Pinetone: Difference between revisions

The porcupine comma is the small step of the scale, so tempering the Pinetone chromatic scale to porcupine leads from 7L 1m 4s = (27/25, 25/24, 250/243) to 7L 1s = (10/9~27/25, 25/24~81/80), which is Porcupine[8]! The Porcupine[7] scale has its large step between G and A, so the eighth note of Porcupine[8] is either G♯ or A♭, adding another small step of Porcupine[7] below A (for G♯) or above G (A♭). Mode -3 or mode 3 of the Pinetone chromatic scale, respectively, are set to D so that this is preserved in The Pinetone System. This leads to the Pinetone octatonic scales: D E F G G♯/A♭ A B C. In Just intonation: 10/9 6/5 4/3 25/18 3/2 5/3 9/5 2/1 with G♯, or 10/9 6/5 4/3 36/25 3/2 5/3 9/5 2/1 with A♭. This scale has 4 large steps of 10/9, 3 medium steps of 27/25, and 1 small step of 25/24. It is not mirror-symmetric, or equivalentely, it is ''[[Chirality|chiral]]'' so it cannot be uniquely defined with a step signature like Meantone[7], Porcupine[7], Porcupine[8], Meantone[12], and the Pinetone diatonic (the Zarlino/Ptolemy just major scale is also not mirror symmetric). Scales that can be uniquely defined by a step signature are called ''step-nested scales''. More on that later. The mirror inverse of any mode of the Pinetone octatonic with G♯ is a mode of the Pinetone octatonic with A♭. The Pinetone octatonic with G♯ is called the left handed porcupine octatonic, and the Pinetone octatonic with A♭ is called the right handed porcupine octatonic (see [[chirality]]).  

|+Modes of the left handed just Pinetone octatonic

|+Modes of the right handed just Pinetone octatonic

Note that the darkest mode of the LH octatonic is the brightest mode of the RH octatonic, etc.

Tempering out 100/99, the large step (174.05488c) represents 10/9~11/10, the medium step (146.63528c) represents 27/25~12/11, and the small step (63.14327c) represents 25/24~33/32. The following tables display the JI intervals approximated by the modes of the ptolemismic Pinetone octatonic scales, along with the scale steps in cents. See [http://x31eq.com/cgi-bin/rt.cgi?ets=4p%263p%261ce&limit=2.3.5.11 TE tuning].

|+Modes of the left handed ptolemismic Pinetone octatonic

|+Modes of the right handed ptolemismic Pinetone octatonic

The following table gives all intervals of the Pinetone octatonic.

|+Intervals of the Pinetone octatonic

The following two tables detail the 3-step stacked triads of the left and right handed Pinetone octatonics:

|+3-step stacked triads of the left handed Pinetone octatonic (G♯-G gamut)

|+3-step stacked triads of the right handed Pinetone octatonic (G-A♭ gamut)

We could alternatively treat the Pinetone 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 left-handed Pinetone octatonic, the G♯ diminished tetrad tunes to 33:40:48:55 (when B is the bottom note). Using the A♭, as in the right-handed Pinetone octatonic, the B diminished tetrad also tunes to 33:40:48:55 (when D is the bottom note).

Unlike the Pinetone diatonic, and chromatic scales, the Pinetone octatonic is chiral, and is therefore not a step-nested scale. As we can see, it is more complex than the Pinetone diatonic. The Pinetone pentatonic and diatonic scales is also wakalix / PWF, and it can be seen that the Pinetone octatonic is more complex than the Pinetone pentatonic as well. It is left as an exercise for the reader to determine the complexity of the Pinetone chromatic, and compare that to the Pinetone octatonic.

=== Pinetone Diminished or Porcupine-Diminished ===

Take the Ultharian dark major mode of the left-handed Pinetone octatonic, for example: LMLsMLML. Raising the sixth and eighth degrees of the scale by the L-M chroma leads to the mode LMLsLMLM. Similarly, taking the Mnarian middle major mode LMLMsLML of the right-handed Pinetone octatonic and lowering the second and the fourth degree by the L-M chroma leads to the mode MLMLsLML. We can see that LMLsLMLM and MLMLsLML are modes of the same scale. We call these modes the dark major diminished and the middle major diminished respectively.

In the Pinetone chromatic with sharps, which contains the left-handed Pinetone octatonic as the naturals plus G♯, the Ultharian dark major mode can be expressed as D E F G G♯ A B C. The Pinetone diminished mode on D, the dark major diminished, is therefore D E F G G♯ A♯ B C♯. In the Pinetone chromatic with flats, which contains the right-handed Pinetone octatonic as the naturals plus A♭, the Mnarian middle major mode can be expressed as D E F G A♭ A B C. The Pinetone diminished mode on D, the middle major diminished, is therefore D E♭ F G♭ A♭ A B C.  

We know that this scale tempers to Porcupine[8]; tempering M=s instead leads to LsLsLsLs, i.e., Diminished[8]; and finally tempering L=s leads to LsLLLsLs, a mod of Father[8]. Like the Pinetone chromatic and diatonic scale, this scale is an SN scale, and is therefor achiral. We may name this scale perhaps the Porcupine-Diminished scale, or we may include it in the Pinetone system as the Pinetone diminished scale. It may be more wise to refer to this scale as the Porcupine-Diminished scale to avoid confusion that might result if, while the octatonic scale and the diminished scale are different names for the same scale, the Pinetone octatonic and the Pinetone diminished scale are not.

Every other step of any mode of the Pinetone Diminished scale gives an inversion of the 5-limit diminished tetrad; therefore every second step of the Pinetone Diminished scales only comes in two different sizes, as opposed to the four different sizes of every second step of the Pinetone octatonic. Although the Pinetone Diminished is simpler in this way, the Pinetone octatonic provides more major and minor triads.

|+Modes of the just Pinetone Diminished scale (Porcupine-Diminished)

|+Modes of the Ptolemismic Pinetone Diminished scale (Porcupine-Diminished)

|+Intervals of the Pinetone diminished (Porcupine-Diminished)

|+3-step stacked triads of the Pinetone diminished (Porcupine-Diminished)

==== Pinetone Diminished chromatic ====

We can extend the Pinetone Diminished into an alterative chromatic scale: Starting with the bright minor diminished scale, MLsLMLML, we add a small step into the bottom or top of every large step, leading to the scales LsMssMLsMLsM and LMssMsLMsLMs respectively, modes of mirror-inversions of one another. In 5-limit just intonation this pair of scales comprises 3 large steps of 27/25, 4 medium steps of 16/15, and 5 small steps of 25/24, i.e., 27/25 9/8 6/5 5/4 125/96 25/18 3/2 25/16 5/3 9/5 15/8 2/1 and 27/25 144/125 6/5 5/4 4/3 25/18 3/2 8/5 5/3 9/5 48/25 2/1 respectively. In other modes, they can be expressed as 25/24 10/9 6/5 5/4 4/3 36/25 3/2 8/5 5/3 125/72 50/27 2/1, and 25/24 10/9 125/108 5/4 4/3 4/3 25/18 3/2 8/5 5/3 9/5 48/25 2/1, i.e., sMLsMLsMssML and sMsLMsLMsLMs respectively.

We could call these scales, which temper to Diminished[12], the left and right-handed Pinetone Diminished chromatic.

Additionally, we have another set of [[Porcupine]][7] modes contained in the Pinetone octatonic: Replacing the G with the G♯ changes the mode of the Porcupine[7] scale represented, and replaces diatonic with harmonic minor modes for the [[Meantone]][7] scale represented, now a MODMOS.  

Starting instead with the Pinetone diminished scale: MLsLMLML, shown in the bright minor mode as Pinetone bright minor diminished. Putting a small step into the bottom of each medium and large step leads to the child SNS of the Pinetone diminished scale: the fifteen note SNS msmLmmLmsmLmsmL, or mLmsmLmsmLmsmLm in it's symmetric mode, comprising 4 large steps of 16/15, 8 medium steps of 25/24 and 3 small steps of 648/625, i.e.,