5L 3s: Difference between revisions

Line 7:

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.

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.

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.

5L 3s has a pentatonic MOS subset [[3L 2s]] (SLSLL). (Note: [[3L 5s]] scales also have 3L 2s subsets.)

Line 294:

Hypohard oneirotonic can be considered "meantone oneirotonic". This is because these 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]]).

* The large step is a "meantone", somewhere between near-10/9 (as in [[13edo]]) and near-9/8 (as in [[18edo]]).

*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 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]].

Line 302:

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.

* 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.

*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.

* 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.

*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.

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

Line 1,119:

| 1178.18

| P

| 98/50, 160/81

| 49/25, 160/81

| +5

|}

Line 1,130:

! style="text-align:center;" |[[Modal UDP Notation|'''UDP''']]

! style="text-align:center;" |'''Name'''

|'''Other names'''

! style="text-align:center;" |'''Other names'''

|-

| |LLsLLsLs

Line 1,221:

!Comments

|-

|3\8|| || || || || || 450.000 ||1||1||1.000||

| 3\8 || || || || || || 450.000 ||1||1||1.000||

|-

| || || || || || 17\45 || 453.333 ||6||5||1.200||

|-

| || || || ||14\37 || || 454.054 ||5||4||1.250||

| || || || || 14\37 || || 454.054 ||5||4||1.250||

|-

| || || || || || 34\59 || 454.545 ||9||7||1.286||

| || || || || || 25\66 || 454.545 ||9||7||1.286||

|-

| || || ||11\29 || || || 455.172 ||4||3||1.333||

| || || || 11\29 || || || 455.172 ||4||3||1.333||

|-

| || || || || || 30\79 || 455.696 ||11||8||1.375||

|-

| || || || ||19\50 || || 456.000 ||7||5||1.400||

| || || || || 19\50 || || 456.000 ||7||5||1.400||

|-

| || || || || || 27\71 || 456.338 ||10||7||1.429||

|-

| || ||8\21|| || || ||457.143 ||3||2||1.500||

| || || 8\21 || || || || 457.143 ||3||2||1.500||

|-

| || || || || || 29\76 || 457.895 ||11||7||1.571||

|-

| || || || ||21\55 || || 458.182 ||8||5||1.600||

| || || || || 21\55 || || 458.182 ||8||5||1.600||

|-

| || || || || || 34\89 || 458.427 ||13||8||1.625||Golden oneirotonic

| || || || || || 34\89 || 458.427 ||13||8||1.625||Golden oneirotonic (458.3592¢)

|-

| || || ||13\34 || || || 458.824 ||5||3||1.667||~~~~

| || || || 13\34 || || || 458.824 ||5||3||1.667||

|-

| || || || || || 31\81 || 459.259 ||12||7||1.714||

|-

| || || || ||18\47 || || 459.574 ||7||4||1.750||

| || || || || 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)

| || 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||

| || || || || 17\44 || || 463.636 ||7||3||2.333||

|-

| || || || || || 29\75 || 464.000 ||12||5||2.400||

|-

| || || ||12\31 || || || 464.516 ||5||2||2.500||~~~~

| || || || 12\31 || || || 464.516 ||5||2||2.500||

|-

| || || || || || 31\80 || 465.000 ||13||5||2.600||

| || || || || || 31\80 || 465.000 ||13||5||2.600||Golden A-Team (465.0841¢)

|-

| || || || ||19\49 || || 465.306 ||8||3||2.667||

| || || || || 19\49 || || 465.306 ||8||3||2.667||

|-

| || || || || || 26\67 || 465.672 ||11||4||2.750||

|-

| || ||7\18|| || || || 466.667 ||3||1||3.000||

| || || 7\18 || || || || 466.667 ||3||1||3.000||

|-

| || || || || || 23\59 || 467.797 ||10||3||3.333||

|-

| || || || ||16\41 || || 468.293 ||7||2||3.500||

| || || || || 16\41 || || 468.293 ||7||2||3.500||

|-

| || || || || || 25\64 || 468.750 ||11||3||3.667||

|-

| || || ||9\23|| || || 469.565 ||4||1||4.000||

| || || || 9\23 || || || 469.565 ||4||1||4.000||

|-

| || || || || || 20\51 || 470.588 ||9||2||4.500||

|-

| || || || ||11\28 || || 471.429 ||5||1||5.000||

| || || || || 11\28 || || 471.429 ||5||1||5.000||

|-

| || || || || || 13\33 || 472.727 ||6||1||6.000||

|-

|2\5|| || || || || || 480.000 ||1||0||→ inf||

| 2\5 || || || || || || 480.000 ||1||0||→ inf||

|}

@@ Line 7: / Line 7: @@
 L 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.
-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.
+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.
 L 3s has a pentatonic MOS subset [[3L 2s]] (SLSLL). (Note: [[3L 5s]] scales also have 3L 2s subsets.)
@@ Line 294: / Line 294: @@
 Hypohard oneirotonic can be considered "meantone oneirotonic". This is because these 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]]).
+* The large step is a "meantone", somewhere between near-10/9 (as in [[13edo]]) and near-9/8 (as in [[18edo]]).
-*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 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]].
@@ Line 302: / Line 302: @@
 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.
+* 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.
-*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.
+* 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.
-*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.
 The sizes of the generator, large step and small step of oneirotonic are as follows in various hypohard oneiro tunings.
@@ Line 1,119: / Line 1,119: @@
 | 1178.18
 | P
-| 98/50, 160/81
+| 49/25, 160/81
 | +5
 |}
@@ Line 1,130: / Line 1,130: @@
 ! style="text-align:center;" |[[Modal UDP Notation|'''UDP''']]
 ! style="text-align:center;" |'''Name'''
-|'''Other names'''
+! style="text-align:center;" |'''Other names'''
 |-
 | |LLsLLsLs
@@ Line 1,221: / Line 1,221: @@
 !Comments
 |-
-|3\8|| || || || ||  || 450.000 ||1||1||1.000||
+| 3\8 || || || || || || 450.000 ||1||1||1.000||
 |-
 | || || || || || 17\45 || 453.333 ||6||5||1.200||
 |-
-| || || || ||14\37 ||  || 454.054 ||5||4||1.250||
+| || || || || 14\37 || || 454.054 ||5||4||1.250||
 |-
-| || || || || || 34\59 || 454.545 ||9||7||1.286||
+| || || || || || 25\66 || 454.545 ||9||7||1.286||
 |-
-| || || ||11\29 ||  || || 455.172 ||4||3||1.333||
+| || || || 11\29 || || || 455.172 ||4||3||1.333||
 |-
 | || || || || || 30\79 || 455.696 ||11||8||1.375||
 |-
-| || || || ||19\50 ||  || 456.000 ||7||5||1.400||
+| || || || || 19\50 || || 456.000 ||7||5||1.400||
 |-
 | || || || || || 27\71 || 456.338 ||10||7||1.429||
 |-
-| || ||8\21|| || || ||457.143 ||3||2||1.500||
+| || || 8\21 || || || || 457.143 ||3||2||1.500||
 |-
-|  || ||  || || || 29\76 || 457.895 ||11||7||1.571||
+| || || || || || 29\76 || 457.895 ||11||7||1.571||
 |-
-| || || || ||21\55 ||  || 458.182 ||8||5||1.600||
+| || || || || 21\55 || || 458.182 ||8||5||1.600||
 |-
-| || || || || || 34\89 || 458.427 ||13||8||1.625||Golden oneirotonic
+| || || || || || 34\89 || 458.427 ||13||8||1.625||Golden oneirotonic (458.3592¢)
 |-
-| || || ||13\34 ||  || || 458.824 ||5||3||1.667||<!--Petrtri is in this region-->
+| || || || 13\34 || || || 458.824 ||5||3||1.667||
 |-
-|  || || || || || 31\81 || 459.259 ||12||7||1.714||
+| || || || || || 31\81 || 459.259 ||12||7||1.714||
 |-
-| || || || ||18\47 ||  || 459.574 ||7||4||1.750||
+| || || || || 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)
+| || 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||
+| || || || || 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-->
+| || || || 12\31 || || || 464.516 ||5||2||2.500||
 |-
-|  || || || || || 31\80 || 465.000 ||13||5||2.600||
+| || || || || || 31\80 || 465.000 ||13||5||2.600||Golden A-Team (465.0841¢)
 |-
-| || || || ||19\49 ||  || 465.306 ||8||3||2.667||
+| || || || || 19\49 || || 465.306 ||8||3||2.667||
 |-
 | || || || || || 26\67 || 465.672 ||11||4||2.750||
 |-
-| || ||7\18|| || || || 466.667 ||3||1||3.000||
+| || || 7\18 || || || || 466.667 ||3||1||3.000||
 |-
-|  || ||  || || || 23\59 || 467.797 ||10||3||3.333||
+| || || || || || 23\59 || 467.797 ||10||3||3.333||
 |-
-| || || || ||16\41 ||  || 468.293 ||7||2||3.500||
+| || || || || 16\41 || || 468.293 ||7||2||3.500||
 |-
 | || || || || || 25\64 || 468.750 ||11||3||3.667||
 |-
-| || || ||9\23|| || || 469.565 ||4||1||4.000||
+| || || || 9\23 || || || 469.565 ||4||1||4.000||
 |-
-|  || ||  ||  || || 20\51 || 470.588 ||9||2||4.500||
+| || || || || || 20\51 || 470.588 ||9||2||4.500||
 |-
-| || || || ||11\28 ||  || 471.429 ||5||1||5.000||
+| || || || || 11\28 || || 471.429 ||5||1||5.000||
 |-
 | || || || || || 13\33 || 472.727 ||6||1||6.000||
 |-
-|2\5||  || || || ||  || 480.000 ||1||0||→ inf||
+| 2\5 || || || || || || 480.000 ||1||0||→ inf||
 |}