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:''For the tritave-equivalent MOS structure with the same step pattern, see [[5L 3s (tritave-equivalent)]].''
{{Interwiki
| en = 5L 3s
| de =
| es =
| ja =
| ko = 5L3s (Korean)
}}
{{Infobox MOS
{{Infobox MOS
| Name = oneirotonic
| Periods = 1
| nLargeSteps = 5
| nSmallSteps = 3
| Equalized = 3
| Paucitonic = 2
| Pattern = LLsLLsLs
| Neutral = 2L 6s
| Neutral = 2L 6s
}}
}}
'''5L 3s'''  refers to the structure of octave-equivalent [[MOS]] scales with generators ranging from 2\5 (two degrees of [[5edo]] = 480¢) to 3\8 (three degrees of [[8edo]] = 450¢). In the case of 8edo, L and s are the same size; in the case of 5edo, s becomes so small it disappears (and all that remains are the five equal L's).
: ''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]]).


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 distorted to the Dylathian mode LLsLLsLs.
== Name ==
{{TAMNAMS name}} 'Oneiro' is sometimes used as a shortened form.


Any edo with an interval between 450¢ and 480¢ has a 5L 3s scale. [[13edo]] is the smallest edo with a (non-degenerate) 5L 3s scale and thus is the most commonly used 5L 3s tuning.
'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). (Note: [[3L 5s]] scales also have 3L 2s subsets.)
== Scale properties ==


== Standing assumptions ==
=== Intervals ===
The [[TAMNAMS]] system is used in this article to name 5L 3s intervals and step size ratios and step ratio ranges.
{{MOS intervals}}


The notation used in this article is J Ultharian (LsLLsLsL) = JKLMNOPQJ, unless specified otherwise. We denote raising and lowering by a chroma (L − s) by & "amp" and @ "at". (Mnemonics: & "and" means additional pitch. @ "at" rhymes with "flat".)
=== Generator chain ===
{{MOS genchain}}


The chain of perfect mosfourths becomes: ... P@ K@ N@ Q L O J M P K N Q& L& O& ...
=== Modes ===
{{MOS mode degrees}}


Thus the [[13edo]] gamut is as follows:
==== Proposed mode names ====
The following names have been proposed for the modes of 5L 3s, and are named after cities in the Dreamlands.
{{MOS modes
| Mode Names=
Dylathian $
Ilarnekian $
Celephaïsian $
Ultharian $
Mnarian $
Kadathian $
Hlanithian $
Sarnathian $
| Collapsed=1
}}


'''J/Q&''' J&/K@ '''K/L@''' '''L/K&''' L&/M@ '''M''' M&/N@ '''N/O@''' '''O/N&''' O&/P@ '''P''' '''Q''' Q&/J@ '''J'''
== Tunings==
=== Simple tunings ===
The simplest tuning for 5L 3s correspond to 13edo, 18edo, and 21edo, with step ratios 2:1, 3:1, and 3:2, respectively.


The [[18edo]] gamut is notated as follows:
{{MOS tunings|JI Ratios=Int Limit: 30; Prime Limit: 19; Tenney Height: 7.7}}


'''J''' Q&/K@ J&/L@ '''K''' '''L''' K&/M@ L& '''M''' N@ M&/O@ '''N''' '''O''' P@ O& '''P''' '''Q''' P&/J@ Q@ '''J'''
=== 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 [[21edo]] gamut:
With step ratios between 5:2 and 2:1, the minor 2-mosstep is close to [[7/6]].


'''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'''
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{{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{{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.
* [[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.


== Names ==
{{MOS tunings|Step Ratios=Hypohard|JI Ratios=Subgroup: 2.5.9.21; Int Limit:40; Complements Only: 1|Tolerance=15}}
The [[TAMNAMS]] system suggests the name '''oneirotonic''' (/oʊnaɪrəˈtɒnɪk/ ''oh-ny-rə-TON-ik'' or /ənaɪrə-/ ''ə-ny-rə-'') or 'oneiro' for short. The name ''oneirotonic'' (from Greek ''oneiros'' 'dream') is coined after the Dreamlands in H.P. Lovecraft's Dream Cycle mythos.  


'Father' is sometimes also used to denote 5L 3s, but it's a misnomer, as [[father]] is technically an abstract [[regular temperament]] (although a very inaccurate one), not a generator range. A more correct way to say it would be 'father[8]' or 'father octatonic'. "Father" is also vague regarding the number of notes, because optimal generators for it also generate [[3L 2s]].
=== Hyposoft tunings ===
[[Hyposoft]] oneirotonic tunings have step ratios between 3:2 and 2:1, which remains relatively unexplored. In these tunings,  
* The large step of oneirotonic tends to be intermediate in size between [[10/9]] and [[11/10]]; the small step size is a semitone close to [[17/16]], about 92{{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}}).


== Intervals ==
* [[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}}).
The table of oneirotonic intervals below takes the flat fourth as the generator. Given the size of the subfourth generator ''g'', any oneirotonic interval can easily be found by noting what multiple of ''g'' it is, and multiplying the size by the number ''k'' of generators it takes to reach the interval and reducing mod 1200 if necessary (so you can use "''k''*''g'' % 1200" for search engines, for plugged-in values of ''k'' and ''g''). For example, since the major mosthird is reached by six subfourth generators, [[18edo]]'s major mosthird is 6*466.67 mod 1200 = 2800 mod 1200 = 400¢, same as the [[12edo]] major third.
* [[34edo]]'s 9:10:11:13 is even better.
{| class="wikitable center-all"
|-
! # generators up
! Notation (1/1 = J)
! [[TAMNAMS]] name
! In L's and s's
! # generators up
! Notation of 2/1 inverse
! [[TAMNAMS]] name
! In L's and s's
|-
| 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 mosfourth
| 2L + 1s
| -1
| O
| perfect mossixth
| 3L + 2s
|-
| 2
| P
| major mosseventh
| 4L + 2s
| -2
| L
| minor mosthird
| 1L + 1s
|-
| 3
| K
| major mossecond
| 1L + 0s
| -3
| Q
| minor moseighth
| 4L + 3s
|-
| 4
| N
| major mosfifth
| 3L + 1s
| -4
| N@
| minor mosfifth
| 2L + 2s
|-
| 5
| Q&
| major moseighth
| 5L + 2s
| -5
| K@
| minor mossecond
| 0L + 1s
|-
| 6
| L&
| major mosthird
| 2L + 0s
| -6
| P@
| minor mosseventh
| 3L + 3s
|-
| 7
| O&
| augmented mossixth
| 4L + 1s
| -7
| M@
| diminished mosfourth
| 1L + 2s
|-
| 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 mosunison (aka moschroma)
| 1L - 1s
| -8
| J@
| diminished mosoctave (aka diminished mosninth)
| 4L + 4s
|-
| 9
| M&
| augmented mosfourth
| 3L + 0s
| -9
| O@
| diminished mossixth
| 2L + 3s
|-
| 10
| P&
| augmented mosseventh
| 5L + 1s
| -10
| L@
| diminished mosthird
| 0L + 2s
|-
| 11
| K&
| augmented mossecond
| 2L - 1s
| -11
| Q@
| diminished moseighth
| 3L + 4s
|-
| 12
| N&
| augmented mosfifth
| 4L + 0s
| -12
| N@@
| diminished mosfifth
| 1L + 3s
|}


== Tuning ranges ==
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.)
=== 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:
{{MOS tunings
* The large step is a "meantone", somewhere between near-10/9 (as in [[13edo]]) and near-9/8 (as in [[18edo]]).
| Step Ratios = Hyposoft
* 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.
| JI Ratios =
1/1;
16/15;
10/9; 11/10;
13/11; 20/17;
11/9;
5/4;
13/10;
18/13; 32/23;
13/9; 23/16;
20/13;
8/5;
18/11;
22/13; 17/10;
9/5;
15/8;
2/1
}}


Also, in [[18edo]] and [[31edo]], the minor mosthird is close to [[7/6]].
=== Parasoft and ultrasoft tunings ===
The range of oneirotonic tunings of step ratio between 6:5 and 3:2 is closely related to [[porcupine]] temperament; these tunings equate three oneirotonic large steps to a diatonic perfect fourth, i.e. they equate the oneirotonic large step to a [[porcupine]] generator. The chord 10:11:13 is very well approximated in 29edo.


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


EDOs that are in the hypohard range include [[13edo]], [[18edo]], and [[31edo]].
=== Parahard tunings ===
* 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.
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).
* 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.
{{MOS tunings
{| class="wikitable right-2 right-3 right-4 right-5"
| JI Ratios =  
|-
1/1;
!
21/17;
! [[13edo]] (basic)
17/16;
! [[18edo]] (hard)
14/11;
! [[31edo]] (semihard)
6/5;
|-
21/16;
| generator (g)
21/17;
| 5\13, 461.54
34/21;
| 7\18, 466.67
32/21;
| 12\31, 464.52
5/3;
|-
11/7;
| L (3g - octave)
32/17;
| 2\13, 184.62
34/21;
| 3\18, 200.00
2/1
| 5\31, 193.55
| Step Ratios = 4/1
|-
}}
| s (-5g + 2 octaves)
| 1\13, 92.31
| 1\18, 66.67
| 2\31, 77.42
|}


==== Intervals ====
=== Ultrahard tunings ===
Sortable table of major and minor intervals in hypohard oneiro tunings:
{{Main|5L 3s/Temperaments#Buzzard}}


{| class="wikitable right-2 right-3 right-4 sortable "
[[Buzzard]] is a rank-2 temperament in the [[Step ratio|pseudocollapsed]] range. It represents the only [[harmonic entropy]] minimum of the oneirotonic spectrum.
|-
! 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 mos2nd
| 1\13, 92.31
| 1\18, 66.67
| 2\31, 77.42
| K@
| 21/20, ''22/21''
| -5
|-
| major mos2nd
| 2\13, 184.62
| 3\18, 200.00
| 5\31, 193.55
| K
| 9/8, 10/9
| +3
|-bgcolor="#eaeaff"
| minor mos3rd
| 3\13, 276.92
| 4\18, 266.67
| 7\31, 270.97
| L
| 7/6
| -2
|-bgcolor="#eaeaff"
| major mos3rd
| 4\13, 369.23
| 6\18, 400.00
| 10\31, 387.10
| L&
| 5/4
| +6
|-
| dim. mos4th
| 4\13, 369.23
| 5\18, 333.33
| 9\31, 348.39
| M@
| ''16/13, 11/9''
| -7
|-
| perf. mos4th
| 5\13, 461.54
| 7\18, 466.67
| 12\31, 464.52
| M
| 21/16, ''13/10'', 17/13
| +1
|-bgcolor="#eaeaff"
| minor mos5th
| 6\13, 553.85
| 8\18, 533.33
| 14\31, 541.94
| N@
| ''11/8''
| -4
|-bgcolor="#eaeaff"
| major mos5th
| 7\13, 646.15
| 10\18, 666.66
| 17\31, 658.06
| N
| ''13/9'', ''16/11''
| +4
|-
| perf. mos6th
| 8\13, 738.46
| 11\18, 733.33
| 19\31, 735.48
| O
| 26/17
| -1
|-
| aug. mos6th
| 9\13, 830.77
| 13\18, 866.66
| 22\31, 851.61
| O&
| ''13/8'', ''18/11''
| +7
|-bgcolor="#eaeaff"
| minor mos7th
| 9\13, 830.77
| 12\18, 800.00
| 21\31, 812.90
| P@
| 8/5
| -6
|-bgcolor="#eaeaff"
| major mos7th
| 10\13, 923.08
| 14\18, 933.33
| 24\31, 929.03
| P
| 12/7
| +2
|-
| minor mos8th
| 11\13, 1015.39
| 15\18, 1000.00
| 26\31, 1006.45
| Q
| 9/5, 16/9
| -3
|-
| major mos8th
| 12\13, 1107.69
| 17\18, 1133.33
| 29\31, 1122.58
| Q&
|
| +5
|}
<references/>


=== Hyposoft ===
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.  
[[Hyposoft]] oneirotonic tunings (with generator between 8\21 and 5\13) have step ratios between 3/2 and 2/1. The 8\21-to-5\13 range of oneirotonic tunings remains relatively unexplored. In these tunings,
* the large step of oneirotonic tends to be intermediate in size between [[10/9]] and [[11/10]]; the small step size is a semitone close to [[17/16]], about 92¢ to 114¢.
* The major mosthird (made of two large steps) in these tunings tends to be more of a neutral third, ranging from 6\21 (342¢) to 4\13 (369¢).
 
* [[21edo]]'s P1-Lmos2-Lmos3-Lmos5 approximates 9:10:11:13 better than the corresponding 13edo chord does. 21edo will serve those who like the combination of neogothic minor thirds (285.71¢) and Baroque diatonic semitones (114.29¢, close to quarter-comma meantone's 117.11¢).
* [[34edo]]'s 9:10:11:13 is even better.
 
This set of JI identifications is associated with [[5L 3s/Temperaments#Petrtri|petrtri]] temperament. (P1-Mmos2-Pmos4 could be said to approximate 5:11:13 in all soft-of-basic tunings, which is what "basic" [[petrtri]] temperament is.)
 
The sizes of the generator, large step and small step of oneirotonic are as follows in various hyposoft oneiro tunings (13edo not shown).
{| 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 mos2nd
| 2\21, 114.29
| 3\34, 105.88
| K@
| 16/15
| -5
|-
| major mos2nd
| 3\21, 171.43
| 5\34, 176.47
| K
| 10/9, 11/10
| +3
|-bgcolor="#eaeaff"
| minor mos3rd
| 5\21, 285.71
| 8\34, 282.35
| L
| 13/11, 20/17
| -2
|-bgcolor="#eaeaff"
| major mos3rd
| 6\21, 342.86
| 10\34, 352.94
| L&
| 11/9
| +6
|-
| dim. mos4th
| 7\21, 400.00
| 11\34, 388.24
| M@
| 5/4
| -7
|-
| perf. mos4th
| 7\18, 457.14
| 12\31, 458.82
| M
| 13/10
| +1
|-bgcolor="#eaeaff"
| minor mos5th
| 10\21, 571.43
| 16\34, 564.72
| N@
| 18/13, 32/23
| -4
|-bgcolor="#eaeaff"
| major mos5th
| 11\21, 628.57
| 18\34, 635.29
| N
| 13/9, 23/16
| +4
|-
| perf. mos6th
| 13\21, 742.86
| 21\34, 741.18
| O
| 20/13
| -1
|-
| aug. mos6th
| 14\21, 800.00
| 23\34, 811.77
| O&
| 8/5
| +7
|-bgcolor="#eaeaff"
| minor mos7th
| 15\21, 857.14
| 24\34, 847.06
| P@
| 18/11
| -6
|-bgcolor="#eaeaff"
| major mos7th
| 16\21, 914.29
| 26\34, 917.65
| P
| 22/13, 17/10
| +2
|-
| minor mos8th
| 18\21, 1028.57
| 29\34, 1023.53
| Q
| 9/5
| -3
|-
| major mos8th
| 19\21, 1085.71
| 31\34, 1094.12
| Q&
| 15/8
| +5
|}
 
=== Parasoft to ultrasoft tunings ===
The range of oneirotonic tunings of step ratio between 6/5 and 3/2 (thus in the [[parasoft]] to [[ultrasoft]] range) may be of interest because it is closely related to [[porcupine]] temperament: these tunings equate three oneirotonic large steps to a diatonic perfect fourth, i.e. they equate the oneirotonic large step to a [[porcupine]] generator. [This identification may come in handy since many altered oneirotonic modes have three consecutive large steps.] The 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, 41.3
| J&
|
| +8
|-
| dim. mos2nd
| 2\29, 82.8
| K@@
|
| -13
|-
| minor mos2nd
| 3\29, 124.1
| K@
| 14/13
| -5
|-
| major mos2nd
| 4\29, 165.5
| K
| 11/10
| +3
|-
| aug. mos2nd
| 5\29, 206.9
| K&
| 9/8
| +11
|-bgcolor="#eaeaff"
| dim. mos3rd
| 6\29, 248.3
| L@
| 15/13
| -10
|-bgcolor="#eaeaff"
| minor mos3rd
| 7\29, 289.7
| L
| 13/11
| -2
|-bgcolor="#eaeaff"
| major mos3rd
| 8\29, 331.0
| L&
|
| +6
|-bgcolor="#eaeaff"
| aug. mos3rd
| 9\29, 372.4
| L&&
|
| +14
|-
| doubly dim. mos4th
| 9\29, 372.4
| M@@
|
| -15
|-
| dim. mos4th
| 10\29, 413.8
| M@
| 14/11
| -7
|-
| perf. mos4th
| 11\29, 455.2
| M
| 13/10
| +1
|-
| aug. mos4th
| 12\29, 496.6
| M&
| 4/3
| +9
|-bgcolor="#eaeaff"
| dim. mos5th
| 13\29, 537.9
| N@@
| 15/11
| -12
|-bgcolor="#eaeaff"
| minor mos5th
| 14\29, 579.3
| N@
| 7/5
| -4
|-bgcolor="#eaeaff"
| major mos5th
| 15\29 620.7
| N
| 10/7
| +4
|-bgcolor="#eaeaff"
| aug. mos5th
| 16\29 662.1
| N&
| 22/15
| +12
|-
| dim. mos6th
| 17\29, 703.4
| O@
| 3/2
| -9
|-
| perf. mos6th
| 18\29, 755.2
| O
| 20/13
| -1
|-
| aug. mos6th
| 19\29, 786.2
| O&
| 11/7
| +7
|-
| doubly aug. mos6th
| 20\29 827.6
| O&&
|
| +15
|-bgcolor="#eaeaff"
| dim. mos7th
| 20\29 827.6
| P@@
|
| -14
|-bgcolor="#eaeaff"
| minor mos7th
| 21\29 869.0
| P@
|
| -6
|-bgcolor="#eaeaff"
| major mos7th
| 22\29, 910.3
| P
| 22/13
| +2
|-bgcolor="#eaeaff"
| aug. mos7th
| 23\29, 951.7
| P&
| 26/15
| +10
|-
| dim. mos8th
| 24\29, 993.1
| Q@
| 16/9
| -11
|-
| minor mos8th
| 25\29, 1034.5
| Q
| 20/11
| -3
|-
| major mos8th
| 26\29, 1075.9
| Q&
| 13/7
| +5
|-
| aug. mos8th
| 27\29, 1117.2
| Q&&
|
| +13
|-bgcolor="#eaeaff"
| dim. mos9th
| 28\29, 1158.6
| J@
|
| -8
|}
 
=== Parahard to ultrahard ===
==== 23edo ====
==== Buzzard ====
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.
{{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
| 3/2 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
| 55/54 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, 9/7
| +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
| 108/55, 160/81
| +5
|}
== Modes ==
Oneirotonic modes are named after cities in the Dreamlands.
{| class="wikitable"
|-
| style="text-align:center;" | '''Mode'''
| style="text-align:center;" | [[Modal UDP Notation|'''UDP''']]
| style="text-align:center;" | '''Name'''
|-
| | LLsLLsLs
| style="text-align:center;" | 7|0
| | Dylathian (də-LA(H)TH-iən)
|-
| | LLsLsLLs
| style="text-align:center;" | 6|1
| | Illarnekian (ill-ar-NEK-iən)
|-
| | LsLLsLLs
| style="text-align:center;" | 5|2
| | Celephaïsian (kel-ə-FAY-zhən)
|-
| | LsLLsLsL
| style="text-align:center;" | 4|3
| | Ultharian (ul-THA(I)R-iən)
|-
| | LsLsLLsL
| style="text-align:center;" | 3|4
| | Mnarian (mə-NA(I)R-iən)
|-
| | sLLsLLsL
| style="text-align:center;" | 2|5
| | Kadathian (kə-DA(H)TH-iən)
|-
| | sLLsLsLL
| style="text-align:center;" | 1|6
| | Hlanithian (lə-NITH-iən)
|-
| | sLsLLsLL
| style="text-align:center;" | 0|7
| | Sarnathian (sar-NA(H)TH-iən), can be shortened to "Sarn"
|}
== Approaches ==
== Approaches ==
* [[5L 3s/Inthar's approach]]
* [[5L&nbsp;3s/Temperaments]]
* [[5L 3s/Temperaments]]


== Samples ==
== Samples ==
[[File:The Angels' Library.mp3]] [[:File:The Angels' Library.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]])


[[File:13edo Prelude in J Oneirominor.mp3]]
[[File:13edo Prelude in J Oneirominor.mp3]]
Line 906: 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 912: 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 center-all"
{{MOS tuning spectrum
! colspan="6" | Generator
| 13/8 = Golden oneirotonic (458.3592{{c}})
! Cents
| 13/5 = Golden A-Team (465.0841{{c}})
! L
}}
! s
! L/s
! Comments
|-
| 3\8 || || || || || || 450.000 || 1 || 1 || 1.000 ||
|-
| || || || || || 17\45 || 453.333 || 6 || 5 || 1.200 ||
|-
| || || || || 14\37 || || 454.054 || 5 || 4 || 1.250 ||
|-
| || || || || || 34\59 || 454.545 || 9 || 7 || 1.286 ||
|-
| || || || 11\29 || || || 455.172 || 4 || 3 || 1.333 ||
|-
| || || || || || 30\79 || 455.696 || 11 || 8 || 1.375 ||
|-
| || || || || 19\50 || || 456.000 || 7 || 5 || 1.400 ||
|-
| || || || || || 27\71 || 456.338 || 10 || 7 || 1.429 ||
|-
| || || 8\21 || || || || 457.143 || 3 || 2 || 1.500 || L/s = 3/2
|-
| || || || || || 29\76 || 457.895 || 11 || 7 || 1.571 ||
|-
| || || || || 21\55 || || 458.182 || 8 || 5 || 1.600 ||
|-
| || || || || || 34\89 || 458.427 || 13 || 8 || 1.625 || Golden oneirotonic
|-
| || || || 13\34 || || || 458.824 || 5 || 3 || 1.667 || <!--Petrtri is in this region-->
|-
| || || || || || 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 || <!--A-Team is in this region-->
|-
| || || || || || 31\80 || 465.000 || 13 || 5 || 2.600 ||
|-
| || || || || 19\49 || || 465.306 || 8 || 3 || 2.667 ||
|-
| || || || || || 26\67 || 465.672 || 11 || 4 || 2.750 ||
|-
| || || 7\18 || || || || 466.667 || 3 || 1 || 3.000 || L/s = 3/1
|-
| || || || || || 23\59 || 467.797 || 10 || 3 || 3.333 ||
|-
| || || || || 16\41 || || 468.293 || 7 || 2 || 3.500 ||
|-
| || || || || || 25\64 || 468.750 || 11 || 3 || 3.667 ||
|-
| || || || 9\23 || || || 469.565 || 4 || 1 || 4.000 ||
|-
| || || || || || 20\51 || 470.588 || 9 || 2 || 4.500 ||
|-
| || || || || 11\28 || || 471.429 || 5 || 1 || 5.000 ||
|-
| || || || || || 13\33 || 472.727 || 6 || 1 || 6.000 ||
|-
| 2\5 || || || || || || 480.000 || 1 || 0 || → inf ||
|}


[[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:Abstract MOS patterns]]
[[Category:Pages with internal sound examples]]
[[Category:Oneirotonic]]
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