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.

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.  

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.  

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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].

<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.