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→The porcutone octatonic: work in progress
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The porcupine comma is the small step of the scale, so tempering the porcutone 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 porcutone chromatic scale, respectively, are set to D so that this is preserved in The Porcutone System. This leads to the porcutone 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 porcutone diatonic (the Zarlio/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 porcutone octatonic with G♯ is a mode of the porcutone octatonic with A♭. The porcutone octatonic with G♯ is called the left handed porcupine octatonic, and the porcutone octatonic with A♭ is called the right handed porcupine octatonic (see [[chirality]]).  
The porcupine comma is the small step of the scale, so tempering the porcutone 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 porcutone chromatic scale, respectively, are set to D so that this is preserved in The Porcutone System. This leads to the porcutone 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 porcutone diatonic (the Zarlio/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 porcutone octatonic with G♯ is a mode of the porcutone octatonic with A♭. The porcutone octatonic with G♯ is called the left handed porcupine octatonic, and the porcutone octatonic with A♭ is called the right handed porcupine octatonic (see [[chirality]]).  


If we temper out the difference between the large and medium steps, we reduce the scale to Porcupine[8]. We get Father[8] 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]. For out modes of the left handed and right handed porcupine octatonic scales we prefix a set of functional mode names for Porcupine[8], introduced in the table below, with the [[oneirotonic]] mode names associated with Father[8].
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. The table below introduces a set of functional mode names for Porcupine[8]. Along with the step pattern and mode number, the modes' ''[[UDP]]'' is show in the table. The UDP show 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.
{| class="wikitable"
|+Porcupine[8] functional mode names
!Mode number
!Step pattern
!UDP
!Mode name
|-
|4
|LLLLLLLs
|<nowiki>7|0</nowiki>
|Bright quartal
|-
|3
|LLLLLLsL
|<nowiki>6|1</nowiki>
|Dark quartal
|-
|2
|LLLLLsLL
|<nowiki>5|2</nowiki>
|Bright major
|-
|1
|LLLLsLLL
|<nowiki>4|3</nowiki>
|Middle major
|-
| -1
|LLLsLLLL
|<nowiki>3|4</nowiki>
|Dark major
|-
| -2
|LLsLLLLL
|<nowiki>2|5</nowiki>
|Bright minor
|-
| -3
|LsLLLLLL
|<nowiki>1|6</nowiki>
|Middle minor
|-
| -4
|sLLLLLLL
|<nowiki>0|7</nowiki>
|Dark minor
|}
 
 
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.
{| class="wikitable"
|+Father[8] oneirotonic mode names
!Mode number
!Step pattern
!UDP
!Mode name
|-
|4
|LLsLLsLs
|<nowiki>7|0</nowiki>
|Dylathian (də-LA(H)TH-iən)
|-
|3
|LLsLsLLs
|<nowiki>6|1</nowiki>
|Illarnekian (ill-ar-NEK-iən)
|-
|2
|LsLLsLLs
|<nowiki>5|2</nowiki>
|Celephaïsian (kel-ə-FAY-zhən)
|-
|1
|LsLLsLsL
|<nowiki>4|3</nowiki>
|Ultharian (ul-THA(I)R-iən)
|-
| -1
|LsLsLLsL
|<nowiki>3|4</nowiki>
|Mnarian (mə-NA(I)R-iən)
|-
| -2
|sLLsLLsL
|<nowiki>2|5</nowiki>
|Kadathian (kə-DA(H)TH-iən)
|-
| -3
|sLLsLsLL
|<nowiki>1|6</nowiki>
|Hlanithian (lə-NITH-iən)
|-
| -4
|sLsLLsLL
|<nowiki>0|7</nowiki>
|Sarnathian (sar-NA(H)TH-iən), can be shortened to "Sarn"
|}
For our modes of the left handed and right handed porcupine octatonic scales we prefix the functional mode names for Porcupine[8], with the [[oneirotonic]] mode names associated with Father[8]. Like in the tables of modes of the porcutone diatonic, the modes are listed in order of brightest, with the brightest mode at the top, and the darkest mode at the bottom.
{| class="wikitable"
|+Left handed just Porcutone octatonic
!mode in JI
!step pattern
!Porcupine[8]
step pattern and UDP
!Porcupine[8]
mode
!Father[8]
step pattern and UDP
!Father[8]
mode
!Porcutone octatonic
mode
|-
|10/9 6/5 4/3 40/27 8/5 16/9 50/27 2/1
|LMLLMLsM
|<nowiki>LLLLLLsL 6|1</nowiki>
|Dark quartal
|<nowiki>LsLLsLLs 5|2</nowiki>
|Celephaïsian
|Celephaïsian dark quartal
|-
|27/25 6/5 162/125 36/25 8/5 216/125 48/25 2/1
|MLMLLMLs
|<nowiki>LLLLLLLs 7|0</nowiki>
|Bright quartal
|<nowiki>sLsLLsLL 0|7</nowiki>
|Sarnathian
|Sarnathian bright quartal
|-
|10/9 100/81 4/3 40/27 125/81 5/3 50/27 2/1
|LLMLsMLM
|<nowiki>LLLLsLLL 4|3</nowiki>
|Middle major
|<nowiki>LLsLLsLs 7|0</nowiki>
|Dylathian
|Dylathian middle major
|-
|27/25 6/5 4/3 36/25 8/5 5/3 9/5 2/1
|MLLMLsML
|<nowiki>LLLLLsLL 5|2</nowiki>
|Bright major
|<nowiki>sLLsLLsL 2|5</nowiki>
|Kadathian
|Kadathian bright major
|-
|10/9 6/5 4/3 25/18 3/2 5/3 9/5 2/1
|LMLsMLML
|<nowiki>LLLsLLLL 3|4</nowiki>
|Dark major
|<nowiki>LsLLsLsL 4|3</nowiki>
|Ultharian
|Ultharian dark major
|-
|10/9 125/108 5/4 25/18 3/2 5/3 50/27 2/1
|LsMLMLLM
|<nowiki>LsLLLLLL 1|6</nowiki>
|Middle minor
|<nowiki>LLsLsLLs 6|1</nowiki>
|Illarnekian
|Illarnekian middle minor
|-
|27/25 6/5 5/4 27/20 3/2 81/50 9/5 2/1
|MLsMLMLL
|<nowiki>LLsLLLLL 2|5</nowiki>
|Bright minor
|<nowiki>sLLsLsLL 1|6</nowiki>
|Hlanithian
|Hlanithian bright minor
|-
|25/24 9/8 5/4 27/20 3/2 5/3 9/5 2/1
|sMLMLLML
|<nowiki>sLLLLLLL 0|7</nowiki>
|Dark minor
|<nowiki>LsLsLLsL 3|4</nowiki>
|Mnarian
|Mnarian dark minor
|}
{| class="wikitable"
|+Right handed just Porcutone octatonic
!mode in JI
!step pattern
!Porcupine[8]
step pattern and UDP
!Porcupine[8]
mode
!Father[8]
step pattern and UDP
!Father[8]
mode
!Porcutone octatonic
mode
|-
|10/9 6/5 4/3 40/27 8/5 16/9 50/27 2/1
|LMLLMLMs
|<nowiki>LLLLLLLs 7|0</nowiki>
|Bright quartal
|<nowiki>LsLLsLsL 4|3</nowiki>
|Ultharian
|Ultharian bright quartal
|-
|27/25 6/5 162/125 36/25 8/5 216/125 48/25 2/1
|LLMLMsLM
|<nowiki>LLLLLsLL 5|2</nowiki>
|Bright major
|<nowiki>LLsLsLLs 6|1</nowiki>
|Illarnekian
|Illarnekian bright major
|-
|10/9 100/81 4/3 40/27 125/81 5/3 50/27 2/1
|MLLMLMsL
|<nowiki>LLLLLLsL 6|1</nowiki>
|Dark quartal
|<nowiki>sLLsLsLL 1|6</nowiki>
|Hlanithian
|Hlanithian dark quartal
|-
|27/25 6/5 4/3 36/25 8/5 5/3 9/5 2/1
|LMLMsLML
|<nowiki>LLLLsLLL 4|3</nowiki>
|Middle major
|<nowiki>LsLsLLsL 3|4</nowiki>
|Mnarian
|Mnarian middle major
|-
|10/9 6/5 4/3 25/18 3/2 5/3 9/5 2/1
|LMsLMLLM
|<nowiki>LLsLLLLL 2|5</nowiki>
|Bright minor
|<nowiki>LsLLsLLs 5|2</nowiki>
|Celephaïsian
|Celephaïsian bright minor
|-
|10/9 125/108 5/4 25/18 3/2 5/3 50/27 2/1
|MLMsLMLL
|<nowiki>LLLsLLLL 3|4</nowiki>
|Middle minor
|<nowiki>sLsLLsLL 0|7</nowiki>
|Sarnathian
|Sarnathian dark major
|-
|27/25 6/5 5/4 27/20 3/2 81/50 9/5 2/1
|sLMLLMLM
|<nowiki>sLLLLLLL 0|7</nowiki>
|Bright minor
|<nowiki>LLsLLsLs 7|0</nowiki>
|Dylathian
|Dylathian dark minor
|-
|25/24 9/8 5/4 27/20 3/2 5/3 9/5 2/1
|MsLMLLML
|<nowiki>LsLLLLLL 1|6</nowiki>
|Dark minor
|<nowiki>sLLsLLsL 2|5</nowiki>
|Kadathian
|Kadathian middle minor
|}


== Summary for xen-math nerds ==
== Summary for xen-math nerds ==
Line 462: Line 718:
Porcutone Chromatic (Gb-B): 146.635 174.055 320.69 467.325 494.745 641.38 704.524 851.159 878.579 1025.214 1171.849 1199.269  
Porcutone Chromatic (Gb-B): 146.635 174.055 320.69 467.325 494.745 641.38 704.524 851.159 878.579 1025.214 1171.849 1199.269  


* Mode 4b:  LMLLMLMs -> Porcupine[8]: LLLLLLLs 7|0, Father[8]: LsLLsLsL 4|3 -> Ultharian bright quartal
*  
* Mode 3b:  LLMLMsLM -> Porcupine[8]: LLLLLsLL 5|2, Father[8]: LLsLsLLs 6|1 -> Illarnekian bright major
* Mode 2b:  MLLMLMsL -> Porcupine[8]: LLLLLLsL 6|1, Father[8]: sLLsLsLL 1|6 -> Hlanithian dark quartal
* Mode 1b:  LMLMsLML -> Porcupine[8]: LLLLsLLL 4|3, Father[8]: LsLsLLsL 2|5 -> Kadathian middle major
* Mode -1b: LMsLMLLM -> Porcupine[8]: LLsLLLLL 2|5, Father[8]: LsLLsLLs 5|2 -> Celephaïsian bright minor
* Mode -2b: MLMsLMLL -> Porcupine[8]: LLLsLLLL 3|4, Father[8]: sLsLLsLL 0|7 -> Sarnathian dark major
* Mode -3b: LsMLLMLM -> Porcupine[8]: sLLLLLLL 0|7, Father[8]: LLsLLsLs 7|0 -> Dylathian dark minor
* Mode -4b: MsLMLLML -> Porcupine[8]: LsLLLLLL 1|6, Father[8]: sLLsLLsL 3|4 -> Mnarian middle minor


== Porcutone harmonic minor and harmonic major ==
== Porcutone harmonic minor and harmonic major ==
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