Pinetone: Difference between revisions

Something to note - the [[Meantone]] diatonic scale is ''generated'' by the perfect fifth, [[3/2]], which means that it can be formed by stacking perfect fifths on top of each other, i.e., F-C-G-D-A-E, and all the notes are connected by perfect fifths. [[Porcupine]][7], on the other hand, is generated by [[10/9]], so all notes are connected by a chain of 10/9s, i.e., A-B-C-D-E-F-G, where the large step of [[9/8]] then separates G from A. The [[Zarlino]]/Ptolemy just major scale 9/8 5/4 4/3 3/2 5/3 15/8 2/1 can be built of two parallel chains of [[3/2]], i.e., 4/3-2/1-3/2-9/8, 5/3-5/4-15/8. Accordingly it is a ''[[Generator-offset property|generator-offset]]'' scale. If the scale is on C, then D-A is not a [[3/2]] perfect fifth, but a wolf fifth of [[40/27]]. The Pinetone diatonic is not a [[generator-offset]] scale. Setting the scale to the naturals, D E F G A B C D, [[3/2]] perfect fifths are available above D, E, F, and C, so there are 1 fewer [[3/2]] perfect fifths in the Pinetone diatonic scale than in the [[Zarlino]]/Ptolemy just major scale, and two fewer than in the typical diatonic scale. [[Porcupine]][7] also has [[3/2]] fifths only above D, E, F, and G. It is because [[3/2]] perfect fifths are available above D, E, F, and G in both [[Meantone]][7] and [[Porcupine]][7] that they are available above D, E, F, and G in the Pinetone diatonic.   

{| class="wikitable"

!Mode number

!Pinetone diatonic mode

=== Intervals and chords ===

{| class="wikitable"

!Interval class

!size

|}

{| class="wikitable"

!Root note

!Triad notes

Porcupine tetrads in the table below are named after the third above the tonic and the third above the fifth, apart from tetrads with a diminished fifth.

{| class="wikitable"

!Root note

!Triad notes

Also of interest are the quartal triads of the Pinetone diatonic. We describe these as stacked 3-step intervals (fourths) of the scale, with major and minor designating the large and small 3-step intervals (fourths) respectively. This may seem an odd way to describe quartal chords, but it is consistent with the naming scheme I introduce for the Porcupine[7] 3-step (quartal) triads along side the quartal triads of the Pinetone diatonic.

{| class="wikitable"

!Root note

!Triad notes

We can see that compared to tertian (2-step stacked) chords, quartal (3-step stacked) chords are much more evenly spread in the Pinetone diatonic. 3-step pentads ala the quartal voicings of modern jazz harmony may also be of interest. Two octaves are spanned by four 3-step intervals and a 2-step remainder. Given that Pinetone cannot be generated by 4/3 perfect fourths like Meantone can these pentads have much more variety in Pinetone than in Meantone. As in modern jazz, variety is found through utilizing all inversions of the pentad. The root position and 2nd inversion are shown, where the second inversion spans an octave with a symmetric pattern of two 3-step intervals on either side of a 2-step interval. Root position pentads on A and B (2nd inversion on G and A) cover the notes of the [[The Pinetone System#Pinetone pentatonic|Pinetone major and minor pentatonics]] respectively, introduced below.  

{| class="wikitable"

!Root note

!Notes

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{| class="wikitable"

!Root note

!Notes

The now familiar Meantone comma of 81/80 separates the medium step (25/24) from the small step (250/243), so our Pinetone chromatic is a ''detempering'' of Meantone[12], the meantone chromatic scale, just like how the Pinetone diatonic is a detempering of Meantone[7], the meantone diatonic scale.  

Mode -3 approximates the JI ratios: 55/54 10/9 6/5 11/9 4/3 11/8 3/2 55/36 5/3 9/5 11/6 2/1.  

The following tables show the (3, 4) and (4, 3) triads available of mode 3 and mode -3 of the Pinetone chromatic scale:

{| class="wikitable"

! Note

!Triad class

41edo with 1200.2039c octave: 7L 1m 4s = (5, 2, 1) = (146.3663c, 58.5465c, 29.2733c)  

[http://x31eq.com/cgi-bin/rt.cgi?ets=7%261ce%264f&limit=2.3.5.11.13 7L 1m 4s = (27/25~12/11~13/12, 25/24~33/32~27/26, 250/243~55/54~121/120~40/39) = (142.77537c, 66.76626c, 33.11646c)],   

[http://x31eq.com/cgi-bin/rt.cgi?ets=7d%261cdde%264f&limit=13 7L 1m 4s = (27/25~15/14~12/11~13/12, 25/24~21/20~33/32~27/26, 250/243~28/27~55/54~121/120~40/39) = (136.27690c, 81.02531c, 40.63434c)],  

[http://x31eq.com/cgi-bin/rt.cgi?ets=7%261ce%264p&limit=2.3.5.11 7L 1m 4s = (27/25~12/11, 25/24~33/32, 250/243~55/54~121/120) = (146.63528c, 63.14327c, 27.41960c)].  

Or we might tune to TE 2.3.5.11.13 Tetracot for  

== Pinetone octatonic scales==

Note that the darkest mode of the major-harmonic octatonic is the mirror-inverse of the brightest mode of the minor-harmonic octatonic, etc.

Tempering out 144/143 as well, the large step is tuned to 175.89183c TE, medium step (142.77537c TE) also represents 13/12, and the small step (66.76626c TE) also represents 27/26. See [http://x31eq.com/cgi-bin/rt.cgi?ets=4f%263f%261ce&limit=2.3.5.11.13 TE tuning].

{| class="wikitable"

!Mode (height order)

!Step pattern

|}

{| class="wikitable"

!Mode (height order)

!Step pattern

The following table gives all intervals of the Pinetone harmonic octatonics.

{| class="wikitable"

!Interval class

!sizes

|1

|}

The following two tables detail the 3-step stacked triads of the Pinetone harmonic octatonics:

{| class="wikitable"

!Mode (rotational order)

!Step pattern

|12:15:20

|}

{| class="wikitable"

!Mode (rotational order)

! Step pattern

|}

We could alternatively treat either Pinetone harmonic octatonic as a bebop scale, using 2-step stacked tetrads. Since the scale has 8 notes, there are only 2 different 2-step stacked tetrads. In 12edo these are the major add 6 and the fully diminished tetrads. The meantone C major add 6 tunes to 45:55:66:75 in Pinetone. Using the G♯, as in the Pinetone major-harmonic octatonic, the G♯ diminished tetrad tunes to 33:40:48:55 (when B is the bottom note). Using the A♭, as in the Pinetone minor-harmonic octatonic, the B diminished tetrad also tunes to 33:40:48:55 (when D is the bottom note).

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{| class="wikitable"

!Interval class

!sizes

|1

|}

{| class="wikitable"

!Mode (rotational order)

!Step pattern

| 12:15:20

|}

The following 13 notes are used in total for these scales: E♭, G♭, A♭ D, E, F, G, A, B, C, G♯, A♯, C♯  

Tempering out 100/99 and 144/143 leads to the simplest pre-image: 25/24 10/9 15/13 6/5 5/4 4/3 11/8 13/9 3/2 8/5 5/3 26/15 9/5 48/25 2/1.

4 large steps of 109.12557c, approximating 16/15;  

8 small steps of 66.76626c, approximating 25/24, 33/32, and 27/26.

66.766 175.892 242.658 318.667 385.433 494.559 561.325 637.334 704.101 813.226 879.993 956.002 1022.768 1131.893 1198.660 as sLsmsLsmsLsmsLs.

Accordingly Pinetone-15 would temper to two step sizes in 19edo (Hanson), 22edo (Porcupine), 34edo (Hanson), and 27edo (Augmented). If we wish to keep the 3-step size structure, we can tune to 26edo or 41edo with (L, m, s) = (3, 2, 1), and (4, 3, 2) respectively.

2.3.5.13 325/324 may be better tuned to 46edo, with (L, m, s) = (4, 2, 1).

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{| class="wikitable"

!Mode (height order)

!Step pattern

|}

{| class="wikitable"

!Interval class

!sizes

!Occurence

|-

|s

|minor step

|minor step

|minor step

|25/24, 33/32, 27/26

|66.766

|2

|-

|L + s

|perfect 2-step

|minor 2-step

|minor 2-step

|15/13 (7/6 or 8/7)

|242.658

| 2

|-

| L + M + s = A + 2s

|minor 3-step

|minor 3-step

|minor 3-step

|5/4

|385.433

|3

|-

|2L + M + s = A + L + 2s

| perfect 4-step

|minor 4-step

|minor 4-step

|25/18, 11/8, 18/13

|561.325

|4

|-

|minor 5-step

|minor 5-step

|minor 5-step

|3/2

|704.101

|4

|-

|3L + 2M + s = A + 2L + M + 2s

|perfect 6-step

|minor 6-step

|minor 6-step

|5/3

|879.992

|6

|-

|minor 7-step

|diminished 7-step

|diminished 7-step

|125/72, 55/32, 45/26

|946.759

|1

{| class="wikitable"

!Mode (rotational order)

!Step pattern

|12:15:20

|}

==Comma pump==

D-F-A-C -> F-A-C-E -> E-G-B-D -> D-F-A-C