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→Other diatonic subsets of the Pinetone chromatic scale: added last table, updated scale names
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|-
|-
|2
|2
|Mixolydian bright minor*
|Mixolydian bright major*
|G
|G
|Lmsmmsm
|Lmsmmsm
Line 1,345: Line 1,345:
7L 1m 4s = (27/25~12/11~13/12, 25/24~33/32~27/26, 81/80~250/243~55/54~121/120~40/39) = (142.6653, 66.6782, 33.3391), which we note is very similar to 2.3.5.11.13 Ptolemismic.  
7L 1m 4s = (27/25~12/11~13/12, 25/24~33/32~27/26, 81/80~250/243~55/54~121/120~40/39) = (142.6653, 66.6782, 33.3391), which we note is very similar to 2.3.5.11.13 Ptolemismic.  


== Other diatonic subsets of the Pinetone chromatic scale ==
== Pinetone Aeolian bright and dark minor, Dorian bright minor, and Locrian dark minor scales and their modes ==
As well as our natural keys, diatonic scales are available in the Pinetone chromatic in 5 other keys (not 11, since we do not temper out the [[Pythagorean comma]]), totally 5 scales that differ in their interval content.  
As well as our natural keys, diatonic scales are available in the Pinetone chromatic in 5 other keys (not 11, since we do not temper out the [[Pythagorean comma]]), totally 5 scales that differ in their interval content.  
{| class="wikitable"
{| class="wikitable"
Line 1,353: Line 1,353:
!JI ratios approximated
!JI ratios approximated
!Step pattern
!Step pattern
!Symmetric minor mode
!Dark minor mode
|-
|-
|D
|D
Line 1,433: Line 1,433:
|12/11 6/5 4/3 3/2 18/11 9/5 2/1
|12/11 6/5 4/3 3/2 18/11 9/5 2/1
|smmLsmm
|smmLsmm
|Phrygian
|Locrian
|-
|-
|C♯
|C♯
Line 1,481: Line 1,481:
To calculate the mode numbers for Tables 4.2-4.5, the mode numbers of their temperings to Porcupine and to Meantone were added, ordered, and renumbered. When two modes are tied, small changes in tuning will affect their order. Given that Ptolemismic Porcupine is more accurate than Ptolemismic Meantone, the Porcupine mode number is weighted more heavily to determine the mode order in the event of any ties.
To calculate the mode numbers for Tables 4.2-4.5, the mode numbers of their temperings to Porcupine and to Meantone were added, ordered, and renumbered. When two modes are tied, small changes in tuning will affect their order. Given that Ptolemismic Porcupine is more accurate than Ptolemismic Meantone, the Porcupine mode number is weighted more heavily to determine the mode order in the event of any ties.
{| class="wikitable"
{| class="wikitable"
|+Table 4.2. Modes of the Ptolemismic Pinetone Mixolydian symmetric minor
|+Table 4.2. Modes of the Ptolemismic Pinetone Aeolian bright minor
!Mode number
!Mode number
!Pinetone diatonic mode
!Pinetone diatonic mode
Line 1,503: Line 1,503:
|-
|-
|1
|1
|Mixolydian symmetric minor
|Mixolydian dark minor
|D
|D
|E♭  
|E♭  
Line 1,539: Line 1,539:
Similarly, lowering B to B♭ lowers by ~55/54, leading to the Aeolian symmetric minor shown in Table 4.2, which I consider to be a really beautiful minor minor mode.
Similarly, lowering B to B♭ lowers by ~55/54, leading to the Aeolian symmetric minor shown in Table 4.2, which I consider to be a really beautiful minor minor mode.
{| class="wikitable"
{| class="wikitable"
|+Table 4.3. Modes of the Ptolemismic Pinetone Aeolian symmetric minor
|+Table 4.3. Modes of the Ptolemismic Pinetone Aeolian dark minor
!Mode number
!Mode number
!Pinetone diatonic mode
!Pinetone diatonic mode
Line 1,575: Line 1,575:
|-
|-
| -1
| -1
|Aeolian symmetric minor*
|Aeolian dark minor*
|D
|D
|C♯
|C♯
Line 1,597: Line 1,597:
Note that the Pinetone melodic major includes a neutral triad 18:22:27 on the root of the Mixolydian symmetric minor, and the Pinetone melodic minor includes it's inverse on the root of the Dorian bright major, a flavour of triad not available in the Pinetone diatonic. Note also though the only one major and one minor triad are available in each of these scales, as opposed to the two each available in the Pinetone diatonic, hence the labelling of these scales as for melodic purposes.  
Note that the Pinetone melodic major includes a neutral triad 18:22:27 on the root of the Mixolydian symmetric minor, and the Pinetone melodic minor includes it's inverse on the root of the Dorian bright major, a flavour of triad not available in the Pinetone diatonic. Note also though the only one major and one minor triad are available in each of these scales, as opposed to the two each available in the Pinetone diatonic, hence the labelling of these scales as for melodic purposes.  
{| class="wikitable"
{| class="wikitable"
|+Table 4.4. Modes of the Ptolemismic Pinetone Ionian symmetric minor
|+Table 4.4. Modes of the Ptolemismic Pinetone Dorian bright minor
!Mode number
!Mode number
!Pinetone diatonic mode
!Pinetone diatonic mode
Line 1,612: Line 1,612:
|-
|-
|2
|2
|Ionian symmetric minor
|Ionian dark minor
|D  
|D  
|E♭
|E♭
Line 1,653: Line 1,653:
|~ 10/9 6/5 4/3 22/15 8/5 9/5 2/1
|~ 10/9 6/5 4/3 22/15 8/5 9/5 2/1
|}
|}
==Pinetone octatonic scales==
{| class="wikitable"
|+Table 4.5. Modes of the Ptolemismic Pinetone Locrian dark minor
!Mode number
!Pinetone diatonic mode
! colspan="2" |Root note
!Step pattern
!Mode as simplest JI pre-image
|-
|3
|Mixolydian dark major*
|F
|E
|mLsmmsm
|~ 10/9 5/4 15/11 3/2 5/3 9/5 2/1
|-
| 2
|Lydian bright minor
|E♭
|D
|mmLsmms
|~ 10/9 11/9 11/8 3/2 11/8 9/5 2/1
|-
|1
|Aeolian bright major
|G
|F♯
|Lsmmsmm
|~ 9/8 27/22 15/11 3/2 18/11 9/5 2/1
|-
|0
|Ionian dark diminished
|B♭
|A
|mmsmmLs
|~ 10/9 11/9 4/3 22/15 44/27 11/6 2/1
|-
| -1
|Dorian bright diminished
|C
|B
|msmmLsm
|~ 10/9 6/5 4/3 22/15 5/3 9/5 2/1
|-
| -2
|Locrian dark minor*
|D
|C♯
|smmLsmm
|~ 12/11 6/5 4/3 3/2 18/11 9/5 2/1
|-
| -3
|Phrygian magical seventh
|A
|G♯
|smmsmmL
|~ 12/11 6/5 4/3 16/11 8/5 16/9 2/1
|}
None of these other diatonic scales are useful as a basis for tonal harmony, since for all of these scales the only two major or minor chords are part of the same minor 7 chord. Instead they may be employed for melodic reasons, or as alternate modes to use with the minor 7 chord.
 
Note that the Dorian bright minor and Locrian dark minors are tetrachordal, along with the Dorian dark minor a.k.a. the Pinetone diatonic.
 
== Pinetone octatonic scales ==
The Porcupine comma is the small step of the Pinetone chromatic, 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 Pinetone. 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 Pinetone octatonic with G♯ is called the Pinetone major-harmonic octatonic, and the Pinetone octatonic with A♭ is called the Pinetone minor-harmonic octatonic. These names will make sense to the reader after further reading on these scales and the chords they contain, and on the Pintone diminished octatonic introduced below. The mirror inverse of any mode of the Pinetone major-harmonic octatonic is a mode of the Pinetone minor-harmonic octatonic (see [[chirality]]). This is true similarly of the familiar harmonic minor and harmonic major scales.  
The Porcupine comma is the small step of the Pinetone chromatic, 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 Pinetone. 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 Pinetone octatonic with G♯ is called the Pinetone major-harmonic octatonic, and the Pinetone octatonic with A♭ is called the Pinetone minor-harmonic octatonic. These names will make sense to the reader after further reading on these scales and the chords they contain, and on the Pintone diminished octatonic introduced below. The mirror inverse of any mode of the Pinetone major-harmonic octatonic is a mode of the Pinetone minor-harmonic octatonic (see [[chirality]]). This is true similarly of the familiar harmonic minor and harmonic major scales.  


Line 1,660: Line 1,721:
If we temper out the difference between the large and medium steps, we reduce the scale to Porcupine[8]. As we discussed above, Porcupine is generated by the interval 10/9~27/25. The table below introduces a set of functional mode names for Porcupine[8]. Along with the step pattern and mode number, the modes' [[UDP]] are show in the table. A mode's UDP shows the number of generators in the direction the brighten the intervals of scale, followed the number of generators in the direction that darkens it, (followed by the number of periods per octave, if it is not one. In this case the scale repeats at the octave, so P = 1, and is not shown). Instead of building chords by stacking thirds (2-step intervals), in octatonic scales we can build major and minor triads by stacking 3-step intervals! Instead of diminished, we get modes with two large fourths making a quartal chord: Accordingly we call these modes 'quartal'. When we stack 3-step intervals of 8-note scales out minor triads come in first inversion, and our major triads come in second inversion, as the 3-step intervals of octatonic scales include 5/4 and 4/3. Hence the brightest modes are quartal, and the darkest are minor. The eighth note of Porcupine[8] is typically called 'H', and is equivalent to the note A♭ of Porcupine[7], but we will show the modes for G# as the eighth note as well, since we may use G# in our Pinetone chromatic and octatonic scales.
If we temper out the difference between the large and medium steps, we reduce the scale to Porcupine[8]. As we discussed above, Porcupine is generated by the interval 10/9~27/25. The table below introduces a set of functional mode names for Porcupine[8]. Along with the step pattern and mode number, the modes' [[UDP]] are show in the table. A mode's UDP shows the number of generators in the direction the brighten the intervals of scale, followed the number of generators in the direction that darkens it, (followed by the number of periods per octave, if it is not one. In this case the scale repeats at the octave, so P = 1, and is not shown). Instead of building chords by stacking thirds (2-step intervals), in octatonic scales we can build major and minor triads by stacking 3-step intervals! Instead of diminished, we get modes with two large fourths making a quartal chord: Accordingly we call these modes 'quartal'. When we stack 3-step intervals of 8-note scales out minor triads come in first inversion, and our major triads come in second inversion, as the 3-step intervals of octatonic scales include 5/4 and 4/3. Hence the brightest modes are quartal, and the darkest are minor. The eighth note of Porcupine[8] is typically called 'H', and is equivalent to the note A♭ of Porcupine[7], but we will show the modes for G# as the eighth note as well, since we may use G# in our Pinetone chromatic and octatonic scales.


The step signature and mapping of 5-limit Porcupine[8] is 7L 1s = (10/9~27/25, 25/24~81/80)
The step signature and mapping of 5-limit Porcupine[8] is 7L 1s = (10/9~27/25, 25/24~81/80).
{| class="wikitable"
{| class="wikitable"
|+Table 5.1. Porcupine[8] modes (G♯-G gamut)
|+Table 5.1. Porcupine[8] modes (G♯-G gamut)
Line 1,758: Line 1,819:
We get Father[8], instead, if we temper out the difference (16/15) between the large step and the small step. Recall that the porcupine pentatonic reduces to Father[5], a subset of Father[8]. Father scales are generated by an interval representing both 5/4 and 4/3 (the 3-step interval of 8-note scales). The modes of Father[8] have names in use already, as an [[oneirotonic]]. These are shown in the table below with the mode number, step patter, and UDP.  
We get Father[8], instead, if we temper out the difference (16/15) between the large step and the small step. Recall that the porcupine pentatonic reduces to Father[5], a subset of Father[8]. Father scales are generated by an interval representing both 5/4 and 4/3 (the 3-step interval of 8-note scales). The modes of Father[8] have names in use already, as an [[oneirotonic]]. These are shown in the table below with the mode number, step patter, and UDP.  


The step signature and mapping of 5-limit Father[8] is 5L 3s = (10/9~25/24~32/27, 27/25~81/80),
The step signature and mapping of 5-limit Father[8] is 5L 3s = (10/9~25/24~32/27, 27/25~81/80).
{| class="wikitable"
{| class="wikitable"
|+Table 5.2. Father[8] oneirotonic modes
|+Table 5.2. Father[8] oneirotonic modes
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