5L 3s: Difference between revisions

:''For the tritave-equivalent MOS structure with the same step pattern, see [[5L 3s (tritave-equivalent)]].''

'''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).

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.

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.

== Standing assumptions ==

The [[TAMNAMS]] system is used in this article to name 5L 3s intervals and step size ratios and step ratio ranges.

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

Thus the [[13edo]] gamut is as follows:

 

 

 

 

 

== Names ==

 

 

 

== Tuning ranges ==

! class="unsortable"| Approximate ratios<ref>The ratio interpretations that are not valid for 18edo are italicized.</ref>

| 0\13, 0.00

| 0\18, 0.00

| minor 1-step

| 2\21, 114.29

| major 1-step

| 2\13, 184.62

| 3\18, 200.00

| -1

| 17\18, 1133.33

[[Hypohard]] oneirotonic tunings (with generator between 5\13 and 7\18) have step ratios between 2/1 and 3/1.

 

Hypohard oneirotonic can be considered "meantone oneirotonic". This is because these tunings share the following features with [[meantone]] diatonic tunings:  

* The major 2-mosstep (made of two large steps) is a [[meantone]]- to [[flattone]]-sized major third, thus is a stand-in for the classical diatonic major third.

 

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

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

EDOs that are in the hypohard range include [[13edo]], [[18edo]], and [[31edo]].

* 13edo has characteristically small 1-mossteps of about 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.

* 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 5-mosstep) and falling fifths (666.7c, a major 4-mosstep) almost equally off from a just perfect fifth. 18edo is also more suited for conventionally jazz styles due to its 6-fold symmetry.

* 31edo  can be used to make the major 2-mosstep a near-just 5/4.

* [[44edo]] (generator 17\44 = 463.64¢), [[57edo]] (generator 22\57 = 463.16¢), and [[70edo]] (generator 27\70 = 462.857¢) offer a compromise between 31edo's major third and 13edo's 11/8 and 13/8. In particular, 70edo has an essentially pure 13/8.

The sizes of the generator, large step and small step of oneirotonic are as follows in various hypohard oneiro tunings.

{| class="wikitable right-2 right-3 right-4 right-5"

! [[13edo]] (basic)

| 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 "

|-bgcolor="#eaeaff"

| 2\31, 77.42

| -5

| 2\13, 184.62

| 9/8, 10/9

| 4\18, 266.67

| 21/16, ''13/10'', 17/13

| +1

|-bgcolor="#eaeaff"

| 7\13, 646.15

|}

=== Hyposoft ===

[[Hyposoft]] oneirotonic tunings (with generator between 8\21 and 5\13) have step ratios between 3/2 and 2/1. The 8\21-to-5\13 range of oneirotonic tunings remains relatively unexplored. In these tunings,  

* the large step of oneirotonic tends to be intermediate in size between [[10/9]] and [[11/10]]; the small step size is a semitone close to [[17/16]], about 92¢ to 114¢.

* The major 2-mosstep (made of two large steps) in these tunings tends to be more of a neutral third, ranging from 6\21 (342¢) to 4\13 (369¢).

* [[21edo]]'s P1-L1ms-L2ms-L4ms approximates 9:10:11:13 better than the corresponding 13edo chord does. 21edo will serve those who like the combination of neogothic minor thirds (285.71¢) and Baroque diatonic semitones (114.29¢, close to quarter-comma meantone's 117.11¢).

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"

|-

| 13\34, 458.82

| 5\34, 176.47

| 2\21, 114.29

|}

==== Intervals ====

Sortable table of major and minor intervals in hyposoft tunings (13edo not shown):

{| class="wikitable right-2 right-3 sortable "

|-bgcolor="#eaeaff"

| 1/1

| 16/15

| 10/9, 11/10

| 8\34, 282.35

| 13/11, 20/17

| 10\34, 352.94

| 11/9

| dim. 3-step

| 11\34, 388.24

| 5/4

| -7

| perf. 3-step

| 13/10

| 18/13, 32/23

| 11\21, 628.57

| 13/9, 23/16

| 20/13

| 8/5

| 15\21, 857.14

| 18/11

| 22/13, 17/10

| +2

| 9/5

|}

=== 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.

{| class="wikitable right-2 right-3 right-4 right-5"

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 "

| 1/1

| 14/13

| 11/10

| 9/8

| +11

| 6\29, 248.3

| 15/13

| 13/11

| 14/11

| 13/10

| 4/3

| 15/11

| 7/5

| 15\29 620.7

| 10/7

| 17\29, 703.4

| 3/2

| 18\29, 755.2

| 20/13

| -1

|-

|-bgcolor="#eaeaff"

| 22/13

| 24\29, 993.1

|}

=== Parahard ===

23edo oneiro combines the sound of neogothic tunings like [[46edo]] and the sounds of "superpyth" and "semaphore" scales. This is because 23edo oneirotonic has a large step of 208.7¢, same as [[46edo]]'s neogothic major second, and is both a warped [[22edo]] [[superpyth]] [[diatonic]] and a warped [[24edo]] [[semaphore]] [[semiquartal]] (and both nearby scales are [[superhard]] MOSes).

==== Intervals ====

{| class="wikitable right-2 right-3 sortable "

| N&

| 34/21, φ

|-bgcolor="#eaeaff"

|}

[[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.  

The sizes of the generator, large step and small step of oneirotonic are as follows in various buzzard tunings.

{| class="wikitable right-2 right-3 right-4 right-5"

! JI intervals represented (2.3.5.7.13 subgroup)

| 21\53, 475.47

| 3/2 21/16

| 7/38, 221.04

| 10/53, 226.41

| 12/63, 228.57

| 8/7

| s (-5g + 2 octaves)

| 1/38 31.57

| 1/53 22.64

| 1/63 19.05

| 21.55

|}

* [[5L 3s/Inthar's approach]]

* [[5L 3s/Temperaments]]

[[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]])

(13edo, L Illarnekian)