5L 3s: Difference between revisions

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| 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)]].''

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 can be seen as a [[Warped diatonic|warped diatonic scale]], because it has one extra small step compared to diatonic ([[5L 2s]]).

== 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]].

=== Proposed mode names ===

== Scale properties ==

=== Intervals ===

=== Generator chain ===

=== Modes ===

==== 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;

{{MOS modes

Ilarnekian;

| Mode Names=

Celephaïsian;

Dylathian $

Ultharian;

Ilarnekian $

Mnarian;

Celephaïsian $

Kadathian;

Ultharian $

Hlanithian;

Mnarian $

Sarnathian;|Collapsed=1}}

Kadathian $

Hlanithian $

~~== Scale properties ==~~

Sarnathian $

~~{{MOS data~~}}

| Collapsed=1

}}

== Tunings==

===Simple 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 simplest tuning for 5L 3s correspond to 13edo, 18edo, and 21edo, with step ratios 2:1, 3:1, and 3:2, respectively.

{{MOS tunings|JI Ratios=Int Limit: 30; Prime Limit: 19; Tenney Height: 7.7}}

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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.

* [[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.

{{MOS tunings|Step Ratios=Hypohard|JI Ratios=Subgroup: 2.5.9.21; Int Limit:40; Complements Only: 1|Tolerance=15}}

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=== 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¢~~ 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.

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.)

{{MOS tunings|Step Ratios=Hyposoft|JI Ratios=1/1;

{{MOS tunings

| Step Ratios = Hyposoft

| JI Ratios =

1/1;

16/15;

10/9; 11/10;

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Line 93:

9/5;

15/8;

2/1}}

2/1

}}

=== 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.

{{MOS tunings|Step Ratios=6/5; 3/2; 4/3|JI Ratios=1/1;

{{MOS tunings

| Step Ratios = 6/5; 3/2; 4/3

| JI Ratios =

1/1;

14/13;

11/10;

Line 104:

Line 123:

20/11;

13/7;

2/1}}

2/1

}}

===Parahard tunings===

=== 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).

{{MOS tunings|JI Ratios=1/1;

{{MOS tunings

| JI Ratios =

1/1;

21/17;

17/16;

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Line 144:

32/17;

34/21;

2/1|Step Ratios=4/1}}

2/1

| Step Ratios = 4/1

}}

=== Ultrahard tunings ===

[[Buzzard]] is a rank-2 temperament in the [[Step ratio|pseudocollapsed]] range. It represents the only [[harmonic entropy]] minimum of the oneirotonic spectrum.

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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.

{{MOS tunings|JI Ratios=1/1;

{{MOS tunings

| JI Ratios =

1/1;

8/7;

13/10;

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26/15;

49/25, 160/81;

2/1|Step Ratios=7/1; 10/1; 12/1|Tolerance=30}}

2/1

| Step Ratios = 7/1; 10/1; 12/1

| Tolerance = 30

}}

== Approaches ==

* [[5L 3s/Temperaments]]

* [[5L 3s/Temperaments]]

== Samples ==

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== Scale tree ==

{{MOS tuning spectrum

| 13/8 = Golden oneirotonic (458.3592{{c}})

| 13/5 = Golden A-Team (465.0841{{c}})

}}

[[Category:Oneirotonic| ]]

[[Category:Pages with internal sound examples]]

~~[[Category:8-tone scales]]~~

@@ Line 9: / Line 9: @@
 | 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&nbsp;3s (3/1-equivalent)]].''
 {{MOS intro}}
-L 3s can be seen as a [[Warped diatonic|warped diatonic scale]], because it has one extra small step compared to diatonic ([[5L 2s]]).
+L&nbsp;3s can be seen as a [[Warped diatonic|warped diatonic scale]], because it has one extra small step compared to diatonic ([[5L&nbsp;2s]]).
 == 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&nbsp;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&nbsp;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&nbsp;2s]].
-=== Proposed mode names ===
+== Scale properties ==
+=== Intervals ===
+{{MOS intervals}}
+=== Generator chain ===
+{{MOS genchain}}
+=== Modes ===
+{{MOS mode degrees}}
+==== 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;
+{{MOS modes
-Ilarnekian;
+| Mode Names=
-Celephaïsian;
+Dylathian $
-Ultharian;
+Ilarnekian $
-Mnarian;
+Celephaïsian $
-Kadathian;
+Ultharian $
-Hlanithian;
+Mnarian $
-Sarnathian;|Collapsed=1}}
+Kadathian $
+Hlanithian $
-== Scale properties ==
+Sarnathian $
-{{MOS data}}
+| Collapsed=1
+}}
 == Tunings==
-===Simple 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 simplest tuning for 5L&nbsp;3s correspond to 13edo, 18edo, and 21edo, with step ratios 2:1, 3:1, and 3:2, respectively.
 {{MOS tunings|JI Ratios=Int Limit: 30; Prime Limit: 19; Tenney Height: 7.7}}
@@ Line 46: / Line 58: @@
 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.
-* [[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.
 {{MOS tunings|Step Ratios=Hypohard|JI Ratios=Subgroup: 2.5.9.21; Int Limit:40; Complements Only: 1|Tolerance=15}}
@@ Line 55: / Line 67: @@
 === 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¢ 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.
 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.)
-{{MOS tunings|Step Ratios=Hyposoft|JI Ratios=1/1;
+{{MOS tunings
+| Step Ratios = Hyposoft
+| JI Ratios =
+/1;
 /15;
 /9; 11/10;
@@ Line 78: / Line 93: @@
 /5;
 /8;
-/1}}
+/1
+}}
 === 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.
-{{MOS tunings|Step Ratios=6/5; 3/2; 4/3|JI Ratios=1/1;
+{{MOS tunings
+| Step Ratios = 6/5; 3/2; 4/3
+| JI Ratios =
+/1;
 /13;
 /10;
@@ Line 104: / Line 123: @@
 /11;
 /7;
-/1}}
+/1
+}}
-===Parahard tunings===
+=== Parahard tunings ===
 edo 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).
-{{MOS tunings|JI Ratios=1/1;
+{{MOS tunings
+| JI Ratios =
+/1;
 /17;
 /16;
@@ Line 122: / Line 144: @@
 /17;
 /21;
-/1|Step Ratios=4/1}}
+/1
+| Step Ratios = 4/1
+}}
 === Ultrahard tunings ===
-{{Main|5L 3s/Temperaments#Buzzard}}
+{{Main|5L&nbsp;3s/Temperaments#Buzzard}}
 [[Buzzard]] is a rank-2 temperament in the [[Step ratio|pseudocollapsed]] range. It represents the only [[harmonic entropy]] minimum of the oneirotonic spectrum.
@@ Line 133: / 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.
-{{MOS tunings|JI Ratios=1/1;
+{{MOS tunings
+| JI Ratios =
+/1;
 /7;
 /10;
@@ Line 141: / Line 167: @@
 /15;
 /25, 160/81;
-/1|Step Ratios=7/1; 10/1; 12/1|Tolerance=30}}
+/1
+| Step Ratios = 7/1; 10/1; 12/1
+| Tolerance = 30
+}}
 == Approaches ==
-* [[5L 3s/Temperaments]]
+* [[5L&nbsp;3s/Temperaments]]
 == Samples ==
@@ Line 176: / Line 205: @@
 == Scale tree ==
-{{Scale tree}}
+{{MOS tuning spectrum
+| 13/8 = Golden oneirotonic (458.3592{{c}})
+| 13/5 = Golden A-Team (465.0841{{c}})
+}}
 [[Category:Oneirotonic| ]] <!-- sort order in category: this page shows above A -->
 [[Category:Pages with internal sound examples]]
-[[Category:8-tone scales]]