Pinetone: Difference between revisions

'''Definition: Pinetone is a system of related rank-3 microtonal scales: pentatonic, diatonic, octatonic, chromatic, and hyperchromatic.'''

* [[Pinetone#Pinetone hyperchromatic scales|Hyperchromatic]]: 1313113123113131131 and 1311313113213113131, 1414114124114141141 and 1411414114214114141

The diatonic scale has a [[Step pattern|step signature]] of [[5L 2s]], meaning it has 5 large steps and 2 small step arranged in the [[step pattern]] LsLLLsL (represented in mode 0, Dorian mode). In [[Meantone]][7], the large step represents both [[9/8]] and [[10/9]], the major and minor tones (''tempering out'' the [[81/80]] interval that separates them) hence the name "Meantone". The small step represents [[16/15]] and [[27/25]] (which differ again by [[81/80]]). We write this in the form of a [[Step pattern|step signature]] and ''step mapping'' as [[5L 2s]] = (9/8~10/9, 16/15~27/25). [[Porcupine]][7] instead has a [[Step pattern|step signature]] and step mapping [[1L 6s]] = (~9/8, 10/9~27/25), hence the difference between [[10/9]] and [[27/25]], i.e., [[250/243]], is tempered out. In mode 0 it has [[step pattern]] sssLsss. [[81/80]] is called the [[Meantone comma]], and [[250/243]] is called the [[Porcupine]] comma.

In Tables 1.-4., Modes marked with '*' have a consonant triad on their root.

{| class="wikitable"

!Mode number

!Mode as simplest JI pre-image

|}

{| class="wikitable"

!Mode number

!Mode as simplest JI pre-image

|}

{| class="wikitable"

!Mode number

!Mode in JI

The modes of the Ptolemismic Pinetone diatonic are shown below in their simplest JI pre-image (the simplest JI ratios each interval above the tonic represents), and in cents, in an optimized tuning called [[TE tuning]].  

{| class="wikitable"

!Mode number

!Pinetone diatonic mode

|-

|3

|mLmsmms

|~ 10/9 5/4 11/8 3/2 5/3 11/6 2/1

|-

|2

|Lmsmmsm

|~ 9/8 5/4 15/11 3/2 5/3 9/5 2/1

|-

|1

|mmsmLms

|~ 10/9 11/9 4/3 22/15 5/3 11/6 2/1

|-

|0

|msmLmsm

|~ 10/9 6/5 4/3 3/2 5/3 9/5 2/1

|-

| -1

|smLmsmm

|~ 12/11 6/5 15/11 3/2 18/11 9/5 2/1

|-

| -2

|msmmsmL

|~ 10/9 6/5 4/3 22/15 8/5 16/9 2/1

|-

| -3

|smmsmLm

|~ 12/11 6/5 4/3 16/11 8/5 9/5 2/1

{| class="wikitable"

!Interval class

!size

Note that there are 3 sizes of interval for each ''interval class''. This property is known as ''trivalence''.

{| class="wikitable"

!Pinetone diatonic mode

!2nd

{| class="wikitable"

!Pinetone diatonic mode

!2nd

{| 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

|}

{| class="wikitable"

!Root note

!Notes

The mode names in the table below link to sequenced examples of the root position 3-step triads, and to root position, 1st inversion, 2nd inversion (with double octave), 3rd inversion, and 4th inversion 3-step pentads, on the tonic of each mode, tuned to [http://x31eq.com/cgi-bin/rt.cgi?ets=1ce_4p_2ce&limit=2_3_5_11 TE 2.3.5.11 Ptolemismic].

{| class="wikitable"

!Mode

!Triad root position

! Pentad fourth inversion

|-

|D-G-C

|D-G-C-F-B

|D-F-B-E-A

|-

|E-A-D

|E-A-D-G-C

|E-G-C-F-B

|-

|F-B-E

|F-B-E-A-D

|F-A-D-G-C

|-

|G-C-F

|G-C-F-B-E

|G-B-E-A-D*

|-

|A-D-G

|A-D-G-C-F

|A-C-F-B-E

|-

|B-E-A

|B-E-A-D-G*

|B-D-G-C-F

|-

|C-F-B

|C-F-B-E-A

The chords marked with '*' are those with the notes of the [[The Pinetone System#Pinetone pentatonic|Pinetone major pentatonic]], which sounds particularly consonant as a chord.

We know the (meantone) pentatonic scale to be a subset of the (meantone) diatonic scale. Similarly, the Pinetone pentatonic is a subset of the Pinetone diatonic. We also know that adding a (meantone) pentatonic to a (meantone) diatonic leads to a (meantone) chromatic, i.e., diatonic on white keys + pentatonic on black keys. We can do this with Pinetone.  

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

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.  

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 step signature and mapping of 5-limit Porcupine[8] is 7L 1s = (10/9~27/25, 25/24~81/80)

{| class="wikitable"

!Mode number

!Step pattern

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"

!Mode number

!Step pattern

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 Pinetone 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"

!Mode in JI (height order)

!Step pattern

|}

{| class="wikitable"

!Mode in JI (height order)

!Step pattern

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

!Comments

|-

|LMLLMLsM

|~ 10/9 6/5 4/3 22/15 8/5 16/9 11/6 2/1

|

|-

|MLMLLMLs

|~ 12/11 6/5 13/10 13/9 8/5 26/15 48/25 2/1

|

|-

|LLMLsMLM

|~ 10/9 11/9 4/3 22/15 20/13 5/3 11/6 2/1

|

|-

|MLLMLsML

|~ 12/11 6/5 4/3 13/9 8/5 5/3 9/5 2/1

|

|-

|LMLsMLML

|~ 10/9 6/5 4/3 11/8 3/2 5/3 9/5 2/1

|Major with 10:12:15 on root

|-

|LsMLMLLM

|~ 10/9 15/13 5/4 11/8 3/2 5/3 11/6 2/1

|Minor with 4:5:6 on root

|-

|MLsMLMLL

|~ 12/11 6/5 5/4 15/11 3/2 13/8 9/5 2/1

|4:5:6 and 10:12:15 on root

|-

|sMLMLLML

|~ 25/24 9/8 5/4 15/11 3/2 5/3 9/5 2/1

|}

{| class="wikitable"

!Mode (height order)

!Step pattern

!Comments

|-

|LMLLMLMs

|~ 10/9 6/5 4/3 22/15 8/5 16/9 48/25 2/1

|

|-

|LLMLMsLM

|~ 10/9 11/9 4/3 22/15 8/5 5/3 11/6 2/1

|

|-

|MLLMLMsL

|~ 12/11 6/5 4/3 13/9 8/5 26/15 10/9 2/1

|

|-

|LMLMsLML

|~ 10/9 6/5 4/3 13/9 3/2 5/3 9/5 2/1

|Major with 10:12:15 on root

|-

|LMsLMLLM

|~ 10/9 6/5 5/4 11/8 3/2 5/3 11/6 2/1

|Minor with 4:5:6 and 10:12:15 on root

|-

|MLMsLMLL

|~ 12/11 6/5 13/10 15/11 3/2 13/8 9/5 2/1

|10:12:15 on root

|-

|sLMLLMLM

|~ 25/24 15/13 5/4 11/8 20/13 5/3 11/6 2/1

|

|-

|MsLMLLML

|~ 12/11 9/8 5/4 15/11 3/2 5/3 9/5 2/1

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

{| class="wikitable"

!Interval class

!sizes

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

{| class="wikitable"

!Mode (rotational order)

!Step pattern

<nowiki>*</nowiki> Non-bracketed JI ratios are those approximated in 2.3.5.11.13 Ptolemismic Pinetone; bracketed JI ratios are shows in pairs: the first interval in each pair is the 2.3.7 interval approximated by additionally tempering out 91/90 or 126/125 (Starling); the second is the 2.3.7 interval approximated by additionally tempering out 105/104 or 245/243 (Supermagic).

{| class="wikitable"

!Mode (rotational order)

!Step pattern

{| class="wikitable"

!Mode in JI (height order)

!Step pattern

|}

{| class="wikitable"

!Mode (height order)

!Step pattern

!Comments

|-

|LMLMLMLs

|~ 10/9 6/5 4/3 13/9 8/5 26/15 48/25 2/1

|

|-

|LMLMLsLM

|~ 10/9 6/5 4/3 13/9 8/5 5/3 11/6 2/1

|

|-

|MLMLMLsL

|~ 12/11 6/5 13/10 13/9 39/25 26/15 9/5 2/1

|

|-

|LMLsLMLM

|~ 10/9 6/5 4/3 13/9 20/13 5/3 11/6 2/1

|

|-

|MLMLsLML

|~ 12/11 6/5 13/10 13/9 3/2 5/3 9/5 2/1

|10:12:15 on root

|-

|LsLMLMLM

|~ 10/9 15/13 50/39 11/8 20/13 5/3 11/6 2/1

|

|-

|MLsLMLML

|~ 12/11 6/5 5/4 11/8 3/2 5/3 9/5 2/1

|Minor root 4:5:6, 10:12:15

|-

|sLMLMLML

|~ 25/24 15/13 5/4 11/8 3/2 5/3 9/5 2/1

|}

{| class="wikitable"

!Interval class

!sizes

{| class="wikitable"

! Mode (rotational order)

!Step pattern

{| class="wikitable"

!Interval class

!size

|}

Pinetone scales are built via step nesting from the 5-limit minor seventh tetrad: 6/5 3/2 9/5 2/1. The bounds for its scales are the set of temperings of the rank-3 step-nested children of the 4-note SNS 6/5 3/2 9/5 2/1. Pinetone diminished scales are built via step-nesting from the 5-limit diminished tetrad: 6/5 36/25 5/3 2/1. The bounds for its scales are the set of temperings of the rank-3 step-nested children of the 4-note MOS scale 6/5 36/25 5/3 2/1.  

The Pinetone diminished scale is a [[step-nested scale]] and a [[Porcupine]][8] x Diminished[8] [[Fokker block]] with [[Unison vector|unison vectors]] of 250/243, 648/625, and 16/15; comprising 4 large steps of 10/9 (''L'' x ''L''), 3 medium steps of 27/25 (''L'' x ''s''), and one small step of 25/24 (''s'' x ''s'').

Additionally, we have another set of [[Porcupine]][7] modes contained in the Pinetone harmonic octatonics: 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.  

The Pintone harmonic minor and harmonic minor have step patterns msmmsLs, and mmsmsLs respectively, or, represented as MODMOS of detempered Meantone[7], LsLLsAs and LLsLsAs, the step patterns of the familiar harmonic minor and harmonic major scales.

Maybe you have a Lumatone, and you're wondering, ok so you can either have sharps or flats? Por queno los dos?

41edo: 7L 1m 11s = (4, 2, 1) = (117.0732c, 58.5366c, 29.2683c).

Alternatively, a 15-note scale can be built from the Pinetone diminished. The resulting scale tempers to Porcupine[15], as well as to Hanson[15].

The first pair of modes temper to Diminished[8] (m = 0), and to Father[8] (L=0). The scale has 1 augmented step of 52/45, 3 large steps of 10/9~11/10, 2 medium steps of 12/11~13/12, and 2 small steps of 25/24~33/32~27/26.  

{| class="wikitable"

! Mode (height order)

!Step pattern

! Comments

|-

|AsLMLMLs

|sLLsLsLL (Hlanithian)

|

|-

|LMLMLsAs

|LsLsLLsL (Mnarian)

|

|-

|LMLsAsLM

|LsLLsLLs (Celephaïsian)

|

|-

|MLMLsAsL

|sLsLLsLL (Sarnathian)

|10:12:15 on the root

|-

|LsAsLMLM

|LLsLLsLs (Dylathian)

|

|-

|MLsAsLML

|sLLsLLsL (Kadathian)

|root 4:5:6,10:12:15

|-

|sAsLMLML

|LsLLsLsL (Ultharian)

|root 4:5:6,10:12:15

|-

|sLMLMLsA

|LLsLsLLs (Illarnekian)

|}

{| class="wikitable"

! Mode (height order)

!Step pattern

!Comments

|-

|AsLMLMLs

|~ 52/45 6/5 4/3 13/9 8/5 26/15 48/25 2/1

|

|-

|LMLMLsAs

|~ 10/9 6/5 4/3 13/9 8/5 5/3 48/25 2/1

|

|-

|LMLsAsLM

|~ 10/9 6/5 4/3 11/8 8/5 5/3 11/6 2/1

|

|-

|MLMLsAsL

|~ 12/11 6/5 13/10 13/9 3/2 26/15 9/5 2/1

|10:12:15 on the root

|-

|LsAsLMLM

|~ 10/9 15/13 4/3 11/8 20/13 5/3 11/6 2/1

|

|-

|MLsAsLML

|~ 12/11 6/5 5/4 13/9 3/2 5/3 9/5 2/1

|root 4:5:6,10:12:15

|-

|sAsLMLML

|~ 25/24 6/5 5/4 11/8 3/2 5/3 9/5 2/1

|root 4:5:6,10:12:15

|-

|sLMLMLsA

|~ 25/24 15/13 5/4 11/8 3/2 5/3 45/26 2/1

|}

{| class="wikitable"

!Interval class

!sizes

{| class="wikitable"

|+

!Mode

!step

|}

{| class="wikitable"

! Mode (rotational order)

!Step pattern

<nowiki>*</nowiki> Non-bracketed JI ratios are those approximated in 2.3.5.11.13 Ptolemismic Pinetone; bracketed JI ratios are the 2.3.7 interval approximated by additionally tempering out 105/104 or 245/243 as in Supermagic temperament. Alternatively, if the consonances of the triads are to be maximised, the scale could be tempered to 2.3.5.7 245/243 i.e., Sensamagic temperament.

We can't use our circle of fifths (Meantone comma pump) or our Porcupine comma pumps here, as both 81/80 and 250/243 are observed. In the Ptolemismic tuning we temper out 100/99 which we can can pump with chord progressions such as  

F-A-D -> F-B-E -> (G-B-E ->) G-B-D -> F-A-D

I find the diatonic and the harmonic diminished to be the most desirable of the albitonic Pinetone scales as the only ones with four 3/2s with with mean variety less than or equal to 3, along with and the major and minor harmonic octatonics as they contain 5 major/minor triads (on 4 roots), the most of any albitonic Pinetone scale (in comparison, the [[diminished scale]] contains 8 major/minor triads on 4 roots, the Meantone diatonic scale contains 6 major/minor triads on 6 roots, and Porcupine[8] contains 6 major/minor triads on 4 roots). The harmonic diminished, as well as the harmonic minor and harmonic major are not built directly by step-nesting, but found as subsets of child step-nested scales that can be modified by chromas to give step-nested scales.