5L 3s: Difference between revisions

Line 470:

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

:''Main article: [[A-Team]]''

A-Team tunings (with generator between 5\13 and 7\18) have step ratios between 2/1 and 3/1. Hence A-Team tunings have [[5L 8s]] chromatic MOSes and [[13L 5s]] superchromatic MOSes.

A short definition of A-Team is "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]].

* 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.

* 31edo is very close to the 2.9.5.21 POTE tuning, and can be used to make the major mos3rd 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 A-Team tunings.

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

|-

!

! [[13edo]]

! [[18edo]]

! [[31edo]]

! Optimal ([[POTE]]) tuning

! JI intervals represented (2.9.5.21 subgroup)

|-

| generator (g)

| 5\13, 461.54

| 7\18, 466.67

| 12\31, 464.52

| 464.39

| 21/16

|-

| L (3g - octave)

| 2\13, 184.62

| 3\18, 200.00

| 5\31, 193.55

| 193.16

| 9/8, 10/9

|-

| s (-5g + 2 octaves)

| 1\13, 92.31

| 1\18, 66.67

| 2\31, 77.42

| 78.07

| 21/20

|}

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) ===

@@ Line 470: / Line 470: @@
 === A-Team (13&18) ===
 :''Main article: [[A-Team]]''
+A-Team tunings (with generator between 5\13 and 7\18) have step ratios between 2/1 and 3/1. Hence A-Team tunings have [[5L 8s]] chromatic MOSes and [[13L 5s]] superchromatic MOSes.
+A short definition of A-Team is "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]].
+* 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.
+* 31edo is very close to the 2.9.5.21 POTE tuning, and can be used to make the major mos3rd 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 A-Team tunings.
+{| class="wikitable right-2 right-3 right-4 right-5"
+|-
+!
+! [[13edo]]
+! [[18edo]]
+! [[31edo]]
+! Optimal ([[POTE]]) tuning
+! JI intervals represented (2.9.5.21 subgroup)
+|-
+| generator (g)
+| 5\13, 461.54
+| 7\18, 466.67
+| 12\31, 464.52
+| 464.39
+| 21/16
+|-
+| L (3g - octave)
+| 2\13, 184.62
+| 3\18, 200.00
+| 5\31, 193.55
+| 193.16
+| 9/8, 10/9
+|-
+| s (-5g + 2 octaves)
+| 1\13, 92.31
+| 1\18, 66.67
+| 2\31, 77.42
+| 78.07
+| 21/20
+|}
+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) ===