Pinetone: Difference between revisions

Line 965:

!UDP

!Mode name

!Mode as simplest JI pre-image

!3-step stacked triad on root (with G♯)

!(with A♭ = H)

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|<nowiki>7|0</nowiki>

|Bright quartal

|~ 10/9 6/5 4/3 36/25 8/5 16/9 48/25 2/1

|G♯-C-F

|A-D-G

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|<nowiki>6|1</nowiki>

|Dark quartal

|~ 10/9 6/5 4/3 36/25 8/5 16/9 9/5 2/1

|A-D-G

|B-E-A♭ = B-E-H

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|<nowiki>5|2</nowiki>

|Bright major

|~ 10/9 6/5 4/3 36/25 8/5 5/3 9/5 2/1

|B-E-G♯

|C-F-A

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|<nowiki>4|3</nowiki>

|Middle major

|~ 10/9 6/5 4/3 36/25 3/2 5/3 9/5 2/1

|C-F-A

|D-G-B

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|<nowiki>3|4</nowiki>

|Dark major

|~ 10/9 6/5 4/3 25/18 3/2 5/3 9/5 2/1

|D-G-B

|E-A♭-C = E-H-C

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|<nowiki>2|5</nowiki>

|Bright minor

|~ 10/9 6/5 5/4 25/18 3/2 5/3 9/5 2/1

|E-G♯-C

|F-A-D

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|<nowiki>1|6</nowiki>

|Middle minor

|~ 10/9 9/8 5/4 25/18 3/2 5/3 9/5 2/1

|F-A-D

|G-B-E

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|<nowiki>0|7</nowiki>

|Dark minor

|~ 25/24 9/8 5/4 25/18 3/2 5/3 9/5 2/1

|G-B-E

|A♭-C-F = H-C-F

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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 Father[8] is 5L 3s = (10/9~25/24, 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"

|+Father[8] oneirotonic modes

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!Pinetone octatonic

mode

!~~Notes~~

!Comments

|-

|10/9 6/5 4/3 40/27 8/5 16/9 50/27 2/1

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!Pinetone octatonic

mode

!~~Notes~~

!Comments

|-

|10/9 6/5 4/3 40/27 8/5 16/9 48/25 2/1

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!Mode as simplest JI pre-image

!Mode in cents

!~~Notes~~

!Comments

|-

|[https://xenpaper.com/#%7B0_175.892c_318.667c_494.559c_670.451c_813.226c_989.118c_1055.884c_1198.660c_1374.551c_1517.327c_1693.219c_1869.117c_2011.886c_2187.777c%7D0_1_2_3_4_5_6_7_8_7_6_5_4_3_2_1_0-.._0-_1-_2-_%5B0_3%5D-_%5B1_4%5D-_%5B2_5%5D-_%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D-_%5B2_5%5D-_%5B1_4%5D-_%5B0_3%5D-_2-_1-_0-.._%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B3_6_9%5D-_%5B4_7_10%5D-_%5B5_8_11%5D-_%5B6_9_12%5D-_%5B7_10_13%5D-_%5B8_11_14%5D-_%5B7_10_13%5D-_%5B6_9_12%5D-_%5B5_8_11%5D-_%5B4_7_10%5D-_%5B3_6_9%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D--- Celephaïsian dark quartal]*

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!Mode as simplest JI pre-image

!Mode in cents

!~~Notes~~

!Comments

|-

|[https://xenpaper.com/#%7B0_175.892c_318.667c_494.559c_670.451c_813.226c_989.118c_1131.983c_1198.660c_1374.551c_1517.327c_1693.219c_1869.117c_2011.886c_2187.777c%7D0_1_2_3_4_5_6_7_8_7_6_5_4_3_2_1_0-.._0-_1-_2-_%5B0_3%5D-_%5B1_4%5D-_%5B2_5%5D-_%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D-_%5B2_5%5D-_%5B1_4%5D-_%5B0_3%5D-_2-_1-_0-.._%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B3_6_9%5D-_%5B4_7_10%5D-_%5B5_8_11%5D-_%5B6_9_12%5D-_%5B7_10_13%5D-_%5B8_11_14%5D-_%5B7_10_13%5D-_%5B6_9_12%5D-_%5B5_8_11%5D-_%5B4_7_10%5D-_%5B3_6_9%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D--- Ultharian bright quartal]*

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!Pinetone diminished

mode

!~~Notes~~

!Comments

|-

|10/9 6/5 4/3 36/25 8/5 216/125 48/25 2/1

Line 1,934:

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!Mode as simplest JI pre-image

!Mode in cents

!~~Notes~~

!Comments

|-

|[https://xenpaper.com/#%7B0c_175.892c_318.667c_494.559c_637.334c_813.226c_956.002c_1131.893c_1198.660c_1374.552c_1517.327c_1693.219c_1835.994c_2011.886c_2154.661c%7D0_1_2_3_4_5_6_7_8_7_6_5_4_3_2_1_0-.._0-_1-_2-_%5B0_3%5D-_%5B1_4%5D-_%5B2_5%5D-_%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D-_%5B2_5%5D-_%5B1_4%5D-_%5B0_3%5D-_2-_1-_0-.._%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B3_6_9%5D-_%5B4_7_10%5D-_%5B5_8_11%5D-_%5B6_9_12%5D-_%5B7_10_13%5D-_%5B8_11_14%5D-_%5B7_10_13%5D-_%5B6_9_12%5D-_%5B5_8_11%5D-_%5B4_7_10%5D-_%5B3_6_9%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D--- Bright quartal diminished]

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From sMsLMsLMsLMs, putting a small step below the top of each large step (replacing ''L'' with ''sm'', where ''m'' is the small step of the 12-note scale, the medium step of the 15-note scale) leads to

mLm(sm)Lm(sm)Lm(sm)Lm

mLm(sm)Lm(sm)Lm(sm)Lm -> mLmsmLmsmLmsmLm, which we later introduce as Pinetone-15.

mLmsmLmsmLmsmLm, which we later introduce as Pinetone-15.

== Summary for xen-math nerds ==

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On D we get the scale:

174.055 320.69 557.888 704.524 878.579 1025.214 1199.269 on the notes D E F G♯ A B C D

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

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

* Lsmsmms Lydian ♯2 bright major

* Lsmsmms Lydian ♯2 bright major starting on F

* mmsLsms Ionian ♯5 symmetric minor

* mmsLsms Ionian ♯5 symmetric minor starting on C

* msLsmsm Ukranian dorian bright minor

* msLsmsm Ukranian dorian bright minor starting on D

* sLsmsmm Phyrgian dominant dark major

* sLsmsmm Phyrgian dominant dark major starting on E

* msmmsLs harmonic minor dark diminished

* msmmsLs harmonic minor dark diminished starting on A

* smmsLsm Locrian ♮6 bright diminished

* smmsLsm Locrian ♮6 bright diminished starting on B

* smsmmsL altered diminished magical seventh

* smsmmsL altered diminished magical seventh starting on G♯

~~Using~~ an A♭ instead, we get the scale:

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

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

Which has ~~Pinetone harmonic major~~ modes:

Which has modes:

* Lsmmsms Lydian Augmented ♯2 bright major

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

* msLsmms Lydian ♭3 bright minor

* msLsmms Lydian ♭3 bright minor starting on F

* sLsmmsm Mixolydian ♭2 dark major

* sLsmmsm Mixolydian ♭2 dark major starting on G

* mmsmsLs harmonic major bright diminished

* mmsmsLs harmonic major bright diminished starting on C

* msmsLsm Dorian ♭5 dark diminished

* msmsLsm Dorian ♭5 dark diminished starting on D

* smsLsmm Phrygian ♭4 symmetric minor

* smsLsmm Phrygian ♭4 symmetric minor starting on E

* smmsmsL Locrian magical ♭♭7

* smmsmsL Locrian magical ♭♭7 starting on B

== Pinetone hyperchromatic scales ==

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Or, from the Pinetone chromatic with flats (mode 3), we add another Pinetone diatonic scale, mode 0 starting on D♯, leading to the right-handed Pinetone hyperchromatic scale, with step pattern, sLssLsLssLmsLssLsLs.

If 81/80 were additionally tempered out (tempering out the difference between the small step and the medium step), these scales would temper to Flattone[19], reflected in their layout on the lumatone. These scale comprises 7 large steps approximating 117/110 (the difference between the large and small steps of the Pinetone chromatic), the medium step of the Pinetone chromatic, approximating 25/24, 33/32, and 27/26, and 11 small steps, the same as the small step of the ~~pocutone~~ chromatic, approximating 250/243, 55/54, 121/120, and 40/39.

If 81/80 were additionally tempered out (tempering out the difference between the small step and the medium step), these scales would temper to Flattone[19], reflected in their layout on the lumatone. These scale comprises 7 large steps approximating 117/110 (the difference between the large and small steps of the Pinetone chromatic), the medium step of the Pinetone chromatic, approximating 25/24, 33/32, and 27/26, and 11 small steps, the same as the small step of the pinetone chromatic, approximating 250/243, 55/54, 121/120, and 40/39.

We note that sLss, the interval from D to E♯, for example, is very near 9/8, and that sLsL, the interval from D to F♭, for an example, is very near 32/27. If we recognize these approximates, we additionally temper out 243/242, or 352/351, leading to Tetracot temperament, in which case the large step approximates 16/15. This also adds 81/80 to the list of intervals approximated by the small step. Adding an additional small step above G, for the left handed hyperchromatic, or below A, for the right handed hyperchromatic, would give us a MODMOS of Tetracot[20], splitting the one medium step into two small steps (we note also that TE 2.3.5.11.13 ptolemismic tunes the medium step to 66.76626, which is almost exactly twice the size of its small step of 33.11646c).

We note that sLss, the interval from D to E♯, for example, is very near 9/8, and that sLsL, the interval from D to F♭, for an example, is very near 32/27. If we recognize these approximates, we additionally temper out 243/242, or 352/351, leading to [[Tetracot]] temperament, in which case the large step approximates 16/15. This also adds 81/80 to the list of intervals approximated by the small step. Adding an additional small step above G, for the left handed hyperchromatic, or below A, for the right handed hyperchromatic, would give us a MODMOS of Tetracot[20], splitting the one medium step into two small steps (we note also that TE 2.3.5.11.13 ptolemismic tunes the medium step to 66.76626, which is almost exactly twice the size of its small step of 33.11646c).

In 2.3.5.11.13 Tetracot, the left handed Pinetone hyperchromatic approximates the JI ratios 40/39 12/11 10/9 32/27 6/5 11/9 13/10 4/3 11/8 22/15 3/2 20/13 13/8 5/3 16/9 9/5 11/6 39/20 2/1, and the right handed Pinetone hyperchromatic approximates the JI ratios 40/39 12/11 10/9 9/8 6/5 11/9 13/10 4/3 15/11 13/9 3/2 20/13 13/8 5/3 27/16 9/5 11/6 39/20 2/1.

In 2.3.5.11.13 Tetracot, the left handed Pinetone hyperchromatic approximates the JI ratios 40/39 12/11 10/9 32/27 6/5 11/9 13/10 4/3 11/8 22/15 3/2 20/13 13/8 5/3 16/9 9/5 11/6 39/20 2/1, and the right-handed Pinetone hyperchromatic approximates the JI ratios 40/39 12/11 10/9 9/8 6/5 11/9 13/10 4/3 15/11 13/9 3/2 20/13 13/8 5/3 27/16 9/5 11/6 39/20 2/1.

Tuned to [http://x31eq.com/cgi-bin/rt.cgi?ets=7%2613cee&limit=2.3.5.11.13 TE 2.3.5.11.13 Tetracot] (with a large step of 109.3262 and a small step of 33.3391c), the left handed Pinetone hyperchromatic in cents is

Tuned to [http://x31eq.com/cgi-bin/rt.cgi?ets=7%2613cee&limit=2.3.5.11.13 TE 2.3.5.11.13 Tetracot] (with a large step of 109.3262 and a small step of 33.3391c), the left-handed Pinetone hyperchromatic in cents is

33.3391 142.6653 176.0044 285.3306 318.6697 352.0088 461.335 494.6741 561.3532 670.6785 704.0176 737.3567 846.6829 880.022 989.3482 1022.6873 1056.0264 1165.3526 1198.6917,

and the right handed Pinetone hyperchromatic in cents is

and the right-handed Pinetone hyperchromatic in cents is

33.3391 142.6653 176.0044 209.3435 318.6697 352.0088 461.335 494.6741 528.0132 637.3394 704.0176 737.3567 846.6829 880.022 913.3611 1022.6873 1056.0264 1165.3526 1198.6917.

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From Pinetone diminished scale: MLsLMLML, shown in the bright minor mode as Pinetone bright minor diminished, putting a small step into the bottom of each medium and large step leads to the child SNS of the Pinetone diminished scale: the fifteen note SNS msmLmmLmsmLmsmL, or mLmsmLmsmLmsmLm in it's symmetric mode, comprising 4 large steps of 16/15, 8 medium steps of 25/24 and 3 small steps of 648/625, i.e.,

25/24 10/9 125/108 6/5 5/4 4/3 25/18 36/25 3/2 8/5 5/3 216/125 48/25 2/1.

25/24 10/9 125/108 6/5 5/4 4/3 25/18 36/25 3/2 8/5 5/3 216/125 9/5 48/25 2/1.

Tempering m = s (tempering out 15625/15552, the Hanson comma) results in sLsssLsssLsssLs, which is Hanson[15];

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

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 48/25 2/1.

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.

With [http://x31eq.com/cgi-bin/rt.cgi?ets=4f%263f%268&limit=2.3.5.11.13 TE 2.3.5.11.13 ptolemismic tuning applied], the sizes of the steps shift enough for the size order to change. Pinetone-15 comprises

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Line 2,410:

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 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.

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).

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

=== Modified Pinetone octatonics ===

As a subset of Pinetone-15 we may find modified Pinetone octatonics built on MODMOS of Porcupine[8]. The Porcupine[8]'s 4M (''minimally modified MODMOS'') is useful given that it still comprises consonant 3-step triads on all notes, but with a more spread-out distribution, so that the triads of each type do not all occur adjacent to each other as in Porcupine[7] and Porcupine[8]. This scale may be found either by lowering G or raising B by a Porcupine[8] chroma, which represents 16/15, the large step of Pinetone-15.

2.3.5.11.13 ptolemismic Pinetone-15 has simplest JI 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 as sLsmsLsmsLsmsLs which may be grouped as a detempered Porcupine[8], large step of ''sL'' or ''sm'', small step of ''s'' in the following ways: 12222222, 21222222, 22122222, 22212222, 22221222, 22222122, 22222212, 22222221. Using

22122222 -> 13122222, 22131222, 22221312 i.e., s(Lsm)s(Ls)(ms)(Ls)(ms)(Ls) or (sL)(sm)(sL)(sm)(sL)s(msL)s, (sL)(sm)s(Lsm)s(Ls)(ms)(Ls) or (sL)(sm)(sL)s(msL)s(ms)(Ls), (sL)(sm)(sL)(sm)s(Lsm)s(Ls) or (sL)s(msL)s(ms)(Ls)(ms)(Ls), 6 modes of 2 scales.

The first pair of modes temper to Diminished[8]. 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, 2 small steps of 25/24~33/32~27/26.

{| class="wikitable"

|+Modes of the just Pinetone harmonic diminished

!Mode name

!Step pattern

!Oneirotonic step pattern

!Porcupine[8] step pattern

!Mode in 5-limit JI

!Comments

|-

|[https://xenpaper.com/#%7B1%2F1_144%2F125_6%2F5_4%2F3_36%2F25_8%2F5_216%2F125_48%2F25_2%2F1_288%2F125_12%2F5_8%2F3_72%2F25_16%2F5_432%2F125%7D0_1_2_3_4_5_6_7_8_7_6_5_4_3_2_1_0-.._0-_1-_2-_%5B0_3%5D-_%5B1_4%5D-_%5B2_5%5D-_%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D-_%5B2_5%5D-_%5B1_4%5D-_%5B0_3%5D-_2-_1-_0-.._%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B3_6_9%5D-_%5B4_7_10%5D-_%5B5_8_11%5D-_%5B6_9_12%5D-_%5B7_10_13%5D-_%5B8_11_14%5D-_%5B7_10_13%5D-_%5B6_9_12%5D-_%5B5_8_11%5D-_%5B4_7_10%5D-_%5B3_6_9%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D--- Bright augmented ♯2 diminished]

|AsLMLMLs

|sLLsLsLL (Hlanithian)

|AsLLLLLs

|144/125 6/5 4/3 36/25 8/5 216/125 48/25 2/1

|

|-

|[https://xenpaper.com/#%7B1%2F1_10%2F9_6%2F5_4%2F3_36%2F25_8%2F5_5%2F3_48%2F25_2%2F1_20%2F9_12%2F5_8%2F3_72%2F25_16%2F5_10%2F3%7D0_1_2_3_4_5_6_7_8_7_6_5_4_3_2_1_0-.._0-_1-_2-_%5B0_3%5D-_%5B1_4%5D-_%5B2_5%5D-_%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D-_%5B2_5%5D-_%5B1_4%5D-_%5B0_3%5D-_2-_1-_0-.._%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B3_6_9%5D-_%5B4_7_10%5D-_%5B5_8_11%5D-_%5B6_9_12%5D-_%5B7_10_13%5D-_%5B8_11_14%5D-_%5B7_10_13%5D-_%5B6_9_12%5D-_%5B5_8_11%5D-_%5B4_7_10%5D-_%5B3_6_9%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D--- Bright major ♯8 diminished]*

|LMLMLsAs

|LsLsLLsL (Mnarian)

|LLLLLsAs

|10/9 6/5 4/3 36/25 8/5 5/3 48/25 2/1

|

|-

|[https://xenpaper.com/#%7B1%2F1_10%2F9_6%2F5_4%2F3_25%2F18_8%2F5_5%2F3_50%2F27_2%2F1_20%2F9_12%2F5_8%2F3_25%2F9_16%2F5_10%2F3%7D0_1_2_3_4_5_6_7_8_7_6_5_4_3_2_1_0-.._0-_1-_2-_%5B0_3%5D-_%5B1_4%5D-_%5B2_5%5D-_%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D-_%5B2_5%5D-_%5B1_4%5D-_%5B0_3%5D-_2-_1-_0-.._%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B3_6_9%5D-_%5B4_7_10%5D-_%5B5_8_11%5D-_%5B6_9_12%5D-_%5B7_10_13%5D-_%5B8_11_14%5D-_%5B7_10_13%5D-_%5B6_9_12%5D-_%5B5_8_11%5D-_%5B4_7_10%5D-_%5B3_6_9%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D--- Bright major ♭5 / Dark major ♯6 diminished]*

|LMLsAsLM

|LsLLsLLs (Celephaïsian)

|LLLsAsLL

|10/9 6/5 4/3 25/18 8/5 5/3 50/27 2/1

|

|-

|[https://xenpaper.com/#%7B1%2F1_10%2F9_125%2F108_4%2F3_25%2F18_125%2F81_5%2F3_50%2F27_2%2F1_20%2F9_125%2F54_8%2F3_25%2F9_250%2F81_10%2F3%7D0_1_2_3_4_5_6_7_8_7_6_5_4_3_2_1_0-.._0-_1-_2-_%5B0_3%5D-_%5B1_4%5D-_%5B2_5%5D-_%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D-_%5B2_5%5D-_%5B1_4%5D-_%5B0_3%5D-_2-_1-_0-.._%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B3_6_9%5D-_%5B4_7_10%5D-_%5B5_8_11%5D-_%5B6_9_12%5D-_%5B7_10_13%5D-_%5B8_11_14%5D-_%5B7_10_13%5D-_%5B6_9_12%5D-_%5B5_8_11%5D-_%5B4_7_10%5D-_%5B3_6_9%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D--- Dark major ♭3 diminished]*

|LsAsLMLM

|LLsLLsLs (Dylathian)

|LsAsLLLL

|10/9 125/108 4/3 25/18 125/81 5/3 50/27 2/1

|

|-

|[https://xenpaper.com/#%7B1%2F1_27%2F25_6%2F5_162%2F125_36%2F25_3%2F2_216%2F125_9%2F5_2%2F1_54%2F25_12%2F5_324%2F125_72%2F25_3%2F1_432%2F125%7D0_1_2_3_4_5_6_7_8_7_6_5_4_3_2_1_0-.._0-_1-_2-_%5B0_3%5D-_%5B1_4%5D-_%5B2_5%5D-_%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D-_%5B2_5%5D-_%5B1_4%5D-_%5B0_3%5D-_2-_1-_0-.._%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B3_6_9%5D-_%5B4_7_10%5D-_%5B5_8_11%5D-_%5B6_9_12%5D-_%5B7_10_13%5D-_%5B8_11_14%5D-_%5B7_10_13%5D-_%5B6_9_12%5D-_%5B5_8_11%5D-_%5B4_7_10%5D-_%5B3_6_9%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D--- Dark augmented ♭6 diminished]†

|MLMLsAsL

|sLsLLsLL (Sarnathian)

|LLLLsAsL

|27/25 6/5 162/125 36/25 3/2 216/125 9/5 2/1

|10:12:15 on the root

|-

|[https://xenpaper.com/#%7B1%2F1_27%2F25%C2%A06%2F5_5%2F4_36%2F25_3%2F2_5%2F3_9%2F5_2%2F1_54%2F25_12%2F5_5%2F2_72%2F25_3%2F1_10%2F3%7D0_1_2_3_4_5_6_7_8_7_6_5_4_3_2_1_0-.._0-_1-_2-_%5B0_3%5D-_%5B1_4%5D-_%5B2_5%5D-_%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D-_%5B2_5%5D-_%5B1_4%5D-_%5B0_3%5D-_2-_1-_0-.._%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B3_6_9%5D-_%5B4_7_10%5D-_%5B5_8_11%5D-_%5B6_9_12%5D-_%5B7_10_13%5D-_%5B8_11_14%5D-_%5B7_10_13%5D-_%5B6_9_12%5D-_%5B5_8_11%5D-_%5B4_7_10%5D-_%5B3_6_9%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D--- Bright minor ♯5 diminished]*††

|MLsAsLML

|sLLsLLsL (Kadathian)

|LLsAsLLL

|27/25 6/5 5/4 36/25 3/2 5/3 9/5 2/1

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

|-

|[https://xenpaper.com/#%7B1%2F1_25%2F24_6%2F5_5%2F4_25%2F18_3%2F2_5%2F3_9%2F5_2%2F1_25%2F12_12%2F5_5%2F2_25%2F9_3%2F1_10%2F3%7D0_1_2_3_4_5_6_7_8_7_6_5_4_3_2_1_0-.._0-_1-_2-_%5B0_3%5D-_%5B1_4%5D-_%5B2_5%5D-_%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D-_%5B2_5%5D-_%5B1_4%5D-_%5B0_3%5D-_2-_1-_0-.._%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B3_6_9%5D-_%5B4_7_10%5D-_%5B5_8_11%5D-_%5B6_9_12%5D-_%5B7_10_13%5D-_%5B8_11_14%5D-_%5B7_10_13%5D-_%5B6_9_12%5D-_%5B5_8_11%5D-_%5B4_7_10%5D-_%5B3_6_9%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D--- Bright minor ♭2 / Dark minor ♯3 diminished]*††

|sAsLMLML

|LsLLsLsL (Ultharian)

|sAsLLLLL

|25/24 6/5 5/4 25/18 3/2 5/3 9/5 2/1

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

|-

|[https://xenpaper.com/#%7B1%2F1_25%2F24_125%2F108_5%2F4_25%2F18_3%2F2_5%2F3_125%2F72_2%2F1_25%2F12_125%2F54_5%2F2_25%2F9_3%2F1_10%2F3%7D0_1_2_3_4_5_6_7_8_7_6_5_4_3_2_1_0-.._0-_1-_2-_%5B0_3%5D-_%5B1_4%5D-_%5B2_5%5D-_%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D-_%5B2_5%5D-_%5B1_4%5D-_%5B0_3%5D-_2-_1-_0-.._%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B3_6_9%5D-_%5B4_7_10%5D-_%5B5_8_11%5D-_%5B6_9_12%5D-_%5B7_10_13%5D-_%5B8_11_14%5D-_%5B7_10_13%5D-_%5B6_9_12%5D-_%5B5_8_11%5D-_%5B4_7_10%5D-_%5B3_6_9%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D--- Dark minor ♭8 diminished]*†

|sLMLMLsA

|LLsLsLLs (Illarnekian)

|sLLLLLsA

|25/24 125/108 5/4 25/18 3/2 5/3 125/72 2/1

|root 4:5:6

|}

{| class="wikitable"

|+Modes of the ptolemismic Pinetone harmonic diminished

!Mode name

!Step pattern

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

!Mode in cents

!Comments

|-

|[https://xenpaper.com/#%7B0c_209.542c_318.667c_494.559c_637.334c_813.226c_956.002c_1131.893c_1198.660c_1408.201c_1517.327c_1693.219c_1835.994c_2011.886c_2154.661c%7D0_1_2_3_4_5_6_7_8_7_6_5_4_3_2_1_0-.._0-_1-_2-_%5B0_3%5D-_%5B1_4%5D-_%5B2_5%5D-_%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D-_%5B2_5%5D-_%5B1_4%5D-_%5B0_3%5D-_2-_1-_0-.._%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B3_6_9%5D-_%5B4_7_10%5D-_%5B5_8_11%5D-_%5B6_9_12%5D-_%5B7_10_13%5D-_%5B8_11_14%5D-_%5B7_10_13%5D-_%5B6_9_12%5D-_%5B5_8_11%5D-_%5B4_7_10%5D-_%5B3_6_9%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D--- Bright augmented ♯2 diminished]

|AsLMLMLs

|~ 56/45 6/5 4/3 13/9 8/5 26/15 48/25 2/1

|209.542 318.667 494.559 637.334 813.226 956.002 1131.893 1198.660

|

|-

|[https://xenpaper.com/#%7B0c_175.892c_318.667c_494.559c_637.334c_813.226c_879.993c_1131.893c_1198.660c_1374.551c_1517.327c_1693.219c_1835.994c_2011.886c_2078.652c%7D0_1_2_3_4_5_6_7_8_7_6_5_4_3_2_1_0-.._0-_1-_2-_%5B0_3%5D-_%5B1_4%5D-_%5B2_5%5D-_%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D-_%5B2_5%5D-_%5B1_4%5D-_%5B0_3%5D-_2-_1-_0-.._%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B3_6_9%5D-_%5B4_7_10%5D-_%5B5_8_11%5D-_%5B6_9_12%5D-_%5B7_10_13%5D-_%5B8_11_14%5D-_%5B7_10_13%5D-_%5B6_9_12%5D-_%5B5_8_11%5D-_%5B4_7_10%5D-_%5B3_6_9%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D--- Bright major ♯8 diminished]*

|LMLMLsAs

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

|175.892 318.667 494.559 637.334 813.226 879.993 1131.893 1198.660

|

|-

|[https://xenpaper.com/#%7B0c_175.892c_318.667c_494.559c_561.325c_813.226c_879.993c_1055.884c_1198.660c_1374.551c_1517.327c_1693.219c_1759.985c_2011.886c_2078.652c%7D0_1_2_3_4_5_6_7_8_7_6_5_4_3_2_1_0-.._0-_1-_2-_%5B0_3%5D-_%5B1_4%5D-_%5B2_5%5D-_%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D-_%5B2_5%5D-_%5B1_4%5D-_%5B0_3%5D-_2-_1-_0-.._%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B3_6_9%5D-_%5B4_7_10%5D-_%5B5_8_11%5D-_%5B6_9_12%5D-_%5B7_10_13%5D-_%5B8_11_14%5D-_%5B7_10_13%5D-_%5B6_9_12%5D-_%5B5_8_11%5D-_%5B4_7_10%5D-_%5B3_6_9%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D--- Bright major ♭5 / Dark major ♯6 diminished]*

|LMLsAsLM

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

|175.892 318.667 494.559 561.325 813.226 879.993 1055.884 1198.660

|

|-

|[https://xenpaper.com/#%7B0c_175.892c_242.658c_494.559c_561.325c_737.217c_879.993c_1055.884c_1198.660c_1374.551c_1441.318c_1693.219c_1759.985c_1935.877c_2078.652c%7D0_1_2_3_4_5_6_7_8_7_6_5_4_3_2_1_0-.._0-_1-_2-_%5B0_3%5D-_%5B1_4%5D-_%5B2_5%5D-_%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D-_%5B2_5%5D-_%5B1_4%5D-_%5B0_3%5D-_2-_1-_0-.._%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B3_6_9%5D-_%5B4_7_10%5D-_%5B5_8_11%5D-_%5B6_9_12%5D-_%5B7_10_13%5D-_%5B8_11_14%5D-_%5B7_10_13%5D-_%5B6_9_12%5D-_%5B5_8_11%5D-_%5B4_7_10%5D-_%5B3_6_9%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D--- Dark major ♭3 diminished]*

|LsAsLMLM

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

|175.892 242.658 494.559 561.325 737.217 879.993 1055.884 1198.660

|

|-

|[https://xenpaper.com/#%7B0c_142.775c_318.667c_461.443c_637.334c_704.101c_956.002c_1022.768c_1198.660c_1341.435c_1517.327c_1660.322c_1835.994c_1902.760c_2154.661c%7D0_1_2_3_4_5_6_7_8_7_6_5_4_3_2_1_0-.._0-_1-_2-_%5B0_3%5D-_%5B1_4%5D-_%5B2_5%5D-_%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D-_%5B2_5%5D-_%5B1_4%5D-_%5B0_3%5D-_2-_1-_0-.._%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B3_6_9%5D-_%5B4_7_10%5D-_%5B5_8_11%5D-_%5B6_9_12%5D-_%5B7_10_13%5D-_%5B8_11_14%5D-_%5B7_10_13%5D-_%5B6_9_12%5D-_%5B5_8_11%5D-_%5B4_7_10%5D-_%5B3_6_9%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D--- Dark augmented ♭6 diminished]†

|MLMLsAsL

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

|142.775 318.667 461.443 637.334 704.101 956.002 1022.768 1198.660

|10:12:15 on the root

|-

|[https://xenpaper.com/#%7B0c_142.775c_318.667c_385.433c_637.334c_704.101c_879.993c_1022.768c_1198.660c_1341.435c_1517.327c_1584.903c_1835.994c_1902.760c_2078.652c%7D0_1_2_3_4_5_6_7_8_7_6_5_4_3_2_1_0-.._0-_1-_2-_%5B0_3%5D-_%5B1_4%5D-_%5B2_5%5D-_%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D-_%5B2_5%5D-_%5B1_4%5D-_%5B0_3%5D-_2-_1-_0-.._%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B3_6_9%5D-_%5B4_7_10%5D-_%5B5_8_11%5D-_%5B6_9_12%5D-_%5B7_10_13%5D-_%5B8_11_14%5D-_%5B7_10_13%5D-_%5B6_9_12%5D-_%5B5_8_11%5D-_%5B4_7_10%5D-_%5B3_6_9%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D--- Bright minor ♯5 diminished]*††

|MLsAsLML

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

|142.775 318.667 385.433 637.334 704.101 879.993 1022.768 1198.660

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

|-

|[https://xenpaper.com/#%7B0c_66.766c_318.667c_385.433c_561.325c_704.101c_879.993c_1022.768c_1198.660c_1265.426c_1517.327c_1584.903c_1759.985c_1902.760c_2078.652c%7D0_1_2_3_4_5_6_7_8_7_6_5_4_3_2_1_0-.._0-_1-_2-_%5B0_3%5D-_%5B1_4%5D-_%5B2_5%5D-_%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D-_%5B2_5%5D-_%5B1_4%5D-_%5B0_3%5D-_2-_1-_0-.._%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B3_6_9%5D-_%5B4_7_10%5D-_%5B5_8_11%5D-_%5B6_9_12%5D-_%5B7_10_13%5D-_%5B8_11_14%5D-_%5B7_10_13%5D-_%5B6_9_12%5D-_%5B5_8_11%5D-_%5B4_7_10%5D-_%5B3_6_9%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D--- Bright minor ♭2 / Dark minor ♯3 diminished]*††

|sAsLMLML

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

|66.766 318.667 385.433 561.325 704.101 879.993 1022.768 1198.660

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

|-

|[https://xenpaper.com/#%7B0c_66.766c_242.658c_385.433c_561.325c_704.101c_879.993c_989.118c_1198.660c_1265.426c_1441.318c_1584.903c_1759.985c_1902.760c_2078.652c%7D0_1_2_3_4_5_6_7_8_7_6_5_4_3_2_1_0-.._0-_1-_2-_%5B0_3%5D-_%5B1_4%5D-_%5B2_5%5D-_%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D-_%5B2_5%5D-_%5B1_4%5D-_%5B0_3%5D-_2-_1-_0-.._%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B3_6_9%5D-_%5B4_7_10%5D-_%5B5_8_11%5D-_%5B6_9_12%5D-_%5B7_10_13%5D-_%5B8_11_14%5D-_%5B7_10_13%5D-_%5B6_9_12%5D-_%5B5_8_11%5D-_%5B4_7_10%5D-_%5B3_6_9%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D--- Dark minor ♭8 diminished]*†

|sLMLMLsA

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

|66.766 242.658 385.433 561.325 704.101 879.993 989.118 1198.660

|root 4:5:6

|}

Noting the proximity of the Augmented step to 9/8, we might tune to Tetracot temperament by equating 56/45 with 9/8. Tuning the scale to 27edo, 34edo or 41edo results in this equivalence.

{| class="wikitable"

|+3-step stacked triads of the Pinetone harmonic diminished

!Mode name

!Step pattern

!Diminished[8] name

!Porcupine[8] name

!Pinetone octatonic name

!JI triad approximated*

|-

|Bright augmented ♯2 diminished

|AsLMLMLs

|[8] major

|[8] augmented

|[8] major augmented

|15:20:26 (12:16:21)

|-

|Bright major ♯8 diminished

|LMLMLsAs

|[8] major

|3:4:5

|-

|Bright major ♭5 / Dark major ♯6 diminished

|LMLsAsLM

|[8] major

|3:4:5

|-

|Dark major ♭3 diminished

|LsAsLMLM

|[8] major

|3:4:5

|-

| Dark augmented ♭6 diminished

|MLMLsAsL

|[8] minor

|[8] augmented

|[8] minor augmented

|30:39:52 (16:21:28)

|-

| Bright minor ♯5 diminished

|MLsAsLML

|[8] minor

|12:15:20

|-

| Bright minor ♭2 / Dark minor ♯3 diminished

|sAsLMLML

|[8] minor

|12:15:20

|-

| Dark minor ♭8 diminished

|sLMLMLsA

|[8] minor

|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.

== Comma pump ==

@@ Line 965: / Line 965: @@
 !UDP
 !Mode name
+!Mode as simplest JI pre-image
 !3-step stacked triad on root (with G♯)
 !(with A♭ = H)
@@ Line 974: / Line 975: @@
 |<nowiki>7|0</nowiki>
 |Bright quartal
+|~ 10/9 6/5 4/3 36/25 8/5 16/9 48/25 2/1
 |G♯-C-F
 |A-D-G
@@ Line 983: / Line 985: @@
 |<nowiki>6|1</nowiki>
 |Dark quartal
+|~ 10/9 6/5 4/3 36/25 8/5 16/9 9/5 2/1
 |A-D-G
 |B-E-A♭ = B-E-H
@@ Line 992: / Line 995: @@
 |<nowiki>5|2</nowiki>
 |Bright major
+|~ 10/9 6/5 4/3 36/25 8/5 5/3 9/5 2/1
 |B-E-G♯
 |C-F-A
@@ Line 1,001: / Line 1,005: @@
 |<nowiki>4|3</nowiki>
 |Middle major
+|~ 10/9 6/5 4/3 36/25 3/2 5/3 9/5 2/1
 |C-F-A
 |D-G-B
@@ Line 1,010: / Line 1,015: @@
 |<nowiki>3|4</nowiki>
 |Dark major
+|~ 10/9 6/5 4/3 25/18 3/2 5/3 9/5 2/1
 |D-G-B
 |E-A♭-C = E-H-C
@@ Line 1,019: / Line 1,025: @@
 |<nowiki>2|5</nowiki>
 |Bright minor
+|~ 10/9 6/5 5/4 25/18 3/2 5/3 9/5 2/1
 |E-G♯-C
 |F-A-D
@@ Line 1,028: / Line 1,035: @@
 |<nowiki>1|6</nowiki>
 |Middle minor
+|~ 10/9 9/8 5/4 25/18 3/2 5/3 9/5 2/1
 |F-A-D
 |G-B-E
@@ Line 1,037: / Line 1,045: @@
 |<nowiki>0|7</nowiki>
 |Dark minor
+|~ 25/24 9/8 5/4 25/18 3/2 5/3 9/5 2/1
 |G-B-E
 |A♭-C-F = H-C-F
@@ Line 1,047: / Line 1,056: @@
 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 Father[8] is 5L 3s = (10/9~25/24, 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"
 |+Father[8] oneirotonic modes
@@ Line 1,128: / Line 1,137: @@
 !Pinetone octatonic
 mode
-!Notes
+!Comments
 |-
 |10/9 6/5 4/3 40/27 8/5 16/9 50/27 2/1
@@ Line 1,216: / Line 1,225: @@
 !Pinetone octatonic
 mode
-!Notes
+!Comments
 |-
 |10/9 6/5 4/3 40/27 8/5 16/9 48/25 2/1
@@ Line 1,301: / Line 1,310: @@
 !Mode as simplest JI pre-image
 !Mode in cents
-!Notes
+!Comments
 |-
 |[https://xenpaper.com/#%7B0_175.892c_318.667c_494.559c_670.451c_813.226c_989.118c_1055.884c_1198.660c_1374.551c_1517.327c_1693.219c_1869.117c_2011.886c_2187.777c%7D0_1_2_3_4_5_6_7_8_7_6_5_4_3_2_1_0-.._0-_1-_2-_%5B0_3%5D-_%5B1_4%5D-_%5B2_5%5D-_%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D-_%5B2_5%5D-_%5B1_4%5D-_%5B0_3%5D-_2-_1-_0-.._%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B3_6_9%5D-_%5B4_7_10%5D-_%5B5_8_11%5D-_%5B6_9_12%5D-_%5B7_10_13%5D-_%5B8_11_14%5D-_%5B7_10_13%5D-_%5B6_9_12%5D-_%5B5_8_11%5D-_%5B4_7_10%5D-_%5B3_6_9%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D--- Celephaïsian dark quartal]*
@@ Line 1,357: / Line 1,366: @@
 !Mode as simplest JI pre-image
 !Mode in cents
-!Notes
+!Comments
 |-
 |[https://xenpaper.com/#%7B0_175.892c_318.667c_494.559c_670.451c_813.226c_989.118c_1131.983c_1198.660c_1374.551c_1517.327c_1693.219c_1869.117c_2011.886c_2187.777c%7D0_1_2_3_4_5_6_7_8_7_6_5_4_3_2_1_0-.._0-_1-_2-_%5B0_3%5D-_%5B1_4%5D-_%5B2_5%5D-_%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D-_%5B2_5%5D-_%5B1_4%5D-_%5B0_3%5D-_2-_1-_0-.._%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B3_6_9%5D-_%5B4_7_10%5D-_%5B5_8_11%5D-_%5B6_9_12%5D-_%5B7_10_13%5D-_%5B8_11_14%5D-_%5B7_10_13%5D-_%5B6_9_12%5D-_%5B5_8_11%5D-_%5B4_7_10%5D-_%5B3_6_9%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D--- Ultharian bright quartal]*
@@ Line 1,862: / Line 1,871: @@
 !Pinetone diminished
 mode
-!Notes
+!Comments
 |-
 |10/9 6/5 4/3 36/25 8/5 216/125 48/25 2/1
@@ Line 1,934: / Line 1,943: @@
 !Mode as simplest JI pre-image
 !Mode in cents
-!Notes
+!Comments
 |-
 |[https://xenpaper.com/#%7B0c_175.892c_318.667c_494.559c_637.334c_813.226c_956.002c_1131.893c_1198.660c_1374.552c_1517.327c_1693.219c_1835.994c_2011.886c_2154.661c%7D0_1_2_3_4_5_6_7_8_7_6_5_4_3_2_1_0-.._0-_1-_2-_%5B0_3%5D-_%5B1_4%5D-_%5B2_5%5D-_%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D-_%5B2_5%5D-_%5B1_4%5D-_%5B0_3%5D-_2-_1-_0-.._%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B3_6_9%5D-_%5B4_7_10%5D-_%5B5_8_11%5D-_%5B6_9_12%5D-_%5B7_10_13%5D-_%5B8_11_14%5D-_%5B7_10_13%5D-_%5B6_9_12%5D-_%5B5_8_11%5D-_%5B4_7_10%5D-_%5B3_6_9%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D--- Bright quartal diminished]
@@ Line 2,287: / Line 2,296: @@
 From sMsLMsLMsLMs, putting a small step below the top of each large step (replacing ''L'' with ''sm'', where ''m'' is the small step of the 12-note scale, the medium step of the 15-note scale) leads to
-mLm(sm)Lm(sm)Lm(sm)Lm
+mLm(sm)Lm(sm)Lm(sm)Lm -> mLmsmLmsmLmsmLm, which we later introduce as Pinetone-15.
-mLmsmLmsmLmsmLm, which we later introduce as Pinetone-15.
 == Summary for xen-math nerds ==
@@ Line 2,313: / Line 2,320: @@
 On D we get the scale:
-.055 320.69 557.888 704.524 878.579 1025.214 1199.269 on the notes D E F G♯ A B C D
+.055 320.69 557.888 704.524 878.579 1025.214 1199.269 as the notes D E F G♯ A B C D
 We get the following 7 modes of Pinetone harmonic minor scale:
-* Lsmsmms Lydian ♯2 bright major
+* Lsmsmms Lydian ♯2 bright major starting on F
-* mmsLsms Ionian ♯5 symmetric minor
+* mmsLsms Ionian ♯5 symmetric minor starting on C
-* msLsmsm Ukranian dorian bright minor
+* msLsmsm Ukranian dorian bright minor starting on D
-* sLsmsmm Phyrgian dominant dark major
+* sLsmsmm Phyrgian dominant dark major starting on E
-* msmmsLs harmonic minor dark diminished
+* msmmsLs harmonic minor dark diminished starting on A
-* smmsLsm Locrian ♮6 bright diminished
+* smmsLsm Locrian ♮6 bright diminished starting on B
-* smsmmsL altered diminished magical seventh
+* smsmmsL altered diminished magical seventh starting on G♯
-Using an A♭ instead, we get the scale:
+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
+.055 320.69 494.745 641.38 878.579 1025.214 1199.269 as the notes D E F G A♭ B C D
-Which has Pinetone harmonic major modes:
+Which has modes:
-* Lsmmsms Lydian Augmented ♯2 bright major
+* Lsmmsms Lydian Augmented ♯2 bright major starting on A♭
-* msLsmms Lydian ♭3 bright minor
+* msLsmms Lydian ♭3 bright minor starting on F
-* sLsmmsm Mixolydian ♭2 dark major
+* sLsmmsm Mixolydian ♭2 dark major starting on G
-* mmsmsLs harmonic major bright diminished
+* mmsmsLs harmonic major bright diminished starting on C
-* msmsLsm Dorian ♭5 dark diminished
+* msmsLsm Dorian ♭5 dark diminished starting on D
-* smsLsmm Phrygian ♭4 symmetric minor
+* smsLsmm Phrygian ♭4 symmetric minor starting on E
-* smmsmsL Locrian magical ♭♭7
+* smmsmsL Locrian magical ♭♭7 starting on B
 == Pinetone hyperchromatic scales ==
@@ Line 2,348: / Line 2,355: @@
 Or, from the Pinetone chromatic with flats (mode 3), we add another Pinetone diatonic scale, mode 0 starting on D♯, leading to the right-handed Pinetone hyperchromatic scale, with step pattern, sLssLsLssLmsLssLsLs.
-If 81/80 were additionally tempered out (tempering out the difference between the small step and the medium step), these scales would temper to Flattone[19], reflected in their layout on the lumatone. These scale comprises 7 large steps approximating 117/110 (the difference between the large and small steps of the Pinetone chromatic), the medium step of the Pinetone chromatic, approximating 25/24, 33/32, and 27/26, and 11 small steps, the same as the small step of the pocutone chromatic, approximating 250/243, 55/54, 121/120, and 40/39.
+If 81/80 were additionally tempered out (tempering out the difference between the small step and the medium step), these scales would temper to Flattone[19], reflected in their layout on the lumatone. These scale comprises 7 large steps approximating 117/110 (the difference between the large and small steps of the Pinetone chromatic), the medium step of the Pinetone chromatic, approximating 25/24, 33/32, and 27/26, and 11 small steps, the same as the small step of the pinetone chromatic, approximating 250/243, 55/54, 121/120, and 40/39.
-We note that sLss, the interval from D to E♯, for example, is very near 9/8, and that sLsL, the interval from D to F♭, for an example, is very near 32/27. If we recognize these approximates, we additionally temper out 243/242, or 352/351, leading to Tetracot temperament, in which case the large step approximates 16/15. This also adds 81/80 to the list of intervals approximated by the small step. Adding an additional small step above G, for the left handed hyperchromatic, or below A, for the right handed hyperchromatic, would give us a MODMOS of Tetracot[20], splitting the one medium step into two small steps (we note also that TE 2.3.5.11.13 ptolemismic tunes the medium step to 66.76626, which is almost exactly twice the size of its small step of 33.11646c).
+We note that sLss, the interval from D to E♯, for example, is very near 9/8, and that sLsL, the interval from D to F♭, for an example, is very near 32/27. If we recognize these approximates, we additionally temper out 243/242, or 352/351, leading to [[Tetracot]] temperament, in which case the large step approximates 16/15. This also adds 81/80 to the list of intervals approximated by the small step. Adding an additional small step above G, for the left handed hyperchromatic, or below A, for the right handed hyperchromatic, would give us a MODMOS of Tetracot[20], splitting the one medium step into two small steps (we note also that TE 2.3.5.11.13 ptolemismic tunes the medium step to 66.76626, which is almost exactly twice the size of its small step of 33.11646c).
-In 2.3.5.11.13 Tetracot, the left handed Pinetone hyperchromatic approximates the JI ratios 40/39 12/11 10/9 32/27 6/5 11/9 13/10 4/3 11/8 22/15 3/2 20/13 13/8 5/3 16/9 9/5 11/6 39/20 2/1, and the right handed Pinetone hyperchromatic approximates the JI ratios 40/39 12/11 10/9 9/8 6/5 11/9 13/10 4/3 15/11 13/9 3/2 20/13 13/8 5/3 27/16 9/5 11/6 39/20 2/1.
+In 2.3.5.11.13 Tetracot, the left handed Pinetone hyperchromatic approximates the JI ratios 40/39 12/11 10/9 32/27 6/5 11/9 13/10 4/3 11/8 22/15 3/2 20/13 13/8 5/3 16/9 9/5 11/6 39/20 2/1, and the right-handed Pinetone hyperchromatic approximates the JI ratios 40/39 12/11 10/9 9/8 6/5 11/9 13/10 4/3 15/11 13/9 3/2 20/13 13/8 5/3 27/16 9/5 11/6 39/20 2/1.
-Tuned to [http://x31eq.com/cgi-bin/rt.cgi?ets=7%2613cee&limit=2.3.5.11.13 TE 2.3.5.11.13 Tetracot] (with a large step of 109.3262 and a small step of 33.3391c), the left handed Pinetone hyperchromatic in cents is
+Tuned to [http://x31eq.com/cgi-bin/rt.cgi?ets=7%2613cee&limit=2.3.5.11.13 TE 2.3.5.11.13 Tetracot] (with a large step of 109.3262 and a small step of 33.3391c), the left-handed Pinetone hyperchromatic in cents is
 .3391 142.6653 176.0044 285.3306 318.6697 352.0088 461.335 494.6741 561.3532 670.6785 704.0176 737.3567 846.6829 880.022 989.3482 1022.6873 1056.0264 1165.3526 1198.6917,
-and the right handed Pinetone hyperchromatic in cents is
+and the right-handed Pinetone hyperchromatic in cents is
 .3391 142.6653 176.0044 209.3435 318.6697 352.0088 461.335 494.6741 528.0132 637.3394 704.0176 737.3567 846.6829 880.022 913.3611 1022.6873 1056.0264 1165.3526 1198.6917.
@@ Line 2,375: / Line 2,382: @@
 From Pinetone diminished scale: MLsLMLML, shown in the bright minor mode as Pinetone bright minor diminished, putting a small step into the bottom of each medium and large step leads to the child SNS of the Pinetone diminished scale: the fifteen note SNS msmLmmLmsmLmsmL, or mLmsmLmsmLmsmLm in it's symmetric mode, comprising 4 large steps of 16/15, 8 medium steps of 25/24 and 3 small steps of 648/625, i.e.,
-/24 10/9 125/108 6/5 5/4 4/3 25/18 36/25 3/2 8/5 5/3 216/125 48/25 2/1.
+/24 10/9 125/108 6/5 5/4 4/3 25/18 36/25 3/2 8/5 5/3 216/125 9/5 48/25 2/1.
 Tempering m = s (tempering out 15625/15552, the Hanson comma) results in sLsssLsssLsssLs, which is Hanson[15];
@@ Line 2,387: / Line 2,394: @@
 tempering out m would lead to ssLsLssLsssL, which is a MODMOS of Diminished[12].
-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 48/25 2/1.
+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.
 With [http://x31eq.com/cgi-bin/rt.cgi?ets=4f%263f%268&limit=2.3.5.11.13 TE 2.3.5.11.13 ptolemismic tuning applied], the sizes of the steps shift enough for the size order to change. Pinetone-15 comprises
@@ Line 2,403: / Line 2,410: @@
 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 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.
+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).
-.3.5.13 325/324 may be better tuned to 46edo, with (L, m, s) = (4, 2, 1)
+=== Modified Pinetone octatonics ===
+As a subset of Pinetone-15 we may find modified Pinetone octatonics built on MODMOS of Porcupine[8]. The Porcupine[8]'s 4M (''minimally modified MODMOS'') is useful given that it still comprises consonant 3-step triads on all notes, but with a more spread-out distribution, so that the triads of each type do not all occur adjacent to each other as in Porcupine[7] and Porcupine[8]. This scale may be found either by lowering G or raising B by a Porcupine[8] chroma, which represents 16/15, the large step of Pinetone-15.
+.3.5.11.13 ptolemismic Pinetone-15 has simplest JI 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 as sLsmsLsmsLsmsLs which may be grouped as a detempered Porcupine[8], large step of ''sL'' or ''sm'', small step of ''s'' in the following ways: 12222222, 21222222, 22122222, 22212222, 22221222, 22222122, 22222212, 22222221. Using
+22122222 -> 13122222, 22131222, 22221312 i.e., s(Lsm)s(Ls)(ms)(Ls)(ms)(Ls) or (sL)(sm)(sL)(sm)(sL)s(msL)s, (sL)(sm)s(Lsm)s(Ls)(ms)(Ls) or (sL)(sm)(sL)s(msL)s(ms)(Ls), (sL)(sm)(sL)(sm)s(Lsm)s(Ls) or (sL)s(msL)s(ms)(Ls)(ms)(Ls), 6 modes of 2 scales.
+The first pair of modes temper to Diminished[8]. 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, 2 small steps of 25/24~33/32~27/26.
+{| class="wikitable"
+|+Modes of the just Pinetone harmonic diminished
+!Mode name
+!Step pattern
+!Oneirotonic step pattern
+!Porcupine[8] step pattern
+!Mode in 5-limit JI
+!Comments
+|-
+|[https://xenpaper.com/#%7B1%2F1_144%2F125_6%2F5_4%2F3_36%2F25_8%2F5_216%2F125_48%2F25_2%2F1_288%2F125_12%2F5_8%2F3_72%2F25_16%2F5_432%2F125%7D0_1_2_3_4_5_6_7_8_7_6_5_4_3_2_1_0-.._0-_1-_2-_%5B0_3%5D-_%5B1_4%5D-_%5B2_5%5D-_%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D-_%5B2_5%5D-_%5B1_4%5D-_%5B0_3%5D-_2-_1-_0-.._%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B3_6_9%5D-_%5B4_7_10%5D-_%5B5_8_11%5D-_%5B6_9_12%5D-_%5B7_10_13%5D-_%5B8_11_14%5D-_%5B7_10_13%5D-_%5B6_9_12%5D-_%5B5_8_11%5D-_%5B4_7_10%5D-_%5B3_6_9%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D--- Bright augmented ♯2 diminished]
+|AsLMLMLs
+|sLLsLsLL (Hlanithian)
+|AsLLLLLs
+|144/125 6/5 4/3 36/25 8/5 216/125 48/25 2/1
+|
+|-
+|[https://xenpaper.com/#%7B1%2F1_10%2F9_6%2F5_4%2F3_36%2F25_8%2F5_5%2F3_48%2F25_2%2F1_20%2F9_12%2F5_8%2F3_72%2F25_16%2F5_10%2F3%7D0_1_2_3_4_5_6_7_8_7_6_5_4_3_2_1_0-.._0-_1-_2-_%5B0_3%5D-_%5B1_4%5D-_%5B2_5%5D-_%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D-_%5B2_5%5D-_%5B1_4%5D-_%5B0_3%5D-_2-_1-_0-.._%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B3_6_9%5D-_%5B4_7_10%5D-_%5B5_8_11%5D-_%5B6_9_12%5D-_%5B7_10_13%5D-_%5B8_11_14%5D-_%5B7_10_13%5D-_%5B6_9_12%5D-_%5B5_8_11%5D-_%5B4_7_10%5D-_%5B3_6_9%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D--- Bright major ♯8 diminished]*
+|LMLMLsAs
+|LsLsLLsL (Mnarian)
+|LLLLLsAs
+|10/9 6/5 4/3 36/25 8/5 5/3 48/25 2/1
+|
+|-
+|[https://xenpaper.com/#%7B1%2F1_10%2F9_6%2F5_4%2F3_25%2F18_8%2F5_5%2F3_50%2F27_2%2F1_20%2F9_12%2F5_8%2F3_25%2F9_16%2F5_10%2F3%7D0_1_2_3_4_5_6_7_8_7_6_5_4_3_2_1_0-.._0-_1-_2-_%5B0_3%5D-_%5B1_4%5D-_%5B2_5%5D-_%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D-_%5B2_5%5D-_%5B1_4%5D-_%5B0_3%5D-_2-_1-_0-.._%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B3_6_9%5D-_%5B4_7_10%5D-_%5B5_8_11%5D-_%5B6_9_12%5D-_%5B7_10_13%5D-_%5B8_11_14%5D-_%5B7_10_13%5D-_%5B6_9_12%5D-_%5B5_8_11%5D-_%5B4_7_10%5D-_%5B3_6_9%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D--- Bright major ♭5 / Dark major ♯6 diminished]*
+|LMLsAsLM
+|LsLLsLLs (Celephaïsian)
+|LLLsAsLL
+|10/9 6/5 4/3 25/18 8/5 5/3 50/27 2/1
+|
+|-
+|[https://xenpaper.com/#%7B1%2F1_10%2F9_125%2F108_4%2F3_25%2F18_125%2F81_5%2F3_50%2F27_2%2F1_20%2F9_125%2F54_8%2F3_25%2F9_250%2F81_10%2F3%7D0_1_2_3_4_5_6_7_8_7_6_5_4_3_2_1_0-.._0-_1-_2-_%5B0_3%5D-_%5B1_4%5D-_%5B2_5%5D-_%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D-_%5B2_5%5D-_%5B1_4%5D-_%5B0_3%5D-_2-_1-_0-.._%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B3_6_9%5D-_%5B4_7_10%5D-_%5B5_8_11%5D-_%5B6_9_12%5D-_%5B7_10_13%5D-_%5B8_11_14%5D-_%5B7_10_13%5D-_%5B6_9_12%5D-_%5B5_8_11%5D-_%5B4_7_10%5D-_%5B3_6_9%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D--- Dark major ♭3 diminished]*
+|LsAsLMLM
+|LLsLLsLs (Dylathian)
+|LsAsLLLL
+|10/9 125/108 4/3 25/18 125/81 5/3 50/27 2/1
+|
+|-
+|[https://xenpaper.com/#%7B1%2F1_27%2F25_6%2F5_162%2F125_36%2F25_3%2F2_216%2F125_9%2F5_2%2F1_54%2F25_12%2F5_324%2F125_72%2F25_3%2F1_432%2F125%7D0_1_2_3_4_5_6_7_8_7_6_5_4_3_2_1_0-.._0-_1-_2-_%5B0_3%5D-_%5B1_4%5D-_%5B2_5%5D-_%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D-_%5B2_5%5D-_%5B1_4%5D-_%5B0_3%5D-_2-_1-_0-.._%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B3_6_9%5D-_%5B4_7_10%5D-_%5B5_8_11%5D-_%5B6_9_12%5D-_%5B7_10_13%5D-_%5B8_11_14%5D-_%5B7_10_13%5D-_%5B6_9_12%5D-_%5B5_8_11%5D-_%5B4_7_10%5D-_%5B3_6_9%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D--- Dark augmented ♭6 diminished]<sup>†</sup>
+|MLMLsAsL
+|sLsLLsLL (Sarnathian)
+|LLLLsAsL
+|27/25 6/5 162/125 36/25 3/2 216/125 9/5 2/1
+|10:12:15 on the root
+|-
+|[https://xenpaper.com/#%7B1%2F1_27%2F25%C2%A06%2F5_5%2F4_36%2F25_3%2F2_5%2F3_9%2F5_2%2F1_54%2F25_12%2F5_5%2F2_72%2F25_3%2F1_10%2F3%7D0_1_2_3_4_5_6_7_8_7_6_5_4_3_2_1_0-.._0-_1-_2-_%5B0_3%5D-_%5B1_4%5D-_%5B2_5%5D-_%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D-_%5B2_5%5D-_%5B1_4%5D-_%5B0_3%5D-_2-_1-_0-.._%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B3_6_9%5D-_%5B4_7_10%5D-_%5B5_8_11%5D-_%5B6_9_12%5D-_%5B7_10_13%5D-_%5B8_11_14%5D-_%5B7_10_13%5D-_%5B6_9_12%5D-_%5B5_8_11%5D-_%5B4_7_10%5D-_%5B3_6_9%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D--- Bright minor ♯5 diminished]*<sup>††</sup>
+|MLsAsLML
+|sLLsLLsL (Kadathian)
+|LLsAsLLL
+|27/25 6/5 5/4 36/25 3/2 5/3 9/5 2/1
+|root 4:5:6,10:12:15
+|-
+|[https://xenpaper.com/#%7B1%2F1_25%2F24_6%2F5_5%2F4_25%2F18_3%2F2_5%2F3_9%2F5_2%2F1_25%2F12_12%2F5_5%2F2_25%2F9_3%2F1_10%2F3%7D0_1_2_3_4_5_6_7_8_7_6_5_4_3_2_1_0-.._0-_1-_2-_%5B0_3%5D-_%5B1_4%5D-_%5B2_5%5D-_%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D-_%5B2_5%5D-_%5B1_4%5D-_%5B0_3%5D-_2-_1-_0-.._%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B3_6_9%5D-_%5B4_7_10%5D-_%5B5_8_11%5D-_%5B6_9_12%5D-_%5B7_10_13%5D-_%5B8_11_14%5D-_%5B7_10_13%5D-_%5B6_9_12%5D-_%5B5_8_11%5D-_%5B4_7_10%5D-_%5B3_6_9%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D--- Bright minor ♭2 / Dark minor ♯3 diminished]*<sup>††</sup>
+|sAsLMLML
+|LsLLsLsL (Ultharian)
+|sAsLLLLL
+|25/24 6/5 5/4 25/18 3/2 5/3 9/5 2/1
+|root 4:5:6,10:12:15
+|-
+|[https://xenpaper.com/#%7B1%2F1_25%2F24_125%2F108_5%2F4_25%2F18_3%2F2_5%2F3_125%2F72_2%2F1_25%2F12_125%2F54_5%2F2_25%2F9_3%2F1_10%2F3%7D0_1_2_3_4_5_6_7_8_7_6_5_4_3_2_1_0-.._0-_1-_2-_%5B0_3%5D-_%5B1_4%5D-_%5B2_5%5D-_%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D-_%5B2_5%5D-_%5B1_4%5D-_%5B0_3%5D-_2-_1-_0-.._%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B3_6_9%5D-_%5B4_7_10%5D-_%5B5_8_11%5D-_%5B6_9_12%5D-_%5B7_10_13%5D-_%5B8_11_14%5D-_%5B7_10_13%5D-_%5B6_9_12%5D-_%5B5_8_11%5D-_%5B4_7_10%5D-_%5B3_6_9%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D--- Dark minor ♭8 diminished]*<sup>†</sup>
+|sLMLMLsA
+|LLsLsLLs (Illarnekian)
+|sLLLLLsA
+|25/24 125/108 5/4 25/18 3/2 5/3 125/72 2/1
+|root 4:5:6
+|}
+{| class="wikitable"
+|+Modes of the ptolemismic Pinetone harmonic diminished
+!Mode name
+!Step pattern
+!Mode as simplest JI pre-image 5-limit JI
+!Mode in cents
+!Comments
+|-
+|[https://xenpaper.com/#%7B0c_209.542c_318.667c_494.559c_637.334c_813.226c_956.002c_1131.893c_1198.660c_1408.201c_1517.327c_1693.219c_1835.994c_2011.886c_2154.661c%7D0_1_2_3_4_5_6_7_8_7_6_5_4_3_2_1_0-.._0-_1-_2-_%5B0_3%5D-_%5B1_4%5D-_%5B2_5%5D-_%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D-_%5B2_5%5D-_%5B1_4%5D-_%5B0_3%5D-_2-_1-_0-.._%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B3_6_9%5D-_%5B4_7_10%5D-_%5B5_8_11%5D-_%5B6_9_12%5D-_%5B7_10_13%5D-_%5B8_11_14%5D-_%5B7_10_13%5D-_%5B6_9_12%5D-_%5B5_8_11%5D-_%5B4_7_10%5D-_%5B3_6_9%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D--- Bright augmented ♯2 diminished]
+|AsLMLMLs
+|~ 56/45 6/5 4/3 13/9 8/5 26/15 48/25 2/1
+|209.542 318.667 494.559 637.334 813.226 956.002 1131.893 1198.660
+|
+|-
+|[https://xenpaper.com/#%7B0c_175.892c_318.667c_494.559c_637.334c_813.226c_879.993c_1131.893c_1198.660c_1374.551c_1517.327c_1693.219c_1835.994c_2011.886c_2078.652c%7D0_1_2_3_4_5_6_7_8_7_6_5_4_3_2_1_0-.._0-_1-_2-_%5B0_3%5D-_%5B1_4%5D-_%5B2_5%5D-_%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D-_%5B2_5%5D-_%5B1_4%5D-_%5B0_3%5D-_2-_1-_0-.._%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B3_6_9%5D-_%5B4_7_10%5D-_%5B5_8_11%5D-_%5B6_9_12%5D-_%5B7_10_13%5D-_%5B8_11_14%5D-_%5B7_10_13%5D-_%5B6_9_12%5D-_%5B5_8_11%5D-_%5B4_7_10%5D-_%5B3_6_9%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D--- Bright major ♯8 diminished]*
+|LMLMLsAs
+|~ 10/9 6/5 4/3 13/9 8/5 5/3 48/25 2/1
+|175.892 318.667 494.559 637.334 813.226 879.993 1131.893 1198.660
+|
+|-
+|[https://xenpaper.com/#%7B0c_175.892c_318.667c_494.559c_561.325c_813.226c_879.993c_1055.884c_1198.660c_1374.551c_1517.327c_1693.219c_1759.985c_2011.886c_2078.652c%7D0_1_2_3_4_5_6_7_8_7_6_5_4_3_2_1_0-.._0-_1-_2-_%5B0_3%5D-_%5B1_4%5D-_%5B2_5%5D-_%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D-_%5B2_5%5D-_%5B1_4%5D-_%5B0_3%5D-_2-_1-_0-.._%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B3_6_9%5D-_%5B4_7_10%5D-_%5B5_8_11%5D-_%5B6_9_12%5D-_%5B7_10_13%5D-_%5B8_11_14%5D-_%5B7_10_13%5D-_%5B6_9_12%5D-_%5B5_8_11%5D-_%5B4_7_10%5D-_%5B3_6_9%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D--- Bright major ♭5 / Dark major ♯6 diminished]*
+|LMLsAsLM
+|~ 10/9 6/5 4/3 11/8 8/5 5/3 11/6 2/1
+|175.892 318.667 494.559 561.325 813.226 879.993 1055.884 1198.660
+|
+|-
+|[https://xenpaper.com/#%7B0c_175.892c_242.658c_494.559c_561.325c_737.217c_879.993c_1055.884c_1198.660c_1374.551c_1441.318c_1693.219c_1759.985c_1935.877c_2078.652c%7D0_1_2_3_4_5_6_7_8_7_6_5_4_3_2_1_0-.._0-_1-_2-_%5B0_3%5D-_%5B1_4%5D-_%5B2_5%5D-_%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D-_%5B2_5%5D-_%5B1_4%5D-_%5B0_3%5D-_2-_1-_0-.._%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B3_6_9%5D-_%5B4_7_10%5D-_%5B5_8_11%5D-_%5B6_9_12%5D-_%5B7_10_13%5D-_%5B8_11_14%5D-_%5B7_10_13%5D-_%5B6_9_12%5D-_%5B5_8_11%5D-_%5B4_7_10%5D-_%5B3_6_9%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D--- Dark major ♭3 diminished]*
+|LsAsLMLM
+|~ 10/9 15/13 4/3 11/8 20/13 5/3 11/6 2/1
+|175.892 242.658 494.559 561.325 737.217 879.993 1055.884 1198.660
+|
+|-
+|[https://xenpaper.com/#%7B0c_142.775c_318.667c_461.443c_637.334c_704.101c_956.002c_1022.768c_1198.660c_1341.435c_1517.327c_1660.322c_1835.994c_1902.760c_2154.661c%7D0_1_2_3_4_5_6_7_8_7_6_5_4_3_2_1_0-.._0-_1-_2-_%5B0_3%5D-_%5B1_4%5D-_%5B2_5%5D-_%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D-_%5B2_5%5D-_%5B1_4%5D-_%5B0_3%5D-_2-_1-_0-.._%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B3_6_9%5D-_%5B4_7_10%5D-_%5B5_8_11%5D-_%5B6_9_12%5D-_%5B7_10_13%5D-_%5B8_11_14%5D-_%5B7_10_13%5D-_%5B6_9_12%5D-_%5B5_8_11%5D-_%5B4_7_10%5D-_%5B3_6_9%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D--- Dark augmented ♭6 diminished]<sup>†</sup>
+|MLMLsAsL
+|~ 12/11 6/5 13/10 13/9 3/2 26/15 9/5 2/1
+|142.775 318.667 461.443 637.334 704.101 956.002 1022.768 1198.660
+|10:12:15 on the root
+|-
+|[https://xenpaper.com/#%7B0c_142.775c_318.667c_385.433c_637.334c_704.101c_879.993c_1022.768c_1198.660c_1341.435c_1517.327c_1584.903c_1835.994c_1902.760c_2078.652c%7D0_1_2_3_4_5_6_7_8_7_6_5_4_3_2_1_0-.._0-_1-_2-_%5B0_3%5D-_%5B1_4%5D-_%5B2_5%5D-_%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D-_%5B2_5%5D-_%5B1_4%5D-_%5B0_3%5D-_2-_1-_0-.._%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B3_6_9%5D-_%5B4_7_10%5D-_%5B5_8_11%5D-_%5B6_9_12%5D-_%5B7_10_13%5D-_%5B8_11_14%5D-_%5B7_10_13%5D-_%5B6_9_12%5D-_%5B5_8_11%5D-_%5B4_7_10%5D-_%5B3_6_9%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D--- Bright minor ♯5 diminished]*<sup>††</sup>
+|MLsAsLML
+|~ 12/11 6/5 5/4 13/9 3/2 5/3 9/5 2/1
+|142.775 318.667 385.433 637.334 704.101 879.993 1022.768 1198.660
+|root 4:5:6,10:12:15
+|-
+|[https://xenpaper.com/#%7B0c_66.766c_318.667c_385.433c_561.325c_704.101c_879.993c_1022.768c_1198.660c_1265.426c_1517.327c_1584.903c_1759.985c_1902.760c_2078.652c%7D0_1_2_3_4_5_6_7_8_7_6_5_4_3_2_1_0-.._0-_1-_2-_%5B0_3%5D-_%5B1_4%5D-_%5B2_5%5D-_%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D-_%5B2_5%5D-_%5B1_4%5D-_%5B0_3%5D-_2-_1-_0-.._%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B3_6_9%5D-_%5B4_7_10%5D-_%5B5_8_11%5D-_%5B6_9_12%5D-_%5B7_10_13%5D-_%5B8_11_14%5D-_%5B7_10_13%5D-_%5B6_9_12%5D-_%5B5_8_11%5D-_%5B4_7_10%5D-_%5B3_6_9%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D--- Bright minor ♭2 / Dark minor ♯3 diminished]*<sup>††</sup>
+|sAsLMLML
+|~ 25/24 6/5 5/4 11/8 3/2 5/3 9/5 2/1
+|66.766 318.667 385.433 561.325 704.101 879.993 1022.768 1198.660
+|root 4:5:6,10:12:15
+|-
+|[https://xenpaper.com/#%7B0c_66.766c_242.658c_385.433c_561.325c_704.101c_879.993c_989.118c_1198.660c_1265.426c_1441.318c_1584.903c_1759.985c_1902.760c_2078.652c%7D0_1_2_3_4_5_6_7_8_7_6_5_4_3_2_1_0-.._0-_1-_2-_%5B0_3%5D-_%5B1_4%5D-_%5B2_5%5D-_%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D-_%5B2_5%5D-_%5B1_4%5D-_%5B0_3%5D-_2-_1-_0-.._%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B3_6_9%5D-_%5B4_7_10%5D-_%5B5_8_11%5D-_%5B6_9_12%5D-_%5B7_10_13%5D-_%5B8_11_14%5D-_%5B7_10_13%5D-_%5B6_9_12%5D-_%5B5_8_11%5D-_%5B4_7_10%5D-_%5B3_6_9%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D--- Dark minor ♭8 diminished]*<sup>†</sup>
+|sLMLMLsA
+|~ 25/24 15/13 5/4 11/8 3/2 5/3 45/26 2/1
+|66.766 242.658 385.433 561.325 704.101 879.993 989.118 1198.660
+|root 4:5:6
+|}
+Noting the proximity of the Augmented step to 9/8, we might tune to Tetracot temperament by equating 56/45 with 9/8. Tuning the scale to 27edo, 34edo or 41edo results in this equivalence.
+{| class="wikitable"
+|+3-step stacked triads of the Pinetone harmonic diminished
+!Mode name
+!Step pattern
+!Diminished[8] name
+!Porcupine[8] name
+!Pinetone octatonic name
+!JI triad approximated*
+|-
+|Bright augmented ♯2 diminished
+|AsLMLMLs
+|[8] major
+|[8] augmented
+|[8] major augmented
+|15:20:26 (12:16:21)
+|-
+|Bright major ♯8 diminished
+|LMLMLsAs
+|[8] major
+|[8] major
+|[8] major
+|3:4:5
+|-
+|Bright major ♭5 / Dark major ♯6 diminished
+|LMLsAsLM
+|[8] major
+|[8] major
+|[8] major
+|3:4:5
+|-
+|Dark major ♭3 diminished
+|LsAsLMLM
+|[8] major
+|[8] major
+|[8] major
+|3:4:5
+|-
+| Dark augmented ♭6 diminished
+|MLMLsAsL
+|[8] minor
+|[8] augmented
+|[8] minor augmented
+|30:39:52 (16:21:28)
+|-
+| Bright minor ♯5 diminished
+|MLsAsLML
+|[8] minor
+|[8] minor
+|[8] minor
+|12:15:20
+|-
+| Bright minor ♭2 / Dark minor ♯3 diminished
+|sAsLMLML
+|[8] minor
+|[8] minor
+|[8] minor
+|12:15:20
+|-
+| Dark minor ♭8 diminished
+|sLMLMLsA
+|[8] minor
+|[8] minor
+|[8] minor
+|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.
 == Comma pump ==