Xenharmonic Wiki

5L 3s: Difference between revisions

← Older editNewer edit →
VisualWikitext
ArrowHead294 (talk | contribs)
14,512 edits
Ganaram inukshuk (talk | contribs)
autopatrolled
4,810 edits
Adopt mos tunings
Line 4: Line 4:
:''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 ==
Line 12: Line 10:


'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]].
=== 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;}}


== Scale properties ==
== Scale properties ==
{{TAMNAMS use}}
{{MOS data}}
 
=== Intervals ===
{{MOS intervals}}


=== Modes ===
== Tunings==
{{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
| 7{{pipe}}0
| Dylathian
|-
| LLsLsLLs
| 6{{pipe}}1
| Ilarnekian
|-
| LsLLsLLs
| 5{{pipe}}2
| Celephaïsian
|-
| LsLLsLsL
| 4{{pipe}}3
| Ultharian
|-
| LsLsLLsL
| 3{{pipe}}4
| Mnarian
|-
| sLLsLLsL
| 2{{pipe}}5
| Kadathian
|-
| sLLsLsLL
| 1{{pipe}}6
| Hlanithian
|-
| sLsLLsLL
| 0{{pipe}}7
| Sarnathian
|}
 
== Tuning ranges==
===Simple tunings ===
===Simple tunings ===
Table of intervals in the simplest oneirotonic tunings:
The simplest tuning for 5L 3s correspond to 13edo, 18edo, and 21edo, with step ratios 2:1, 3:1, and 3:2, respectively.
{| class="wikitable right-2 right-3 right-4 sortable "
|-
! class="unsortable" |Degree
! Size in 13edo (basic)
! Size in 18edo (hard)
! Size in 21edo (soft)
! #Gens up
|- style="background-color: #eaeaff;"
| unison
| 0\13, 0.00
| 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 ===
{{MOS tunings}}
[[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:  
=== Hypohard tunings ===
* The large step is a "meantone", somewhere between near-10/9 (as in [[13edo]]) and near-9/8 (as in [[18edo]]).
[[Hypohard]] oneirotonic tunings have step ratios between 2:1 and 3:1 and can be considered "meantone oneirotonic". This is because these tunings share the following features with [[meantone]] diatonic tunings:  
* 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.
* 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.
Also, in [[18edo]] and [[31edo]], the minor 2-mosstep is close to [[7/6]].
* With step ratios between 5:2 and 2:1, 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.
The set of identifications above is associated with [[5L 3s/Temperaments#A-Team|A-Team]] temperament.
Line 180: Line 44:
* 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 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.
{{MOS tunings|Step Ratios=Hypohard|JI Ratios=1/1;
21/20; 22/21;
9/8; 10/9;
7/6;
5/4;
16/13; 11/9;
21/16; 13/10; 17/13;
11/8;
13/9; 16/11;
26/17;
13/8; 18/11;
8/5;
12/7;
9/5; 16/9;}}


The sizes of the generator, large step and small step of oneirotonic are as follows in various hypohard oneiro tunings.
=== Hyposoft tunings ===
{| class="wikitable right-2 right-3 right-4 right-5"
[[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¢.
!
! [[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" | 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]] 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¢).
* 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¢).


Line 334: Line 69:
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|Step Ratios=Hyposoft|JI Ratios=1/1;
{| class="wikitable right-2 right-3 right-4 right-5"
16/15;
|-
10/9; 11/10;
!
13/11; 20/17;
! [[21edo]] (soft)
11/9;
! [[34edo]] (semisoft)
5/4;
|-
13/10;
| generator (g)
18/13; 32/23;
| 8\21, 457.14
13/9; 23/16;
| 13\34, 458.82
20/13;
|-
8/5;
| L (3g - octave)
18/11;
| 3\21, 171.43
22/13; 17/10;
| 5\34, 176.47
9/5;
|-
15/8;}}
| 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" | Approximate ratios
! #Gens up
|- style="background-color: #eaeaff;"
| unison
| 0\21, 0.00
| 0\34, 0.00
| 1/1
| 0
|-
| minor step
| 2\21, 114.29
| 3\34, 105.88
| 16/15
| -5
|-
| major step
| 3\21, 171.43
| 5\34, 176.47
| 10/9, 11/10
| +3
|- 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 ===
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.
=== Parasoft and ultrasoft tunings ===
{| class="wikitable right-2 right-3 right-4 right-5"
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.
|-
!
! [[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====
{{MOS tunings|Step Ratios=6/5; 3/2; 4/3|JI Ratios=1/1;
The intervals of the extended generator chain (-15 to +15 generators) are as follows in various softer-than-soft oneirotonic tunings.
14/13;
{| class="wikitable right-2 right-3 sortable "
11/10;
|-
9/8;
! class="unsortable" | Degree
15/13;
! Size in 29edo (supersoft)
13/11;
! class="unsortable" | Approximate ratios (29edo)
14/11;
! #Gens up
13/10;
|- style="background-color: #eaeaff;"
4/3;
| unison
15/11;
| 0\29, 0.00
7/5;
| 1/1
10/7;
| 0
22/15;
|- style="background-color: #eaeaff;"
3/2;
| oneirochroma
20/13;
| 1\29, 41.4
11/7;
|
22/13;
| +8
26/15;
|-
16/9;
| dim. step
20/11;
| 2\29, 82.8
13/7;}}
|
| -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===
===Parahard tunings===
23edo oneiro combines the sound of neogothic tunings like [[46edo]] and the sounds of "superpyth" and "semaphore" scales. This is because 23edo oneirotonic has a large step of 208.7¢, same as [[46edo]]'s neogothic major second, and is both a warped [[22edo]] [[superpyth]] [[diatonic]] and a warped [[24edo]] [[semaphore]] [[semiquartal]] (and both nearby scales are [[superhard]] MOSes).
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====
{{MOS tunings|JI Ratios=1/1;
The intervals of the extended generator chain (-12 to +12 generators) are as follows in various oneirotonic tunings close to [[23edo]].
21/17;
{| class="wikitable right-2 right-3 sortable "
17/16;
|-
14/11;
! class="unsortable" | Degree
6/5;
! Size in 23edo (superhard)
21/16;
! class="unsortable" | Approximate ratios (23edo)
21/17;
! #Gens up
34/21;
|- style="background-color: #eaeaff;"
32/21;
| unison
5/3;
| 0\23, 0.0
11/7;
| 1/1
32/17;
| 0
34/21;|Step Ratios=4/1}}
|- 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 ===
=== 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.
[[Buzzard]] is a rank-2 temperament in the [[Step ratio|pseudocollapsed]] range. It represents the only [[harmonic entropy]] minimum of the oneirotonic spectrum.


In the broad sense, Buzzard can be viewed as any tuning that divides the 3rd harmonic into 4 equal parts. [[23edo]], [[28edo]] and [[33edo]] can nominally be viewed as supporting it, but are still very flat and in an ambiguous zone between 18edo and true Buzzard in terms of harmonies. [[38edo]] & [[43edo]] are good compromises between melodic utility and harmonic accuracy, as the small step is still large enough to be obvious to the untrained ear, but [[48edo]] is where it really comes into its own in terms of harmonies, providing not only an excellent [[3/2]], but also [[7/4]] and [[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 790: Line 134:
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|JI Ratios=1/1;
{| class="wikitable right-2 right-3 right-4 right-5"
8/7;
|-
13/10;
!
21/16;
! [[38edo]]
3/2;
! [[53edo]]
12/7, 22/13;
! [[63edo]]
26/15;
! Optimal ([[POTE]]) Buzzard tuning
49/25, 160/81;|Step Ratios=7/1; 10/1; 12/1}}
! 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
! 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 ==
Line 933: Line 176:


== Scale tree ==
== 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)
{{Scale tree}}
{{Scale tree}}


Retrieved from "https://en.xen.wiki/w/5L_3s"