Petrtri: Difference between revisions

Line 2:

== Notation==

The notation used in this article is described in [[5L 3s#Notation]].

== Tuning range ==

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,

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

* [[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.

* [[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.

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

|-

!

! [[13edo]]

! [[21edo]]

! [[34edo]]

! Optimal (2.5.9.11.13.17 [[POTE]]) tuning

! JI intervals represented (2.5.9.11.13.17 subgroup)

|-

| generator (g)

| 5\13, 461.54

| 8\21, 457.14

| 13\34, 458.82

| 459.15

| 13/10, 17/13, 22/17

|-

| L (3g - octave)

| 2\13, 184.62

| 3\21, 171.43

| 5\34, 176.47

| 177.45

| 10/9, 11/10

|-

| s (-5g + 2 octaves)

| 1\13, 92.31

| 2\21, 114.29

| 3\34, 105.88

| 104.25

| 18/17, 17/16

|}

== Temperament data ==

@@ Line 2: / Line 2: @@
 == Notation==
 The notation used in this article is described in [[5L 3s#Notation]].
+== Tuning range ==
+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,
+* 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 three major edos in this range, [[13edo]], [[21edo]] and [[34edo]], all nominally support petrtri.
+* [[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.
+* [[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.
+{| class="wikitable right-2 right-3 right-4 right-5"
+|-
+!
+! [[13edo]]
+! [[21edo]]
+! [[34edo]]
+! Optimal (2.5.9.11.13.17 [[POTE]]) tuning
+! JI intervals represented (2.5.9.11.13.17 subgroup)
+|-
+| generator (g)
+| 5\13, 461.54
+| 8\21, 457.14
+| 13\34, 458.82
+| 459.15
+| 13/10, 17/13, 22/17
+|-
+| L (3g - octave)
+| 2\13, 184.62
+| 3\21, 171.43
+| 5\34, 176.47
+| 177.45
+| 10/9, 11/10
+|-
+| s (-5g + 2 octaves)
+| 1\13, 92.31
+| 2\21, 114.29
+| 3\34, 105.88
+| 104.25
+| 18/17, 17/16
+|}
 == Temperament data ==