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{{Interwiki
| en = 5L 3s
| de =
| es =
| ja =
| ko = 5L3s (Korean)
}}
{{Infobox MOS
{{Infobox MOS
| Neutral = 2L 6s
| Neutral = 2L 6s
}}
}}
:''For the tritave-equivalent MOS structure with the same step pattern, see [[5L 3s (3/1-equivalent)]].''
: ''For the tritave-equivalent MOS structure with the same step pattern, see [[5L 3s (3/1-equivalent)]].''
{{MOS intro}}
{{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 can be seen as a [[Warped diatonic|warped diatonic scale]], because it has one extra small step compared to diatonic ([[5L 2s]]).
 
5L 3s has a pentatonic MOS subset [[3L 2s]] (SLSLL). (Note: [[3L 5s]] scales also have 3L 2s subsets.)


== Name ==
== Name ==
{{TAMNAMS name}}
{{TAMNAMS name}} '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]].
'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 ==
== Scale properties ==
{{TAMNAMS use}}


=== Intervals ===
=== Intervals ===
{{MOS intervals}}
{{MOS intervals}}
=== Generator chain ===
{{MOS genchain}}


=== Modes ===
=== Modes ===
{{MOS mode degrees}}
{{MOS mode degrees}}
====Proposed names====
Oneirotonic modes are named after cities in the Dreamlands.
{| class="wikitable center-all"
|-
! Mode
![[Modal UDP Notation|UDP]]
! Name
|-
| LLsLLsLs
|<nowiki>7|0</nowiki>
| Dylathian
|-
| 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
|-
| sLLsLsLL
|<nowiki>1|6</nowiki>
| Hlanithian
|-
| sLsLLsLL
|<nowiki>0|7</nowiki>
| Sarnathian
|}


== Tuning ranges==
==== Proposed mode names ====
===Simple tunings ===
The following names have been proposed for the modes of 5L 3s, and are named after cities in the Dreamlands.
Table of intervals in the simplest oneirotonic tunings:
{{MOS modes
{| class="wikitable right-2 right-3 right-4 sortable "
| Mode Names=
|-
Dylathian $
! class="unsortable" |Degree
Ilarnekian $
! Size in 13edo (basic)
Celephaïsian $
! Size in 18edo (hard)
Ultharian $
! Size in 21edo (soft)
Mnarian $
! #Gens up
Kadathian $
|- style="background-color: #eaeaff;"
Hlanithian $
| unison
Sarnathian $
| 0\13, 0.00
| Collapsed=1
| 0\18, 0.00
}}
| 0\21, 0.00
| 0
|-
| minor step
| 1\13, 92.31
| 1\18, 66.67
| 2\21, 114.29
| -5
|-
| major step
| 2\13, 184.62
| 3\18, 200.00
| 3\21, 171.43
| +3
|- style="background-color: #eaeaff;"
| minor 2-step
| 3\13, 276.92
| 4\18, 266.67
| 5\21, 285.71
| -2
|- style="background-color: #eaeaff;"
| major 2-step
| 4\13, 369.23
| 6\18, 400.00
| 6\21, 342.86
| +6
|-
| dim. 3-step
| 4\13, 369.23
| 5\18, 333.33
| 7\21, 400.00
| -7
|-
| perf. 3-step
| 5\13, 461.54
| 7\18, 466.67
| 8\21, 457.14
| +1
|- style="background-color: #eaeaff;"
| minor 4-step
| 6\13, 553.85
| 8\18, 533.33
| 10\21, 571.43
| -4
|- style="background-color: #eaeaff;"
| major 4-step
| 7\13, 646.15
| 10\18, 666.66
| 11\31, 628.57
| +4
|-
| perf. 5-step
| 8\13, 738.46
| 11\18, 733.33
| 13\21, 742.86
| -1
|-
| aug. 5-step
| 9\13, 830.77
| 13\18, 866.66
| 14\21, 800.00
| +7
|- style="background-color: #eaeaff;"
| minor 6-step
| 9\13, 830.77
| 12\18, 800.00
| 15\21, 857.14
|  -6
|- style="background-color: #eaeaff;"
| major 6-step
| 10\13, 923.08
| 14\18, 933.33
| 16\21, 914.29
| +2
|-
| minor 7-step
| 11\13, 1015.39
| 15\18, 1000.00
| 18\21, 1028.57
| -3
|-
| major 7-step
| 12\13, 1107.69
| 17\18, 1133.33
| 19\21, 1085.71
| +5
|}


===Hypohard===
== Tunings==
[[Hypohard]] oneirotonic tunings (with generator between 5\13 and 7\18) have step ratios between 2/1 and 3/1.  
=== Simple tunings ===
The simplest tuning for 5L&nbsp;3s correspond to 13edo, 18edo, and 21edo, with step ratios 2:1, 3:1, and 3:2, respectively.


Hypohard oneirotonic can be considered "meantone oneirotonic". This is because these tunings share the following features with [[meantone]] diatonic tunings:  
{{MOS tunings|JI Ratios=Int Limit: 30; Prime Limit: 19; Tenney Height: 7.7}}
* 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]].
=== 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.


The set of identifications above is associated with [[5L 3s/Temperaments#A-Team|A-Team]] temperament.
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]].
EDOs that are in the hypohard range include [[13edo]], [[18edo]], and [[31edo]], and are associated with [[5L 3s/Temperaments#A-Team|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.
* 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.
* 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.
* 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.
* 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.
* [[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.
 
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 "
{{MOS tunings|Step Ratios=Hypohard|JI Ratios=Subgroup: 2.5.9.21; Int Limit:40; Complements Only: 1|Tolerance=15}}
|-
! class="unsortable" |Degree
! Size in 13edo (basic)
! Size in 18edo (hard)
! Size in 31edo (semihard)
! class="unsortable" |Approximate ratios<ref>The ratio interpretations that are not valid for 18edo are italicized.</ref>
! #Gens up
|- style="background-color: #eaeaff;"
| unison
| 0\13, 0.00
| 0\18, 0.00
| 0\31, 0.00
| 1/1
| 0
|-
| minor step
| 1\13, 92.31
| 1\18, 66.67
| 2\31, 77.42
| 21/20, ''22/21''
| -5
|-
| major step
| 2\13, 184.62
| 3\18, 200.00
| 5\31, 193.55
| 9/8, 10/9
| +3
|- style="background-color: #eaeaff;"
| minor 2-step
| 3\13, 276.92
| 4\18, 266.67
| 7\31, 270.97
| 7/6
| -2
|- style="background-color: #eaeaff;"
| major 2-step
| 4\13, 369.23
| 6\18, 400.00
| 10\31, 387.10
| 5/4
| +6
|-
| dim. 3-step
| 4\13, 369.23
| 5\18, 333.33
| 9\31, 348.39
| ''16/13, 11/9''
| -7
|-
| perf. 3-step
| 5\13, 461.54
| 7\18, 466.67
| 12\31, 464.52
| 21/16, ''13/10'', 17/13
| +1
|- style="background-color: #eaeaff;"
| minor 4-step
| 6\13, 553.85
| 8\18, 533.33
| 14\31, 541.94
| ''11/8''
| -4
|- style="background-color: #eaeaff;"
| major 4-step
| 7\13, 646.15
| 10\18, 666.66
| 17\31, 658.06
| ''13/9'', ''16/11''
| +4
|-
| perf. 5-step
| 8\13, 738.46
| 11\18, 733.33
| 19\31, 735.48
| 26/17
| -1
|-
| aug. 5-step
| 9\13, 830.77
| 13\18, 866.66
| 22\31, 851.61
| ''13/8'', ''18/11''
| +7
|- style="background-color: #eaeaff;"
| minor 6-step
| 9\13, 830.77
| 12\18, 800.00
| 21\31, 812.90
| 8/5
| -6
|- style="background-color: #eaeaff;"
| major 6-step
| 10\13, 923.08
| 14\18, 933.33
| 24\31, 929.03
| 12/7
| +2
|-
| minor 7-step
| 11\13, 1015.39
| 15\18, 1000.00
| 26\31, 1006.45
| 9/5, 16/9
| -3
|-
| major 7-step
| 12\13, 1107.69
| 17\18, 1133.33
| 29\31, 1122.58
|
| +5
|}
<references />


===Hyposoft===
=== Hyposoft tunings ===
[[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,  
[[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 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¢) to 4\13 (369¢).
* 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}}).


* [[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¢).
* [[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}}).
* [[34edo]]'s 9:10:11:13 is even better.
* [[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.)
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).
{{MOS tunings
{| class="wikitable right-2 right-3 right-4 right-5"
| Step Ratios = Hyposoft
|-
| JI Ratios =
!
1/1;
! [[21edo]] (soft)
16/15;
! [[34edo]] (semisoft)
10/9; 11/10;
|-
13/11; 20/17;
| generator (g)
11/9;
| 8\21, 457.14
5/4;
| 13\34, 458.82
13/10;
|-
18/13; 32/23;
| L (3g - octave)
13/9; 23/16;
| 3\21, 171.43
20/13;
| 5\34, 176.47
8/5;
|-
18/11;
| s (-5g + 2 octaves)
22/13; 17/10;
| 2\21, 114.29
9/5;
| 3\34, 105.88
15/8;
|}
2/1
}}


====Intervals====
=== Parasoft and ultrasoft tunings ===
Sortable table of major and minor intervals in hyposoft tunings (13edo not shown):
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="wikitable right-2 right-3 sortable "
{{MOS tunings
|-
| Step Ratios = 6/5; 3/2; 4/3
! class="unsortable" |Degree
| JI Ratios =  
! Size in 21edo (soft)
1/1;
! Size in 34edo (semisoft)
14/13;
! class="unsortable" |Approximate ratios
11/10;
! #Gens up
9/8;
|- style="background-color: #eaeaff;"
15/13;
| unison
13/11;
| 0\21, 0.00
14/11;
| 0\34, 0.00
13/10;
| 1/1
4/3;
| 0
15/11;
|-
7/5;
| minor step
10/7;
| 2\21, 114.29
22/15;
| 3\34, 105.88
3/2;
| 16/15
20/13;
| -5
11/7;
|-
22/13;
| major step
26/15;
| 3\21, 171.43
16/9;
| 5\34, 176.47
20/11;
| 10/9, 11/10
13/7;
| +3
2/1
|- style="background-color: #eaeaff;"
}}
| minor 2-step
| 5\21, 285.71
| 8\34, 282.35
| 13/11, 20/17
| -2
|- style="background-color: #eaeaff;"
| major 2-step
| 6\21, 342.86
| 10\34, 352.94
| 11/9
| +6
|-
| dim. 3-step
| 7\21, 400.00
| 11\34, 388.24
| 5/4
| -7
|-
| perf. 3-step
| 8\21, 457.14
| 12\31, 458.82
| 13/10
| +1
|- style="background-color: #eaeaff;"
| minor 4-step
| 10\21, 571.43
| 16\34, 564.72
| 18/13, 32/23
| -4
|- style="background-color: #eaeaff;"
| major 4-step
| 11\21, 628.57
| 18\34, 635.29
| 13/9, 23/16
| +4
|-
| perf. 5-step
| 13\21, 742.86
| 21\34, 741.18
| 20/13
| -1
|-
| aug. 5-step
| 14\21, 800.00
| 23\34, 811.77
| 8/5
| +7
|- style="background-color: #eaeaff;"
| minor 6-step
| 15\21, 857.14
| 24\34, 847.06
| 18/11
| -6
|- style="background-color: #eaeaff;"
| major 6-step
| 16\21, 914.29
| 26\34, 917.65
| 22/13, 17/10
| +2
|-
| minor 7-step
| 18\21, 1028.57
| 29\34, 1023.53
| 9/5
| -3
|-
| major 7-step
| 19\21, 1085.71
| 31\34, 1094.12
| 15/8
| +5
|}


===Parasoft to ultrasoft tunings===
=== Parahard 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.
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).


The sizes of the generator, large step and small step of oneirotonic are as follows in various tunings in this range.
{{MOS tunings
{| class="wikitable right-2 right-3 right-4 right-5"
| JI Ratios =  
|-
1/1;
!
21/17;
! [[29edo]] (supersoft)
17/16;
! [[37edo]]
14/11;
|-
6/5;
| generator (g)
21/16;
| 11\29, 455.17
21/17;
| 14\37, 454.05
34/21;
|-
32/21;
| L (3g - octave)
5/3;
| 4\29, 165.52
11/7;
| 5\37, 162.16
32/17;
|-
34/21;
| s (-5g + 2 octaves)
2/1
| 3\29, 124.14
| Step Ratios = 4/1
| 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" | Approximate ratios (29edo)
! #Gens up
|- style="background-color: #eaeaff;"
| unison
| 0\29, 0.00
| 1/1
| 0
|- style="background-color: #eaeaff;"
| oneirochroma
| 1\29, 41.4
|
| +8
|-
| dim. step
| 2\29, 82.8
|
| -13
|-
| minor step
| 3\29, 124.1
| 14/13
| -5
|-
| major step
| 4\29, 165.5
| 11/10
| +3
|-
| aug. step
| 5\29, 206.9
| 9/8
| +11
|- style="background-color: #eaeaff;"
| dim. 2-step
| 6\29, 248.3
| 15/13
| -10
|- style="background-color: #eaeaff;"
| minor 2-step
| 7\29, 289.7
| 13/11
| -2
|- style="background-color: #eaeaff;"
| major 2-step
| 8\29, 331.0
|
| +6
|- style="background-color: #eaeaff;"
| aug. 2-step
| 9\29, 372.4
|
| +14
|-
| doubly dim. 3-step
| 9\29, 372.4
|
| -15
|-
| dim. 3-step
| 10\29, 413.8
| 14/11
| -7
|-
| perf. 3-step
| 11\29, 455.2
| 13/10
| +1
|-
| aug. 3-step
| 12\29, 496.6
| 4/3
| +9
|- style="background-color: #eaeaff;"
| dim. 4-step
| 13\29, 537.9
| 15/11
| -12
|- style="background-color: #eaeaff;"
| minor 4-step
| 14\29, 579.3
| 7/5
| -4
|- style="background-color: #eaeaff;"
| major 4-step
| 15\29 620.7
| 10/7
| +4
|- style="background-color: #eaeaff;"
| aug. 4-step
| 16\29 662.1
| 22/15
| +12
|-
| dim. 5-step
| 17\29, 703.4
| 3/2
| -9
|-
| perf. 5-step
| 18\29, 755.2
| 20/13
| -1
|-
| aug. 5-step
| 19\29, 786.2
| 11/7
| +7
|-
| doubly aug. 5-step
| 20\29 827.6
|
| +15
|- style="background-color: #eaeaff;"
| dim. 6-step
| 20\29 827.6
|
| -14
|- style="background-color: #eaeaff;"
| minor 6-step
| 21\29 869.0
|
| -6
|- style="background-color: #eaeaff;"
| major 6-step
| 22\29, 910.3
| 22/13
| +2
|- style="background-color: #eaeaff;"
| aug. 6-step
| 23\29, 951.7
| 26/15
| +10
|-
| dim. 7-step
| 24\29, 993.1
| 16/9
| -11
|-
| minor 7-step
| 25\29, 1034.5
| 20/11
| -3
|-
| major 7-step
| 26\29, 1075.9
| 13/7
| +5
|-
| aug. 7-step
| 27\29, 1117.2
|
| +13
|- style="background-color: #eaeaff;"
| dim. 8-step
| 28\29, 1158.6
|
| -8
|}


===Parahard===
=== Ultrahard 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).
{{Main|5L&nbsp;3s/Temperaments#Buzzard}}
====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" |Approximate ratios (23edo)
! #Gens up
|- style="background-color: #eaeaff;"
| unison
| 0\23, 0.0
| 1/1
| 0
|- style="background-color: #eaeaff;"
| oneirochroma
| 3\23, 156.5
|
| +8
|-
| minor step
| 1\23, 52.2
|
| -5
|-
| major step
| 4\23, 208.7
|
| +3
|-
| aug. step
| 7\23, 365.2
| 21/17, inverse φ
| +11
|- style="background-color: #eaeaff;"
| dim. 2-step
| 2\23, 104.3
| 17/16
| -10
|- style="background-color: #eaeaff;"
| minor 2-step
| 5\23, 260.9
|
| -2
|- style="background-color: #eaeaff;"
| major 2-step
| 8\23, 417.4
| 14/11
| +6
|-
| dim. 3-step
| 6\23, 313.0
| 6/5
| -7
|-
| perf. 3-step
| 9\23, 469.6
| 21/16
| +1
|-
| aug. 3-step
| 12\23, 626.1
|
| +9
|- style="background-color: #eaeaff;"
| dim. 4-step
| 7\23, 365.2
| 21/17, inverse φ
| -12
|- style="background-color: #eaeaff;"
| minor 4-step
| 10\23, 521.7
|
| -4
|- style="background-color: #eaeaff;"
| major 4-step
| 13\23, 678.3
|
| +4
|- style="background-color: #eaeaff;"
| aug. 4-step
| 16\23, 834.8
| 34/21, φ
| +12
|-
| dim. 5-step
| 11\23, 573.9
|
| -9
|-
| perf. 5-step
| 14\23, 730.4
| 32/21
| -1
|-
| aug. 5-step
| 17\23, 887.0
| 5/3
| +7
|- style="background-color: #eaeaff;"
| minor 6-step
| 15\23 782.6
| 11/7
| -6
|- style="background-color: #eaeaff;"
| major 6-step
| 18\23, 939.1
|
| +2
|- style="background-color: #eaeaff;"
| aug. 6-step
| 21\23, 1095.7
| 32/17
| +10
|-
| dim. 7-step
| 16\23, 834.8
| 34/21, φ
| -11
|-
| minor 7-step
| 19\23, 991.3
|
| -3
|-
| major 7-step
| 22\23, 1147.8
|
| +5
|-
|- style="background-color: #eaeaff;"
| dim. 8-step
| 20\23, 1043.5
|
| -8
|}


===Ultrahard===
[[Buzzard]] is a rank-2 temperament in the [[Step ratio|pseudocollapsed]] range. It represents the only [[harmonic entropy]] minimum of the oneirotonic spectrum.
[[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.  
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.  
Line 788: Line 157:
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.
{{MOS tunings
{| class="wikitable right-2 right-3 right-4 right-5"
| JI Ratios =  
|-
1/1;
!
8/7;
! [[38edo]]
13/10;
! [[53edo]]
21/16;
! [[63edo]]
3/2;
! Optimal ([[POTE]]) Buzzard tuning
12/7, 22/13;
! JI intervals represented (2.3.5.7.13 subgroup)
26/15;
|-
49/25, 160/81;
| generator (g)
2/1
| 15\38, 473.68
| Step Ratios = 7/1; 10/1; 12/1
| 21\53, 475.47
| Tolerance = 30
| 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
! class="unsortable" |Approximate ratios
! #Gens up
|-
| 1
| 0\38, 0.00
| 0\53, 0.00
| 0\63, 0.00
| 0.00
| 1/1
| 0
|-
| 2
| 7\38, 221.05
| 10\53, 226.42
| 12\63, 228.57
| 227.07
| 8/7
| +3
|-
| 3
| 14\38, 442.10
| 20\53, 452.83
| 24\63, 457.14
| 453.81
| 13/10
| +6
|-
| 4
| 15\38, 473.68
| 21\53, 475.47
| 25\63, 476.19
| 475.63
| 21/16
| +1
|-
| 5
| 22\38, 694.73
| 31\53, 701.89
| 37\63, 704.76
| 702.54
| 3/2
| +4
|-
| 6
| 29\38, 915.78
| 41\53, 928.30
| 49\63, 933.33
| 929.45
| 12/7, 22/13
| +7
|-
| 7
| 30\38, 947.36
| 42\53, 950.94
| 50\63, 952.38
| 951.27
| 26/15
| +2
|-
| 8
| 37\38, 1168.42
| 52\53, 1177.36
| 62\63, 1180.95
| 1178.18
| 49/25, 160/81
| +5
|}


== Approaches ==
== Approaches ==
* [[5L 3s/Temperaments]]
* [[5L&nbsp;3s/Temperaments]]


==Samples==
== 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: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]])


Line 930: Line 204:
* [[File:Inthar-13edo Oneirotonic Studies 8 Kadathian.mp3]]: Tonal Study in Kadathian
* [[File:Inthar-13edo Oneirotonic Studies 8 Kadathian.mp3]]: Tonal Study in Kadathian


==Scale tree==
== Scale tree ==
Generator ranges:
{{MOS tuning spectrum
* Bright generator: 450 cents (3\8) to 480 cents (2\5)
| 13/8 = Golden oneirotonic (458.3592{{c}})  
* Dark generator: 720 cents (3\5) to 750 cents (5\8)
| 13/5 = Golden A-Team (465.0841{{c}})
 
}}
{| 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:Oneirotonic| ]] <!-- sort order in category: this page shows above A -->
[[Category:Pages with internal sound examples]]
[[Category:Pages with internal sound examples]]
[[Category:8-tone scales]]
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