Pinetone: Difference between revisions

Line 361:

Using the familiar Bosanquet 12-note keyboard mapping (the preset for 12edo), we set the porcutone diatonic scale to the white keys, starting on D. We than add, on F♯/G♭, the porcutone penatonic as a set of 5 chromatic keys. There are two options for the chromatic keys, either all sharps or all flats. All sharps makes the porcutone harmonic minor available, and all flats makes the porcutone harmonic major available. These scales will be discussed below. In either case, in the just tuning, the chromatic keys give the scale 9/8 5/4 3/2 5/3 2/1, starting from F♯/G♭, tuned to 100/81 (F♯) or 162/125 (G♭) from D. This scale has step pattern msLsL, with step signature and step mapping 2L 1m 2s = (6/5, 9/8, 10/9). The same scale is also available as G-A-B-D-E.

We are familiar with this scale as the just pentatonic. If we temper m and s together, we get Meantone[5]: ssLsL. If we temper m and L together instead we get a scale called Father[5], tempering out the diatonic semitone 16/15. This mode of Father[5] has step pattern LsLsL. Keep the connection to Father[5] in the back of your minds for now, we'll come back to it. The porcutone pentatonic is also a subset of the porcutone diatonic, since Meantone[5] is a subset of Meantone[7]. It is available as G-A-B-D-E. The scale F-G-A-C-D is a mode of the inverse of G-A-B-D-E. The porcutone ~~pentantic~~ is [[Chirality|''chiral'']] (i.e., it is not symmetric, unlike the porcutone diatonic and procutone chromatic scales, which are ''achiral''). There is a pair of porcutone pentatonic scales, the right handed porcutone pentatonic, 9/8 5/4 3/2 5/3 2/1 in JI, it's mirror inverse the left handed porcutone pentatonic, 10/9 5/4 3/2 5/3 2/1 in JI, which tempers to ssLsL in Meantone, but to sLLsL in Father. F-G-A-C-D gives a left handed porcutone pentatonic, while G-A-B-D-E, F#-G#-A#-C#-D#, and Gb-Ab-Bb-Db-Eb are right handed porcutone pentatonics.

We are familiar with this scale as the just pentatonic. If we temper m and s together, we get Meantone[5]: ssLsL. If we temper m and L together instead we get a scale called Father[5], tempering out the diatonic semitone 16/15. This mode of Father[5] has step pattern LsLsL. Keep the connection to Father[5] in the back of your minds for now, we'll come back to it. The porcutone pentatonic is also a subset of the porcutone diatonic, since Meantone[5] is a subset of Meantone[7]. It is available as G-A-B-D-E. The scale F-G-A-C-D is a mode of the inverse of G-A-B-D-E. The porcutone pentatonic is [[Chirality|''chiral'']] (i.e., it is not symmetric, unlike the porcutone diatonic and procutone chromatic scales, which are ''achiral''). There is a pair of porcutone pentatonic scales, the right handed porcutone pentatonic, 9/8 5/4 3/2 5/3 2/1 in JI, it's mirror inverse the left handed porcutone pentatonic, 10/9 5/4 3/2 5/3 2/1 in JI, which tempers to ssLsL in Meantone, but to sLLsL in Father. F-G-A-C-D gives a left handed porcutone pentatonic, while G-A-B-D-E, F#-G#-A#-C#-D#, and Gb-Ab-Bb-Db-Eb are right handed porcutone pentatonics.

Adding the right handed porcutone chromatic (on F♯/G♭) to the just porcutone chromatic, a 12-note mirror-symmetric scale with step signature and step mapping of 7L 1m 4s = (27/25, 25/24, 250/243) = (133.2376c, 70.6724c, 49.1661c), i.e., 7 large steps of what was the small step of the just porcutone diatonic, 1 medium step of the chromatic semitone 25/24, the distance between 6/5 and 5/4, and 4 small steps of 250/243, the porcupine comma, that separates 10/9 from 27/25. For the all sharps scale, we set mode -3 on D (for all flats we set mode 3 on D): 250/243 10/9 6/5 100/81 4/3 25/18 3/2 125/81 5/3 9/5 50/27 2/1, with step pattern sLLsLmLsLLsL.

Line 379:

The ptolemismic porcutone chromatic scale is distinctly xenharmonic, and yet is related to the familiar chromatic scale.

Mode -3 has 3/2 perfect fifths available above D, D♯, E, F, F♯, G, and G♯.

Mode -3 has 3/2 perfect fifths available above D, D♯, E, F, F♯, G, G♯, and C♯.

Mode 3 has 3/2 perfect fifths available above D, E♭, E, F, G♭, G, and D♭.

Mode 3 has 3/2 perfect fifths available above D, E♭, E, F, G♭, G, A♭, and D♭.

Mode -3 has 4:5:6 major triads available above E, F, F♯, G, and G♯

Line 409:

|10:12:15

18:22:27

|

|320.690, 704.524

348.110, 704.524

|-

|D♯

Line 422:

Line 424:

|10:12:15

110:144:165

|

|320.690, 704.524

467.325, 704.524

|-

|E

Line 435:

Line 439:

|10:12:15

4:5:6

|

|320.690, 704.524

383.834, 704.524

|-

|F

Line 448:

Line 454:

|48:55:72

4:5:6

|

|237.198, 704.524

383.834, 704.524

|-

|F♯

Line 461:

Line 469:

|22:27:33

4:5:6

|

|356.414, 704.524

383.834, 704.524

|-

|G

Line 468:

Line 478:

|aug 2

major

|~~aug~~ 2

|maj 2

major

|aug 2

|aug maj 2

major

|64:75:~~128~~

|48:55:72

4:5:6

|

|237.198, 704.524

383.834, 704.524

|-

|G♯

Line 487:

Line 499:

|22:27:33

20:27:30

|

|356.414, 704.524

530.469, 704.524

|-

|A

Line 500:

Line 514:

|15:18:22

27:33:40

|

|320.690, 668.800

348.110, 668.800

|-

|A♯

Line 513:

Line 529:

|275:330:432

275:360:432

|

|320.690, 788.015

467.325, 788.015

|-

|B

Line 526:

Line 544:

|15:18:22

27:33:40

|

|320.690, 668.800

348.110, 668.800

|-

|C

Line 539:

Line 559:

|243:275:360

243:297:360

|

|176.055, 668.800

348.110, 668.800

|-

|C♯

Line 552:

Line 574:

|10:12:15

110:144:165

|

|320.690, 704.524

467.325, 704.524

|}

As with the porcutone diatonic, tuning the porcutone chromatic to 19edo or 31edo collapses it to the Meantone[12] (Meanenneadecal[12]) chromatic scale. Tuning it to 15edo, 22edo, or 29edo collapses it to Porcupine[8]. Step patterns, mappings and sizes for tunings to 27edo, 34edo, and 41edo are as follows:

@@ Line 361: / Line 361: @@
 Using the familiar Bosanquet 12-note keyboard mapping (the preset for 12edo), we set the porcutone diatonic scale to the white keys, starting on D. We than add, on F♯/G♭, the porcutone penatonic as a set of 5 chromatic keys. There are two options for the chromatic keys, either all sharps or all flats. All sharps makes the porcutone harmonic minor available, and all flats makes the porcutone harmonic major available. These scales will be discussed below. In either case, in the just tuning, the chromatic keys give the scale 9/8 5/4 3/2 5/3 2/1, starting from F♯/G♭, tuned to 100/81 (F♯) or 162/125 (G♭) from D. This scale has step pattern msLsL, with step signature and step mapping 2L 1m 2s = (6/5, 9/8, 10/9). The same scale is also available as G-A-B-D-E.
-We are familiar with this scale as the just pentatonic. If we temper m and s together, we get Meantone[5]: ssLsL. If we temper m and L together instead we get a scale called Father[5], tempering out the diatonic semitone 16/15. This mode of Father[5] has step pattern LsLsL. Keep the connection to Father[5] in the back of your minds for now, we'll come back to it. The porcutone pentatonic is also a subset of the porcutone diatonic, since Meantone[5] is a subset of Meantone[7]. It is available as G-A-B-D-E. The scale F-G-A-C-D is a mode of the inverse of G-A-B-D-E. The porcutone pentantic is [[Chirality|''chiral'']] (i.e., it is not symmetric, unlike the porcutone diatonic and procutone chromatic scales, which are ''achiral''). There is a pair of porcutone pentatonic scales, the right handed porcutone pentatonic, 9/8 5/4 3/2 5/3 2/1 in JI, it's mirror inverse the left handed porcutone pentatonic, 10/9 5/4 3/2 5/3 2/1 in JI, which tempers to ssLsL in Meantone, but to sLLsL in Father. F-G-A-C-D gives a left handed porcutone pentatonic, while G-A-B-D-E, F#-G#-A#-C#-D#, and Gb-Ab-Bb-Db-Eb are right handed porcutone pentatonics.
+We are familiar with this scale as the just pentatonic. If we temper m and s together, we get Meantone[5]: ssLsL. If we temper m and L together instead we get a scale called Father[5], tempering out the diatonic semitone 16/15. This mode of Father[5] has step pattern LsLsL. Keep the connection to Father[5] in the back of your minds for now, we'll come back to it. The porcutone pentatonic is also a subset of the porcutone diatonic, since Meantone[5] is a subset of Meantone[7]. It is available as G-A-B-D-E. The scale F-G-A-C-D is a mode of the inverse of G-A-B-D-E. The porcutone pentatonic is [[Chirality|''chiral'']] (i.e., it is not symmetric, unlike the porcutone diatonic and procutone chromatic scales, which are ''achiral''). There is a pair of porcutone pentatonic scales, the right handed porcutone pentatonic, 9/8 5/4 3/2 5/3 2/1 in JI, it's mirror inverse the left handed porcutone pentatonic, 10/9 5/4 3/2 5/3 2/1 in JI, which tempers to ssLsL in Meantone, but to sLLsL in Father. F-G-A-C-D gives a left handed porcutone pentatonic, while G-A-B-D-E, F#-G#-A#-C#-D#, and Gb-Ab-Bb-Db-Eb are right handed porcutone pentatonics.
 Adding the right handed porcutone chromatic (on F♯/G♭) to the just porcutone chromatic, a 12-note mirror-symmetric scale with step signature and step mapping of 7L 1m 4s = (27/25, 25/24, 250/243) = (133.2376c, 70.6724c, 49.1661c), i.e., 7 large steps of what was the small step of the just porcutone diatonic, 1 medium step of the chromatic semitone 25/24, the distance between 6/5 and 5/4, and 4 small steps of 250/243, the porcupine comma, that separates 10/9 from 27/25. For the all sharps scale, we set mode -3 on D (for all flats we set mode 3 on D): 250/243 10/9 6/5 100/81 4/3 25/18 3/2 125/81 5/3 9/5 50/27 2/1, with step pattern sLLsLmLsLLsL.
@@ Line 379: / Line 379: @@
 The ptolemismic porcutone chromatic scale is distinctly xenharmonic, and yet is related to the familiar chromatic scale.
-Mode -3 has 3/2 perfect fifths available above D, D♯, E, F, F♯, G, and G♯.
+Mode -3 has 3/2 perfect fifths available above D, D♯, E, F, F♯, G, G♯, and C♯.
-Mode 3 has 3/2 perfect fifths available above D, E♭, E, F, G♭, G, and D♭.
+Mode 3 has 3/2 perfect fifths available above D, E♭, E, F, G♭, G, A♭, and D♭.
 Mode -3 has 4:5:6 major triads available above E, F, F♯, G, and G♯
@@ Line 409: / Line 409: @@
 |10:12:15
 :22:27
-|
+|320.690, 704.524
+.110, 704.524
 |-
 |D♯
@@ Line 422: / Line 424: @@
 |10:12:15
 :144:165
-|
+|320.690, 704.524
+.325, 704.524
 |-
 |E
@@ Line 435: / Line 439: @@
 |10:12:15
 :5:6
-|
+|320.690, 704.524
+.834, 704.524
 |-
 |F
@@ Line 448: / Line 454: @@
 |48:55:72
 :5:6
-|
+|237.198, 704.524
+.834, 704.524
 |-
 |F♯
@@ Line 461: / Line 469: @@
 |22:27:33
 :5:6
-|
+|356.414, 704.524
+.834, 704.524
 |-
 |G
@@ Line 468: / Line 478: @@
 |aug 2
 major
-|aug 2
+|maj 2
 major
-|aug 2
+|aug maj 2
 major
-|64:75:128
+|48:55:72
 :5:6
-|
+|237.198, 704.524
+.834, 704.524
 |-
 |G♯
@@ Line 487: / Line 499: @@
 |22:27:33
 :27:30
-|
+|356.414, 704.524
+.469, 704.524
 |-
 |A
@@ Line 500: / Line 514: @@
 |15:18:22
 :33:40
-|
+|320.690, 668.800
+.110, 668.800
 |-
 |A♯
@@ Line 513: / Line 529: @@
 |275:330:432
 :360:432
-|
+|320.690, 788.015
+.325, 788.015
 |-
 |B
@@ Line 526: / Line 544: @@
 |15:18:22
 :33:40
-|
+|320.690, 668.800
+.110, 668.800
 |-
 |C
@@ Line 539: / Line 559: @@
 |243:275:360
 :297:360
-|
+|176.055, 668.800
+.110, 668.800
 |-
 |C♯
@@ Line 552: / Line 574: @@
 |10:12:15
 :144:165
-|
+|320.690, 704.524
+.325, 704.524
 |}
 As with the porcutone diatonic, tuning the porcutone chromatic to 19edo or 31edo collapses it to the Meantone[12] (Meanenneadecal[12]) chromatic scale. Tuning it to 15edo, 22edo, or 29edo collapses it to Porcupine[8]. Step patterns, mappings and sizes for tunings to 27edo, 34edo, and 41edo are as follows: