"
| major 4-step
| 15\29 620.7
| N
| 10/7
| +4
|- bgcolor="#eaeaff"
| 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
|- bgcolor="#eaeaff"
| dim. 6-step
| 20\29 827.6
| P@@
|
| -14
|- bgcolor="#eaeaff"
| minor 6-step
| 21\29 869.0
| P@
|
| -6
|- bgcolor="#eaeaff"
| major 6-step
| 22\29, 910.3
| P
| 22/13
| +2
|- bgcolor="#eaeaff"
| 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
|- bgcolor="#eaeaff"
| dim. 8-step
| 28\29, 1158.6
| J@
|
| -8
|}

===Parahard===
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).
====Intervals====
The intervals of the extended generator chain (-12 to +12 generators) are as follows in various oneirotonic tunings close to [[23edo]]. 
{| class="wikitable right-2 right-3 sortable "
|-
! class="unsortable" |Degree
! Size in 23edo (superhard)
! class="unsortable" |Note name on J
! class="unsortable" |Approximate ratios (23edo)
! #Gens up
|- bgcolor="#eaeaff"
| unison
| 0\23, 0.0
| J
| 1/1
| 0
|- bgcolor="#eaeaff"
| oneirochroma
| 3\23, 156.5
| J&
|
| +8
|-
| 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
|- bgcolor="#eaeaff"
| dim. 2-step
| 2\23, 104.3
| L@
| 17/16
| -10
|- bgcolor="#eaeaff"
| minor 2-step
| 5\23, 260.9
| L
|
| -2
|- bgcolor="#eaeaff"
| 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
|- bgcolor="#eaeaff"
| dim. 4-step
| 7\23, 365.2
| N@@
| 21/17, inverse φ
| -12
|- bgcolor="#eaeaff"
| minor 4-step
| 10\23, 521.7
| N@
|
| -4
|- bgcolor="#eaeaff"
| major 4-step
| 13\23, 678.3
| N
|
| +4
|- bgcolor="#eaeaff"
| 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
|- bgcolor="#eaeaff"
| minor 6-step
| 15\23 782.6
| P@
| 11/7
| -6
|- bgcolor="#eaeaff"
| major 6-step
| 18\23, 939.1
| P
|
| +2
|- bgcolor="#eaeaff"
| 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
|-
|- bgcolor="#eaeaff"
| dim. 8-step
| 20\23, 1043.5
| J@
|
| -8
|}

===Ultrahard===
[[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 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. 

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.
{| class="wikitable right-2 right-3 right-4 right-5"
|-
!
! [[38edo]]
! [[53edo]]
! [[63edo]]
! Optimal ([[POTE]]) Buzzard tuning
! JI intervals represented (2.3.5.7.13 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
|}

==== Intervals====
Sortable table of intervals in the Dylathian mode and their Buzzard interpretations:

{| class="wikitable right-2 right-3 right-4 right-5 sortable"
|-
! Degree
! Size in 38edo
! Size in 53edo
! Size in 63edo
! Size in POTE tuning
! Note name on Q
! class="unsortable" |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
|}

== Modes ==
Oneirotonic modes are named after cities in the Dreamlands.
{| class="wikitable center-all"
|-
! Mode
! [[Modal UDP Notation|UDP]]
! Name
! Other Names
|-
| LLsLLsLs
| <nowiki>7|0</nowiki>
| Dylathian
| Dylian
|-
| LLsLsLLs
| <nowiki>6|1</nowiki>
| Ilarnekian
| 
|-
| LsLLsLLs
| |<nowiki>5|2</nowiki>
| Celephaïsian
| 
|-
| LsLLsLsL
| <nowiki>4|3</nowiki>
| Ultharian
| 
|-
| LsLsLLsL
| <nowiki>3|4</nowiki>
| Mnarian
| 
|-
| sLLsLLsL
| <nowiki>2|5</nowiki>
| Kadathian
| Kadian
|-
| sLLsLsLL
| <nowiki>1|6</nowiki>
| Hlanithian
| Hlanian
|-
| sLsLLsLL
| <nowiki>0|7</nowiki>
| Sarnathian
| Sarnian
|}

==Approaches==
* [[5L 3s/Inthar's approach]]
* [[5L 3s/Temperaments]]

==Samples==
[[File:The Angels' Library edo.mp3]] [[:File:The Angels' Library edo.mp3|The Angels' Library]] by [[Inthar]] in the Sarnathian (23233233) mode of 21edo oneirotonic ([[:File:The Angels' Library Score.pdf|score]])

[[File:13edo Prelude in J Oneirominor.mp3]]

[[WT13C]] [[:File:13edo Prelude in J Oneirominor.mp3|Prelude II (J Oneirominor)]] ([[:File:13edo Prelude in J Oneirominor Score.pdf|score]]) – Simple two-part Baroque piece. It stays in oneirotonic even though it modulates to other keys a little.

[[File:13edo_1MC.mp3]] 

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

[[File:A Moment of Respite.mp3]]

(13edo, L Ilarnekian)

[[File:Lunar Approach.mp3]]

(by [[Igliashon Jones]], 13edo, J Celephaïsian)

=== 13edo Oneirotonic Modal Studies ===
* [[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

==See also==
* [[Well-Tempered 13-Tone Clavier]] (collab project to create 13edo oneirotonic keyboard pieces in a variety of keys and modes)
==Scale tree==
Generator ranges:
* Bright generator: 450 cents (3\8) to 480 cents (2\5)
* Dark generator: 720 cents (3\5) to 750 cents (5\8)

{| class="wikitable center-all"
! colspan="6" |Bright 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||
|-
| || || || || || 25\66 || 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||
|-
| || || || || || 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 (458.3592¢)
|-
| || || || 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 <br>(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||Golden A-Team (465.0841¢)
|-
| || || || || 19\49 || || 465.306 ||8||3||2.667||
|-
| || || || || || 26\67 || 465.672 ||11||4||2.750||
|-
| || || 7\18 || || || || 466.667 ||3||1||3.000||
|-
| || || || || || 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||
|}

[[Category:Oneirotonic| ]] <!-- sort order in category: this page shows above A -->
[[Category:Pages with internal sound examples]]
[[Category:8-tone scales]]