Pinetone: Difference between revisions

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

Are you interested in microtonal music with wild and wacky harmonies but want some familiarity to guide you? Heard about this Porcupine thing but not sure how to get 12 notes of it? Wish you had something like Porcupine but more accurate or with more interesting scales? Introducing Pinetone. The scales you know and love, with a new-age quirky spin. The perfect mix of consonant and dissonant harmonies, familiar and newfangled. Try it on your keyboard straight away (if you can retune your keyboard using Scala files, grab [[Pinetone chromatic (sharps)|this one]]! Copy the text into notepad and save as a .scl file).

"Are you interested in microtonal music with wild and wacky harmonies but want some familiarity to guide you? Heard about this 'Porcupine' thing but not sure how to get 12 notes of it? Wish you had something like Porcupine but more accurate or with more interesting scales? Introducing Pinetone. The scales you know and love, with a new-age quirky spin. The perfect mix of consonant and dissonant harmonies, familiar and newfangled. Try it on your keyboard straight away (if you can retune your keyboard using Scala files, grab [[Pinetone chromatic (sharps)|this one]]! Copy the text into notepad and save as a .scl file)."

Pinetone combines [[Porcupine]] – arguably the best way to add the 11th harmonic to major and minor harmonies in a seven-note scale – with [[Meantone]] – the system underpinning most common practice music from the last several hundred years, so all the same scales (diatonic, harmonic minor, pentatonic, chromatic, etc.) are still available, just with a new Porcupine spin, and the 11th harmonic (and the 13th harmonic as well!).

Line 10:

As opposed to in [[12edo]], each key is distinctly different in Pinetone scales, both a blessing and a curse.

Additionally available in Pinetone are a two sets of octatonic modes with their own Porcupine-like functional harmony that combine [[Porcupine]][8] with the [[oneirotonic]] modes that are gaining popularity at the moment. Finally, Pinetone diminished scales combine Porcupine with the familiar diminished scale.

Additionally available in Pinetone are two sets of octatonic modes with their own Porcupine-like functional harmony that combine [[Porcupine]][8] with the [[oneirotonic]] modes that are gaining popularity at the moment. Finally, Pinetone diminished scales combine Porcupine with the familiar diminished scale.

If you have a [[Lumatone]], you can use the standard Bosanquet mapping for 12edo. The white keys are the Pinetone diatonic, a cross between the Meantone diatonic scale and Porcupine[7], and the black keys give the Pinetone pentatonic, which approximates the [[just intonation]] pentatonic scale 9/8 5/4 3/2 5/3 2/1. I've chosen to colour the G♯/A♭ key pink, and the other chromatic keys blue, because I'm a proud trans woman and a big nerd. You can use any colours, but I find it helps to colour the G♯/A♭ key a different colour since that's the one chromatic key used along with the diatonic keys to make the Pinetone octatonic. The white keys and the pink key together make a Pinetone octatonic scale - major-harmonic with G♯ and minor-harmonic for A♭. This is what I call the ''Pinetone System.''

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* [[Pinetone#Pinetone pentatonic|Pentatonic minor]]: 95695, B67B6 (subset of diatonic, harmonic diminished)

* [[Pinetone#Pinetone diatonic|Diatonic]]: 5456545, 6567656 (subset of major and minor-harmonic octatonics)

* [[Pinetone#Pinetone harmonic minor and harmonic major|Harmonic minor]]: 5455475, 6566585 (subset of major and minor-harmonic octatonic)

* [[Pinetone#Pinetone harmonic minor and harmonic major|Harmonic major]]: 5545474, 6656585 (subset of major and minor-harmonic octatonic)

* [[Pinetone#Additional heptatonic subsets of the Pinetone chromatic|Azurian bright minor]]: 5465455, 6576566 (subset of chromatic)

* [[Pinetone#Additional heptatonic subsets of the Pinetone chromatic|Duradian dark minor]]: 5456455, 6567566 (subset of chromatic)

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|}

{| class="wikitable"

|+Table 2.3. Modes of the just Pinetone diatonic

|+Table 2.3. Modes of the [[SNS (2/1, 3/2, 6/5)-7|just Pinetone diatonic]]

!Mode number

!Mode in JI

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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"

|+Table 2.4. Modes of the Ptolemismic Pinetone diatonic

|+Table 2.4. Modes of the [[SNS (2/1, 3/2, 6/5: 100/99)-7|Ptolemismic Pinetone diatonic]]

!Mode number

!Pinetone diatonic mode

Line 1,061:

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The TE tuning in cents is: 146.636 174.055 320.690 467.326 494.745 641.381 704.524 851.159 878.579 1025.214 1171.849 1199.269

{| class="wikitable"

|+(2.3.5.11) Ptolemismic Pinetone Chromatic

|+(2.3.5.11) [[Ptolemismic Pinetone chromatic|Ptolemismic Pinetone Chromatic]]

!Mode number

!Mode as simplest JI pre-image

Line 1,509:

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{| class="wikitable"

|+Table 4.1.

! colspan="8" |Scale (mode -3 subset)

! colspan="8" |Scale (mode 3 subset)

~~! colspan="8" |Scale (mode -3 subset)~~

!JI ratios approximated

!Step pattern

Line 1,637:

Line 1,639:

The scale D E F G A B C has step pattern MsMLMsM, which tempers to sssLsss under Porcupine (M=s) and LsLLLsL under Meantone (M=L).

Raising an F to an F♯ replaces ~6/5 with ~11/9, i.e., it raises the F by ~55/54, which is tempered out in [[Porcupine]], so the scale D E F♯ G A B C tempers to sssLsss under Porcupine as before, but to LLsLLsL under Meantone (Mixolydian mode rather than Dorian as before)~~, so will call the mode approximating the ratios 10/9 11/9~~ 4/3 3/~~2 5/3 9/5 2/1 Mixolydian symmetric minor~~, ~~and use~~ the ~~mode~~ to ~~name the scale as~~ a ~~whole~~.

Raising an F to an F♯ replaces ~6/5 with ~11/9, i.e., it raises the F by ~55/54, which is tempered out in [[Porcupine]], so the scale D E F♯ G A B C tempers to sssLsss under Porcupine as before, but to LLsLLsL under Meantone (Mixolydian mode rather than Dorian as before). The modes of this scale are detailed in Table 4.3. Similarly, lowering B to B♭ lowers by ~55/54, leading to the Duradian ♯5 shown in Table 4.4, which I consider to be a really beautiful minor mode.

To calculate the mode numbers for Tables 4.3-4.5, 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. Modes are named via the [[Tetracot]][7] [[MODMOS]] they temper to when [[243/242]] is tempered out (i.e., the difference between [[11/9]] and [[27/22]]), as in [[27edo]], [[34edo]], and [[41edo]]. Table 4.2 introduces the modes of Tetracot[7]. The Pinetone diatonic, therefore, is also a detempering of a Tetracot MODMOS, with generator chain equivalent to that of the double harmonic major scale (a MODMOS of the Meantone diatonic scale). Tetracot[7] mode names used the Archeotonic mode names [[6L 1s#Proposed names|here]] as a basis, with the substitution of Azurian, Duradian, and Phyradian as Tetracot specific mode names from [https://www.youtube.com/watch?v=xYZwye9PWSo here].

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|~ 10/9 6/5 4/3 22/15 44/27 16/9 2/1

|}

~~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"

|+Table 4.4. Modes of the Ptolemismic Pinetone Duradian dark minor

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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 chromatic is a 12-note rank-3 [[Meantone]][12] x [[Ripple]][12] [[Fokker block]], a [[step-nested scale]] that also tempers to [[Porcupine]][8], comprising a diatonic [[Meantone]][7]-[[Porcupine]][7]-[[Dicot]][7] [[wakalix]] / 3-[[Step-nested scale|SNS]] on the white keys, and a pentatonic [[Meantone]][5]-[[Father]][5]-[[Bug]][5] [[wakalix]] on the 'black' keys.

For the accompanying mapping for the Lumatone keyboard the G♯ / A♭ key is coloured pink (and the remaining chromatic keys blue), and along with the white keys makes a [[Porcupine]][8] / [[Father]][8] [[Fokker block]] (any colours could be chosen instead of white, pink, and blue).

The Pinetone diatonic is a [[wakalix]] (pairwise well-formed scale) and a [[step-nested scale]]: A detempering of [[Meantone]][7] and [[Porcupine]][7], (and also of [[Dicot]][7]), a [[Fokker block]] (product word) with [[Unison vector|unison vectors]] of [[81/80]] and [[250/243]] (and [[25/24]]) comprising 1 large step of 9/8 (''L'' x ''L''), 3 medium steps of 10/9 (''L'' x ''s''), and 3 small steps of 27/25 (''s'' x ''s'').

The Pinetone diatonic is a [[wakalix]] (pairwise well-formed scale) and a [[step-nested scale]]: A detempering of [[Meantone]][7] and [[Porcupine]][7], (and also of [[Dicot]][7]), a [[Fokker block]] (product word) with [[Unison vector|unison vectors]] of [[81/80]] and [[250/243]] (and [[25/24]]) comprising 1 large step of 9/8 (''L'' x ''L''), 3 medium steps of 10/9 (''L'' x ''s''), and 3 small steps of 27/25 (''s'' x ''s''). As a product word of 5L 2s (standard diatonic scale form) and 1L 6s (Porcupine[7] scale form) the Pinetone diatonic in it's symmetric mode msmLmsm = LsLLLsL x sssLsss, a product of Dorian mode of the standard diatonic scale, and the symmetric minor mode of the Porcupine[7] scale form.

The Azurian bright minor, Duradian dark minor, Karakalian bright minor, Phyradian dark minor are alternative heptatonic subsets of the Pinetone chromatic — Fokker blocks of the same unison vectors and step sizes as the Pinetone diatonic arranged in different patterns, corresponding to product words of different combinations of modes of Porcupine[7] and Meantone[7]. Unlike the Pinetone diatonic, these scales are not also Fokker blocks of Dicot[7] with Porcupine[7] or Meantone[7].

Azurian bright minor msLmsmm = LLLLsLL x ssLssss

Duradian dark minor msmLsmm = LsLLLLL x sssLsss

Karakalian bright minor msLmmsm = LLLLLsL x ssLssss

Phyradian dark minor smmLsmm = sLLLLLL x sssLsss

The Pinetone major and minor-harmonic octatonics are the 8-note rank-3 [[Porcupine]][8] x [[Father]][8] [[Fokker block|Fokker blocks]] with [[Unison vector|unison vectors]] of 250/243, 16/15, and 648/625; 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 ''L'').

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'').

==Pinetone harmonic minor and harmonic major==

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.

We note that there are fewer consonant triads available in these scales than in the Pinetone diatonic and octatonic scales, so they may be useful for melody only.

On D we get the scale:

174.055 320.69 557.888 704.524 878.579 1025.214 1199.269 as the notes D E F G♯ A B C D, representing 10/9 6/5 25/18~11/8 3/2 5/3 9/5 2/1

We get the following 7 modes of Pinetone harmonic minor scale:

*Lsmsmms Lydian ♯2 bright major starting on F

*mmsLsms Ionian ♯5 symmetric minor starting on C

*msLsmsm Ukranian dorian bright minor starting on D

*sLsmsmm Phyrgian dominant dark major starting on E

*msmmsLs harmonic minor dark diminished starting on A

*smmsLsm Locrian ♮6 bright diminished starting on B

*smsmmsL altered diminished magical seventh starting on G♯

Replacing the A with an A♭ instead, we get the modes of the Pinetone harmonic major scale. Starting on D we get the mode:

174.055 320.69 494.745 641.38 878.579 1025.214 1199.269 as the notes D E F G A♭ B C D, representing 10/9 6/5 4/3 36/25~13/9 5/3 9/5 2/1.

Which has modes:

*Lsmmsms Lydian Augmented ♯2 bright major starting on A♭

*msLsmms Lydian ♭3 bright minor starting on F

*sLsmmsm Mixolydian ♭2 dark major starting on G

*mmsmsLs harmonic major bright diminished starting on C

*msmsLsm Dorian ♭5 dark diminished starting on D

*smsLsmm Phrygian ♭4 symmetric minor starting on E

*smmsmsL Locrian magical ♭♭7 starting on B

We can see that this scale differs from the Pinetone diminished heptatonic by only a single note - 9/5 instead of 50/27.

The augmented step of the Pinetone harmonic minor and major scales is the same as of the Pinetone diminished heptatonic, representing 15/13 when 325/324 is tempered out (the difference between 100/99 and 144/143).

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.

==Pinetone hyperchromatic scales==

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tempering L = s (tempering out 250/243, the Porcupine comma) results in sLsLsLsLsLsLsLs, which is Porcupine[15].

tempering out s would lead to ~~sLLsLsL~~, which is ~~Dicot~~[7];

tempering out s would lead to ssLsLssLsssL, which is a MODMOS of Diminished[12].

tempering out m would lead to ~~ssLsLssLsssL~~, which is ~~a MODMOS of Diminished~~[12].

tempering out m would lead to sLLsLsL, which is Dicot[7];

Tempering out 100/99 and 144/143 leads to the simplest pre-image: 25/24 10/9 15/13 6/5 5/4 4/3 11/8 13/9 3/2 8/5 5/3 26/15 9/5 48/25 2/1.

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Accordingly Pinetone-15 would temper to two step sizes in 19edo (Hanson), 22edo (Porcupine), 34edo (Hanson), and 27edo (Augmented). If we wish to keep the 3-step size structure, we can tune to 26edo or 41edo with (L, m, s) = (3, 2, 1), and (4, 3, 2) respectively.

Tempering out the 325/324, the difference between 100/99 and 144/143 rather than both of 100/99 and 144/143 leads to a more accurate temperament that does not include the whole 2.3.5.11.13 subgroup., rather just the 2.3.5.13 subgroup. The simplest JI pre-image in this temperament would be 25/24 10/9 15/13 6/5 5/4 4/3 18/13 13/9 3/2 8/5 5/3 26/15 9/5 48/25 2/1, which differs only from the simplest pre-image of the scale under 2.3.5.11.13 Ptolemismic tempering by the inclusion of 18/13 rather than 11/8.

2.3.5.13 325/324 may be ~~better~~ tuned to 46edo, with (L, m, s) = (4, 2, 1).

In this tuning the medium and small steps are within 1c of the same size. Tempering them together results in Cata temperament, an extension of Hanson. 2.3.5.13 325/324 may alternatively be tuned to 46edo, with (L, m, s) = (4, 2, 3), or as Cata[15] in 53edo, 72edo, or 87edo as (5, 3, 3), (7, 4, 4), or (8, 5, 5).

Pinetone-15 may be represented with note names representing the corresponding Meantone notes as: D D♯ E E♯ F F♯ G G♯ A♭ A B♭ B C♭ C D♭ D, though we note that these notes do not necessarily correspond to the notes of the Pinetone chromatic or hyperchromatic of the same name.

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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"

|+Table 8.1. Modes of the just Pinetone harmonic diminished

|+Table 9.1. Modes of the just Pinetone harmonic diminished

! Mode ~~(height order)~~

!Mode number

! Mode name

!Step pattern

!Oneirotonic step pattern

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! Comments

|-

|4

|Hlanithian diminished

|AsLMLMLs

Line 3,476:

Line 3,523:

|

|-

|3

|Mnarian diminished*

|LMLMLsAs

Line 3,483:

Line 3,531:

|

|-

|2

|Celephaïsian diminished*

|LMLsAsLM

Line 3,490:

Line 3,539:

|

|-

|1

|Sarnathian diminished†

|MLMLsAsL

Line 3,497:

Line 3,547:

|10:12:15 on the root

|-

| -1

|Dylathian diminished*

|LsAsLMLM

Line 3,504:

Line 3,555:

|

|-

| -2

|Kadathian diminished*††

|MLsAsLML

Line 3,511:

Line 3,563:

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

|-

| -3

|Ultharian diminished*††

|sAsLMLML

Line 3,518:

Line 3,571:

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

|-

| -4

|Illarnekian diminished*†

|sLMLMLsA

Line 3,527:

Line 3,581:

For Tables 9.1. and 9.2. modes with major and minor [8] triads (3-step) triads on the root are marked with '*'; modes with major and minor tertian (2-step) triads on the root are marked with '†' ('††' for both major and minor).

{| class="wikitable"

|+Table 8.2. Modes of the Ptolemismic Pinetone harmonic diminished

|+Table 9.2. Modes of the Ptolemismic Pinetone harmonic diminished

! Mode ~~(height order)~~

!Mode number

! Mode name

!Step pattern

!Mode as simplest JI pre-image 5-limit JI

Line 3,534:

Line 3,589:

!Comments

|-

|4

|Hlanithian diminished

|AsLMLMLs

Line 3,540:

Line 3,596:

|

|-

|3

|Mnarian diminished*

|LMLMLsAs

Line 3,546:

Line 3,603:

|

|-

|2

|Celephaïsian diminished*

|LMLsAsLM

Line 3,552:

Line 3,610:

|

|-

|1

|Sarnathian diminished†

|MLMLsAsL

Line 3,558:

Line 3,617:

|10:12:15 on the root

|-

| -1

|Dylathian diminished*

|LsAsLMLM

Line 3,564:

Line 3,624:

|

|-

| -2

|Kadathian diminished*††

|MLsAsLML

Line 3,570:

Line 3,631:

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

|-

| -3

|Ultharian diminished*††

|sAsLMLML

Line 3,576:

Line 3,638:

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

|-

| -4

|Illarnekian diminished*†

|sLMLMLsA

Line 3,583:

Line 3,646:

|}

{| class="wikitable"

|+Table 8.3. Intervals of the Ptolemismic Pinetone harmonic diminished (variety = 20)

|+Table 9.3. Intervals of the Ptolemismic Pinetone harmonic diminished (variety = 20)

!Interval class

!sizes

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{| class="wikitable"

|+

Table 8.4. Intervals of modes of the Pinetone harmonic diminished

Table 9.4. Intervals of modes of the Pinetone harmonic diminished

!Mode

!Mode (height order)

!step

!2-step

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|}

{| class="wikitable"

|+Table 8.5. 3-step stacked triads of the Ptolemismic Pinetone harmonic diminished

|+Table 9.5. 3-step stacked triads of the Ptolemismic Pinetone harmonic diminished

! Mode (rotational order)

!Step pattern

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Line 3,954:

|12:15:20

|}

<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~~.

<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 as in Supermagic temperament. Alternatively, if the consonances of the triads are to be maximised, the scale could be tempered 7-limit rather than 13-limit Supermagic.

==Comma pump==

@@ Line 2: / Line 2: @@
 '''Definition: Pinetone is a system of related rank-3 microtonal scales: pentatonic, diatonic, octatonic, chromatic, and hyperchromatic.'''
-Are you interested in microtonal music with wild and wacky harmonies but want some familiarity to guide you? Heard about this Porcupine thing but not sure how to get 12 notes of it? Wish you had something like Porcupine but more accurate or with more interesting scales? Introducing Pinetone. The scales you know and love, with a new-age quirky spin. The perfect mix of consonant and dissonant harmonies, familiar and newfangled. Try it on your keyboard straight away (if you can retune your keyboard using Scala files, grab [[Pinetone chromatic (sharps)|this one]]! Copy the text into notepad and save as a .scl file).
+"Are you interested in microtonal music with wild and wacky harmonies but want some familiarity to guide you? Heard about this 'Porcupine' thing but not sure how to get 12 notes of it? Wish you had something like Porcupine but more accurate or with more interesting scales? Introducing Pinetone. The scales you know and love, with a new-age quirky spin. The perfect mix of consonant and dissonant harmonies, familiar and newfangled. Try it on your keyboard straight away (if you can retune your keyboard using Scala files, grab [[Pinetone chromatic (sharps)|this one]]! Copy the text into notepad and save as a .scl file)."
 Pinetone combines [[Porcupine]] – arguably the best way to add the 11th harmonic to major and minor harmonies in a seven-note scale – with [[Meantone]] – the system underpinning most common practice music from the last several hundred years, so all the same scales (diatonic, harmonic minor, pentatonic, chromatic, etc.) are still available, just with a new Porcupine spin, and the 11th harmonic (and the 13th harmonic as well!).
@@ Line 10: / Line 10: @@
 As opposed to in [[12edo]], each key is distinctly different in Pinetone scales, both a blessing and a curse.
-Additionally available in Pinetone are a two sets of octatonic modes with their own Porcupine-like functional harmony that combine [[Porcupine]][8] with the [[oneirotonic]] modes that are gaining popularity at the moment. Finally, Pinetone diminished scales combine Porcupine with the familiar diminished scale.
+Additionally available in Pinetone are two sets of octatonic modes with their own Porcupine-like functional harmony that combine [[Porcupine]][8] with the [[oneirotonic]] modes that are gaining popularity at the moment. Finally, Pinetone diminished scales combine Porcupine with the familiar diminished scale.
 If you have a [[Lumatone]], you can use the standard Bosanquet mapping for 12edo. The white keys are the Pinetone diatonic, a cross between the Meantone diatonic scale and Porcupine[7], and the black keys give the Pinetone pentatonic, which approximates the [[just intonation]] pentatonic scale 9/8 5/4 3/2 5/3 2/1. I've chosen to colour the G♯/A♭ key pink, and the other chromatic keys blue, because I'm a proud trans woman and a big nerd. You can use any colours, but I find it helps to colour the G♯/A♭ key a different colour since that's the one chromatic key used along with the diatonic keys to make the Pinetone octatonic. The white keys and the pink key together make a Pinetone octatonic scale - major-harmonic with G♯ and minor-harmonic for A♭. This is what I call the ''Pinetone System.''
@@ Line 23: / Line 23: @@
 * [[Pinetone#Pinetone pentatonic|Pentatonic minor]]: 95695, B67B6 (subset of diatonic, harmonic diminished)
 * [[Pinetone#Pinetone diatonic|Diatonic]]: 5456545, 6567656 (subset of major and minor-harmonic octatonics)
+* [[Pinetone#Pinetone harmonic minor and harmonic major|Harmonic minor]]: 5455475, 6566585 (subset of major and minor-harmonic octatonic)
+* [[Pinetone#Pinetone harmonic minor and harmonic major|Harmonic major]]: 5545474, 6656585 (subset of major and minor-harmonic octatonic)
 * [[Pinetone#Additional heptatonic subsets of the Pinetone chromatic|Azurian bright minor]]: 5465455, 6576566 (subset of chromatic)
 * [[Pinetone#Additional heptatonic subsets of the Pinetone chromatic|Duradian dark minor]]: 5456455, 6567566 (subset of chromatic)
@@ Line 148: / Line 150: @@
 |}
 {| class="wikitable"
-|+Table 2.3. Modes of the just Pinetone diatonic
+|+Table 2.3. Modes of the [[SNS (2/1, 3/2, 6/5)-7|just Pinetone diatonic]]
 !Mode number
 !Mode in JI
@@ Line 237: / Line 239: @@
 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"
-|+Table 2.4. Modes of the Ptolemismic Pinetone diatonic
+|+Table 2.4. Modes of the [[SNS (2/1, 3/2, 6/5: 100/99)-7|Ptolemismic Pinetone diatonic]]
 !Mode number
 !Pinetone diatonic mode
@@ Line 1,061: / Line 1,063: @@
 The TE tuning in cents is: 146.636 174.055 320.690 467.326 494.745 641.381 704.524 851.159 878.579 1025.214 1171.849 1199.269
 {| class="wikitable"
-|+(2.3.5.11) Ptolemismic Pinetone Chromatic
+|+(2.3.5.11) [[Ptolemismic Pinetone chromatic|Ptolemismic Pinetone Chromatic]]
 !Mode number
 !Mode as simplest JI pre-image
@@ Line 1,509: / Line 1,511: @@
 {| class="wikitable"
 |+Table 4.1.
+! colspan="8" |Scale (mode -3 subset)
 ! colspan="8" |Scale (mode 3 subset)
-! colspan="8" |Scale (mode -3 subset)
 !JI ratios approximated
 !Step pattern
@@ Line 1,637: / Line 1,639: @@
 The scale D E F G A B C has step pattern MsMLMsM, which tempers to sssLsss under Porcupine (M=s) and LsLLLsL under Meantone (M=L).
-Raising an F to an F♯ replaces ~6/5 with ~11/9, i.e., it raises the F by ~55/54, which is tempered out in [[Porcupine]], so the scale D E F♯ G A B C tempers to sssLsss under Porcupine as before, but to LLsLLsL under Meantone (Mixolydian mode rather than Dorian as before), so will call the mode approximating the ratios 10/9 11/9 4/3 3/2 5/3 9/5 2/1 Mixolydian symmetric minor, and use the mode to name the scale as a whole.
+Raising an F to an F♯ replaces ~6/5 with ~11/9, i.e., it raises the F by ~55/54, which is tempered out in [[Porcupine]], so the scale D E F♯ G A B C tempers to sssLsss under Porcupine as before, but to LLsLLsL under Meantone (Mixolydian mode rather than Dorian as before). The modes of this scale are detailed in Table 4.3. Similarly, lowering B to B♭ lowers by ~55/54, leading to the Duradian ♯5 shown in Table 4.4, which I consider to be a really beautiful minor mode.
 To calculate the mode numbers for Tables 4.3-4.5, 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. Modes are named via the [[Tetracot]][7] [[MODMOS]] they temper to when [[243/242]] is tempered out (i.e., the difference between [[11/9]] and [[27/22]]), as in [[27edo]], [[34edo]], and [[41edo]]. Table 4.2 introduces the modes of Tetracot[7]. The Pinetone diatonic, therefore, is also a detempering of a Tetracot MODMOS, with generator chain equivalent to that of the double harmonic major scale (a MODMOS of the Meantone diatonic scale). Tetracot[7] mode names used the Archeotonic mode names [[6L 1s#Proposed names|here]] as a basis, with the substitution of Azurian, Duradian, and Phyradian as Tetracot specific mode names from [https://www.youtube.com/watch?v=xYZwye9PWSo here].
@@ Line 1,740: / Line 1,742: @@
 |~ 10/9 6/5 4/3 22/15 44/27 16/9 2/1
 |}
-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"
 |+Table 4.4. Modes of the Ptolemismic Pinetone Duradian dark minor
@@ Line 3,368: / Line 3,370: @@
 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 chromatic is a 12-note rank-3 [[Meantone]][12] x [[Ripple]][12] [[Fokker block]], a [[step-nested scale]] that also tempers to [[Porcupine]][8], comprising a diatonic [[Meantone]][7]-[[Porcupine]][7]-[[Dicot]][7] [[wakalix]] / 3-[[Step-nested scale|SNS]] on the white keys, and a pentatonic [[Meantone]][5]-[[Father]][5]-[[Bug]][5] [[wakalix]] on the 'black' keys.
 For the accompanying mapping for the Lumatone keyboard the G♯ / A♭ key is coloured pink (and the remaining chromatic keys blue), and along with the white keys makes a [[Porcupine]][8] / [[Father]][8] [[Fokker block]] (any colours could be chosen instead of white, pink, and blue).
-The Pinetone diatonic is a [[wakalix]] (pairwise well-formed scale) and a [[step-nested scale]]: A detempering of [[Meantone]][7] and [[Porcupine]][7], (and also of [[Dicot]][7]), a [[Fokker block]] (product word) with [[Unison vector|unison vectors]] of [[81/80]] and [[250/243]] (and [[25/24]]) comprising 1 large step of 9/8 (''L'' x ''L''), 3 medium steps of 10/9 (''L'' x ''s''), and 3 small steps of 27/25 (''s'' x ''s'').
+The Pinetone diatonic is a [[wakalix]] (pairwise well-formed scale) and a [[step-nested scale]]: A detempering of [[Meantone]][7] and [[Porcupine]][7], (and also of [[Dicot]][7]), a [[Fokker block]] (product word) with [[Unison vector|unison vectors]] of [[81/80]] and [[250/243]] (and [[25/24]]) comprising 1 large step of 9/8 (''L'' x ''L''), 3 medium steps of 10/9 (''L'' x ''s''), and 3 small steps of 27/25 (''s'' x ''s''). As a product word of 5L 2s (standard diatonic scale form) and 1L 6s (Porcupine[7] scale form) the Pinetone diatonic in it's symmetric mode msmLmsm = LsLLLsL x sssLsss, a product of Dorian mode of the standard diatonic scale, and the symmetric minor mode of the Porcupine[7] scale form.
 The Azurian bright minor, Duradian dark minor, Karakalian bright minor, Phyradian dark minor are alternative heptatonic subsets of the Pinetone chromatic — Fokker blocks of the same unison vectors and step sizes as the Pinetone diatonic arranged in different patterns, corresponding to product words of different combinations of modes of Porcupine[7] and Meantone[7]. Unlike the Pinetone diatonic, these scales are not also Fokker blocks of Dicot[7] with Porcupine[7] or Meantone[7].
+Azurian bright minor msLmsmm =  LLLLsLL x ssLssss
+Duradian dark minor msmLsmm = LsLLLLL x sssLsss
+Karakalian bright minor msLmmsm = LLLLLsL x ssLssss
+Phyradian dark minor smmLsmm = sLLLLLL x sssLsss
 The Pinetone major and minor-harmonic octatonics are the 8-note rank-3 [[Porcupine]][8] x [[Father]][8] [[Fokker block|Fokker blocks]] with [[Unison vector|unison vectors]] of 250/243, 16/15, and 648/625; 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 ''L'').
 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'').
+==Pinetone harmonic minor and harmonic major==
+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.
+We note that there are fewer consonant triads available in these scales than in the Pinetone diatonic and octatonic scales, so they may be useful for melody only.
+On D we get the scale:
+.055 320.69 557.888 704.524 878.579 1025.214 1199.269 as the notes D E F G♯ A B C D, representing 10/9 6/5 25/18~11/8 3/2 5/3 9/5 2/1
+We get the following 7 modes of Pinetone harmonic minor scale:
+*Lsmsmms Lydian ♯2 bright major starting on F
+*mmsLsms Ionian ♯5 symmetric minor starting on C
+*msLsmsm Ukranian dorian bright minor starting on D
+*sLsmsmm Phyrgian dominant dark major starting on E
+*msmmsLs harmonic minor dark diminished starting on A
+*smmsLsm Locrian ♮6 bright diminished starting on B
+*smsmmsL altered diminished magical seventh starting on G♯
+Replacing the A with an A♭ instead, we get the modes of the Pinetone harmonic major scale. Starting on D we get the mode:
+.055 320.69 494.745 641.38 878.579 1025.214 1199.269 as the notes D E F G A♭ B C D, representing 10/9 6/5 4/3 36/25~13/9 5/3 9/5 2/1.
+Which has modes:
+*Lsmmsms Lydian Augmented ♯2 bright major starting on A♭
+*msLsmms Lydian ♭3 bright minor starting on F
+*sLsmmsm Mixolydian ♭2 dark major starting on G
+*mmsmsLs harmonic major bright diminished starting on C
+*msmsLsm Dorian ♭5 dark diminished starting on D
+*smsLsmm Phrygian ♭4 symmetric minor starting on E
+*smmsmsL Locrian magical ♭♭7 starting on B
+We can see that this scale differs from the Pinetone diminished heptatonic by only a single note -  9/5 instead of 50/27.
+The augmented step of the Pinetone harmonic minor and major scales is the same as of the Pinetone diminished heptatonic, representing 15/13 when 325/324 is tempered out (the difference between 100/99 and 144/143).
+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.
 ==Pinetone hyperchromatic scales==
@@ Line 3,424: / Line 3,469: @@
 tempering L = s (tempering out 250/243, the Porcupine comma) results in sLsLsLsLsLsLsLs, which is Porcupine[15].
-tempering out s would lead to sLLsLsL, which is Dicot[7];
+tempering out s would lead to ssLsLssLsssL, which is a MODMOS of Diminished[12].
-tempering out m would lead to ssLsLssLsssL, which is a MODMOS of Diminished[12].
+tempering out m would lead to sLLsLsL, which is Dicot[7];
 Tempering out 100/99 and 144/143 leads to the simplest pre-image: 25/24 10/9 15/13 6/5 5/4 4/3 11/8 13/9 3/2 8/5 5/3 26/15 9/5 48/25 2/1.
@@ Line 3,444: / Line 3,489: @@
 Accordingly Pinetone-15 would temper to two step sizes in 19edo (Hanson), 22edo (Porcupine), 34edo (Hanson), and 27edo (Augmented). If we wish to keep the 3-step size structure, we can tune to 26edo or 41edo with (L, m, s) = (3, 2, 1), and (4, 3, 2) respectively.
 Tempering out the 325/324, the difference between 100/99 and 144/143 rather than both of 100/99 and 144/143 leads to a more accurate temperament that does not include the whole 2.3.5.11.13 subgroup., rather just the 2.3.5.13 subgroup. The simplest JI pre-image in this temperament would be 25/24 10/9 15/13 6/5 5/4 4/3 18/13 13/9 3/2 8/5 5/3 26/15 9/5 48/25 2/1, which differs only from the simplest pre-image of the scale under 2.3.5.11.13 Ptolemismic tempering by the inclusion of 18/13 rather than 11/8.
-.3.5.13 325/324 may be better tuned to 46edo, with (L, m, s) = (4, 2, 1).
+In this tuning the medium and small steps are within 1c of the same size. Tempering them together results in Cata temperament, an extension of Hanson. 2.3.5.13 325/324 may alternatively be tuned to 46edo, with (L, m, s) = (4, 2, 3), or as Cata[15] in 53edo, 72edo, or 87edo as (5, 3, 3), (7, 4, 4), or (8, 5, 5).
 Pinetone-15 may be represented with note names representing the corresponding Meantone notes as: D D♯ E E♯ F F♯ G G♯ A♭ A B♭ B C♭ C D♭ D, though we note that these notes do not necessarily correspond to the notes of the Pinetone chromatic or hyperchromatic of the same name.
@@ Line 3,461: / Line 3,506: @@
 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"
-|+Table 8.1. Modes of the just Pinetone harmonic diminished
+|+Table 9.1. Modes of the just Pinetone harmonic diminished
-! Mode (height order)
+!Mode number
+! Mode name
 !Step pattern
 !Oneirotonic step pattern
@@ Line 3,469: / Line 3,515: @@
 ! Comments
 |-
+|4
 |Hlanithian diminished
 |AsLMLMLs
@@ Line 3,476: / Line 3,523: @@
 |
 |-
+|3
 |Mnarian diminished*
 |LMLMLsAs
@@ Line 3,483: / Line 3,531: @@
 |
 |-
+|2
 |Celephaïsian diminished*
 |LMLsAsLM
@@ Line 3,490: / Line 3,539: @@
 |
 |-
+|1
 |Sarnathian diminished<sup>†</sup>
 |MLMLsAsL
@@ Line 3,497: / Line 3,547: @@
 |10:12:15 on the root
 |-
+| -1
 |Dylathian diminished*
 |LsAsLMLM
@@ Line 3,504: / Line 3,555: @@
 |
 |-
+| -2
 |Kadathian diminished*<sup>††</sup>
 |MLsAsLML
@@ Line 3,511: / Line 3,563: @@
 |root 4:5:6,10:12:15
 |-
+| -3
 |Ultharian diminished*<sup>††</sup>
 |sAsLMLML
@@ Line 3,518: / Line 3,571: @@
 |root 4:5:6,10:12:15
 |-
+| -4
 |Illarnekian diminished*<sup>†</sup>
 |sLMLMLsA
@@ Line 3,527: / Line 3,581: @@
 For Tables 9.1. and 9.2. modes with major and minor [8] triads (3-step) triads on the root are marked with '*'; modes with major and minor tertian (2-step) triads on the root are marked with '<sup>†</sup>' ('<sup>††</sup>' for both major and minor).
 {| class="wikitable"
-|+Table 8.2. Modes of the Ptolemismic Pinetone harmonic diminished
+|+Table 9.2. Modes of the Ptolemismic Pinetone harmonic diminished
-! Mode (height order)
+!Mode number
+! Mode name
 !Step pattern
 !Mode as simplest JI pre-image 5-limit JI
@@ Line 3,534: / Line 3,589: @@
 !Comments
 |-
+|4
 |Hlanithian diminished
 |AsLMLMLs
@@ Line 3,540: / Line 3,596: @@
 |
 |-
+|3
 |Mnarian diminished*
 |LMLMLsAs
@@ Line 3,546: / Line 3,603: @@
 |
 |-
+|2
 |Celephaïsian diminished*
 |LMLsAsLM
@@ Line 3,552: / Line 3,610: @@
 |
 |-
+|1
 |Sarnathian diminished<sup>†</sup>
 |MLMLsAsL
@@ Line 3,558: / Line 3,617: @@
 |10:12:15 on the root
 |-
+| -1
 |Dylathian diminished*
 |LsAsLMLM
@@ Line 3,564: / Line 3,624: @@
 |
 |-
+| -2
 |Kadathian diminished*<sup>††</sup>
 |MLsAsLML
@@ Line 3,570: / Line 3,631: @@
 |root 4:5:6,10:12:15
 |-
+| -3
 |Ultharian diminished*<sup>††</sup>
 |sAsLMLML
@@ Line 3,576: / Line 3,638: @@
 |root 4:5:6,10:12:15
 |-
+| -4
 |Illarnekian diminished*<sup>†</sup>
 |sLMLMLsA
@@ Line 3,583: / Line 3,646: @@
 |}
 {| class="wikitable"
-|+Table 8.3. Intervals of the Ptolemismic Pinetone harmonic diminished (variety = 20)
+|+Table 9.3. Intervals of the Ptolemismic Pinetone harmonic diminished (variety = 20)
 !Interval class
 !sizes
@@ Line 3,764: / Line 3,827: @@
 {| class="wikitable"
 |+
-Table 8.4. Intervals of modes of the Pinetone harmonic diminished
+Table 9.4. Intervals of modes of the Pinetone harmonic diminished
-!Mode
+!Mode (height order)
 !step
 !2-step
@@ Line 3,827: / Line 3,890: @@
 |}
 {| class="wikitable"
-|+Table 8.5. 3-step stacked triads of the Ptolemismic Pinetone harmonic diminished
+|+Table 9.5. 3-step stacked triads of the Ptolemismic Pinetone harmonic diminished
 ! Mode (rotational order)
 !Step pattern
@@ Line 3,891: / Line 3,954: @@
 |12:15:20
 |}
-<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.
+<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 as in Supermagic temperament. Alternatively, if the consonances of the triads are to be maximised, the scale could be tempered 7-limit rather than 13-limit Supermagic.
 ==Comma pump==