5L 3s: Difference between revisions

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|}

== Tuning ranges ==

=== ~~A-Team (13&18)~~ ===

=== Hypohard ===

~~A-Team~~ tunings (with generator between 5\13 and 7\18) have step ratios between 2/1 and 3/1.

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

~~A short definition of A-Team is~~ "meantone oneirotonic". This is because A-Team tunings share the following features with [[meantone]] diatonic tunings:

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

* The large step is a "meantone", somewhere between near-10/9 (as in [[13edo]]) and near-9/8 (as in [[18edo]]). Thus A-Team tempers out [[81/80]] like meantone does.

* 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.

EDOs that ~~support A-Team~~ include [[13edo]], [[18edo]], and [[31edo]].

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

* 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 A-team 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 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.

Line 481:

* [[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 ~~A-Team~~ tunings.

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"

|-

Line 515:

Trivia: A-Team can be tuned by ear, by tuning a chain of pure harmonic sevenths and taking every other note. This corresponds to using a generator of 64/49 = 462.34819 cents. A chain of fourteen 7/4's are needed to tune the 8-note oneirotonic MOS. This produces a tuning close to 13edo.

=== ~~Petrtri (13&21)~~ ===

=== Hyposoft ===

~~Petrtri~~ tunings (with generator between 8\21 and 5\13) have less extreme step ratios than A-Team tunings, 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 (with generator between 8\21 and 5\13) have less extreme step ratios than A-Team tunings, 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¢)~~, and the temperament interprets it as both [[11/9]] and [[16/13]]~~.

* 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¢).

~~The three major edos in this range, [[13edo]],~~ [[21edo]] ~~and [[34edo]], all nominally support petrtri.~~

* [[21edo]]'s P1-M2-M3-M5 (in oneiro interval classes) 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¢).

* [[13edo]] nominally supports it, but its approximation of 9:10:11:13 ~~is quite weak and tempers 11/9 to a 369¢ submajor third, which may not be desirable.~~

* [[34edo]] is close to optimal for [[5L 3s#Petrtri|petrtri]] temperament, with a generator only 0.33¢ flat of the 2.5.9.11.13.17 [[POTE]] petrtri generator of 459.1502¢ and 0.73¢ sharp of the 2.9/5.11/5.13/5 POTE (i.e. optimal for the chord 9:10:11:13, spelled as R-M2-M3-M5 in oneirotonic intervals) petrtri generator of 458.0950¢.

* [[21edo]] is a much better ~~petrtri tuning~~ than 13edo~~, in terms of approximating 9:10:11:13~~. 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]] is close to optimal for ~~the~~ temperament, with a generator only 0.33¢ flat of the 2.5.9.11.13.17 [[POTE]] petrtri generator of 459.1502¢ and 0.73¢ sharp of the 2.9/5.11/5.13/5 POTE (i.e. optimal for the chord 9:10:11:13, spelled as R-M2-M3-M5 in oneirotonic intervals) petrtri generator of 458.0950¢.

* If you only care about optimizing 9:10:11:13, then [[55edo]]'s 21\55 (458.182¢) is even better, but 55 is a bit big for a usable edo.

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

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

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

|-

Line 558:

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|}

=== ~~Tridec (29&37)~~ ===

=== Softer-than-soft tunings ===

~~In the broad sense, Tridec can be viewed as any oneirotonic tuning that~~ equates three oneirotonic large steps to a ~~[[4/3]]~~ perfect fourth, i.e. equates the oneirotonic large step to a [[porcupine]] generator. [This identification may come in handy since many altered oneirotonic modes have three consecutive large steps.] Based on the JI interpretations of the [[29edo]] and [[37edo]] tunings, it can in fact be viewed as a 2.3.7/5.11/5.13/5 temperament, i.e. a [[Non-over-1 temperament|non-over-1 temperament]] that approximates the chord 5:7:11:13:15. Since it is the same as Petrtri when you only care about the 9:10:11:13 (R-M2-M3-M5), it can be regarded as a flatter variant of Petrtri (analogous to how septimal meantone and flattone are the same when you only consider how it maps 8:9:10:12).

Oneirotonic tunings of step ratio between 6/5 and 3/2 equates three oneirotonic large steps to a diatonic perfect fourth, i.e. equates 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 optimal generator is 455.2178¢, which is very close to 29edo's 11\29 (455.17¢), but we could accept any generator between 17\45 (453.33¢) and 8\21 (457.14¢), if we stipulate that the 3/2 has to be between [[7edo]]'s fifth and [[5edo]]'s fifth.

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

~~Based on the EDOs that support it, Tridec is essentially the same as 13-limit [[Ammonite]].~~

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

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

|-

Line 572:

Line 566:

! [[29edo]]

! [[37edo]]

! ~~Optimal (~~[[~~POTE~~]]~~) tuning~~

! [[45edo]]

! JI intervals represented ~~(2.3.7/5.11/5.13/5 subgroup)~~

! JI intervals represented

|-

| generator (g)

Line 579:

Line 573:

| 11\29, 455.17

| 14\37, 454.05

| ~~455~~.22

| 17\45, 453.33

| 13/10

|-

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Line 580:

| 4\29, 165.52

| 5\37, 162.16

| ~~165~~.65

| 6\45, 160.00

| 11/10

|-

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Line 587:

| 3\29, 124.14

| 4\37, 129.73

| ~~123~~.91

| 5\45, 133.33

| 14/13~~, 15/14~~

| 15/14 to 14/13

|}

=== Buzzard ~~(48&53)~~ ===

=== Buzzard ===

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 A-Team 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.

Buzzard is an oneirotonic temperament in the ultrahard-to-[[Step ratio|pseudopaucitonic]] range.

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 A-Team 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.

@@ Line 467: / Line 467: @@
 |}
-==  Tuning ranges ==
+== Tuning ranges ==
-=== A-Team (13&18) ===
+=== Hypohard ===
-A-Team tunings (with generator between 5\13 and 7\18) have step ratios between 2/1 and 3/1.
+Hypohard oneirotonic tunings (with generator between 5\13 and 7\18) have step ratios between 2/1 and 3/1.
-A short definition of A-Team is "meantone oneirotonic". This is because A-Team tunings share the following features with [[meantone]] diatonic tunings:
+Hypohard oneirotonic can be considered "meantone oneirotonic". This is because A-Team tunings share the following features with [[meantone]] diatonic tunings:
 * The large step is a "meantone", somewhere between near-10/9 (as in [[13edo]]) and near-9/8 (as in [[18edo]]). Thus A-Team tempers out [[81/80]] like meantone does.
 * 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.
-EDOs that support A-Team include [[13edo]], [[18edo]], and [[31edo]].
+EDOs that are in the hypohard range include [[13edo]], [[18edo]], and [[31edo]].
 * 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 A-team 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 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.
@@ Line 481: / Line 481: @@
 * [[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 A-Team tunings.
+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"
 |-
@@ Line 515: / Line 515: @@
 Trivia: A-Team can be tuned by ear, by tuning a chain of pure harmonic sevenths and taking every other note. This corresponds to using a generator of 64/49 = 462.34819 cents. A chain of fourteen 7/4's are needed to tune the 8-note oneirotonic MOS. This produces a tuning close to 13edo.
-=== Petrtri (13&21) ===
+=== Hyposoft ===
-Petrtri tunings (with generator between 8\21 and 5\13) have less extreme step ratios than A-Team tunings, 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 (with generator between 8\21 and 5\13) have less extreme step ratios than A-Team tunings, 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¢), and the temperament interprets it as both [[11/9]] and [[16/13]].
+* 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¢).
-The three major edos in this range, [[13edo]], [[21edo]] and [[34edo]], all nominally support petrtri.
+* [[21edo]]'s P1-M2-M3-M5 (in oneiro interval classes) 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¢).
-* [[13edo]] nominally supports it, but its approximation of 9:10:11:13 is quite weak and tempers 11/9 to a 369¢ submajor third, which may not be desirable.
+* [[34edo]] is close to optimal for [[5L 3s#Petrtri|petrtri]] temperament, with a generator only 0.33¢ flat of the 2.5.9.11.13.17 [[POTE]] petrtri generator of 459.1502¢ and 0.73¢ sharp of the 2.9/5.11/5.13/5 POTE (i.e. optimal for the chord 9:10:11:13, spelled as R-M2-M3-M5 in oneirotonic intervals) petrtri generator of 458.0950¢.
-* [[21edo]] is a much better petrtri tuning than 13edo, in terms of approximating 9:10:11:13. 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]] is close to optimal for the temperament, with a generator only 0.33¢ flat of the 2.5.9.11.13.17 [[POTE]] petrtri generator of 459.1502¢ and 0.73¢ sharp of the 2.9/5.11/5.13/5 POTE (i.e. optimal for the chord 9:10:11:13, spelled as R-M2-M3-M5 in oneirotonic intervals) petrtri generator of 458.0950¢.
 * If you only care about optimizing 9:10:11:13, then [[55edo]]'s 21\55 (458.182¢) is even better, but 55 is a bit big for a usable edo.
-The sizes of the generator, large step and small step of oneirotonic are as follows in various petrtri tunings.
+The sizes of the generator, large step and small step of oneirotonic are as follows in various hyposoft oneiro tunings.
 {| class="wikitable right-2 right-3 right-4 right-5"
 |-
@@ Line 558: / Line 556: @@
 |}
-=== Tridec (29&37) ===
+=== Softer-than-soft tunings ===
-In the broad sense, Tridec can be viewed as any oneirotonic tuning that equates three oneirotonic large steps to a [[4/3]] perfect fourth, i.e. equates the oneirotonic large step to a [[porcupine]] generator. [This identification may come in handy since many altered oneirotonic modes have three consecutive large steps.] Based on the JI interpretations of the [[29edo]] and [[37edo]] tunings, it can in fact be viewed as a 2.3.7/5.11/5.13/5 temperament, i.e. a [[Non-over-1 temperament|non-over-1 temperament]] that approximates the chord 5:7:11:13:15. Since it is the same as Petrtri when you only care about the 9:10:11:13 (R-M2-M3-M5), it can be regarded as a flatter variant of Petrtri (analogous to how septimal meantone and flattone are the same when you only consider how it maps 8:9:10:12).
+Oneirotonic tunings of step ratio between 6/5 and 3/2 equates three oneirotonic large steps to a diatonic perfect fourth, i.e. equates 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 optimal generator is 455.2178¢, which is very close to 29edo's 11\29 (455.17¢), but we could accept any generator between 17\45 (453.33¢) and 8\21 (457.14¢), if we stipulate that the 3/2 has to be between [[7edo]]'s fifth and [[5edo]]'s fifth.
+The sizes of the generator, large step and small step of oneirotonic are as follows in various parahard tunings.
-Based on the EDOs that support it, Tridec is essentially the same as 13-limit [[Ammonite]].
-The sizes of the generator, large step and small step of oneirotonic are as follows in various tridec tunings.
 {| class="wikitable right-2 right-3 right-4 right-5"
 |-
@@ Line 572: / Line 566: @@
 ! [[29edo]]
 ! [[37edo]]
-! Optimal ([[POTE]]) tuning
+! [[45edo]]
-! JI intervals represented (2.3.7/5.11/5.13/5 subgroup)
+! JI intervals represented
 |-
 | generator (g)
@@ Line 579: / Line 573: @@
 | 11\29, 455.17
 | 14\37, 454.05
-| 455.22
+| 17\45, 453.33
 | 13/10
 |-
@@ Line 586: / Line 580: @@
 | 4\29, 165.52
 | 5\37, 162.16
-| 165.65
+| 6\45, 160.00
 | 11/10
 |-
@@ Line 593: / Line 587: @@
 | 3\29, 124.14
 | 4\37, 129.73
-| 123.91
+| 5\45, 133.33
-| 14/13, 15/14
+| 15/14 to 14/13
 |}
-=== Buzzard (48&53) ===
+=== Buzzard ===
-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 A-Team 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.
+Buzzard is an oneirotonic temperament in the ultrahard-to-[[Step ratio|pseudopaucitonic]] range.
+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 A-Team 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.