Pinetone: Difference between revisions

!8ve

|-

| rowspan="2" |Major

| rowspan="4" |Major

|-

|-

|-

| rowspan="4" |Minor

| rowspan="2" |Major

| rowspan="2" |Minor-major

|-

|-

| rowspan="2" |Minor

| rowspan="2" |Minor

|-

|}

!8ve

|-

| rowspan="2" |Large

| rowspan="4" |Large

|-

|-

| rowspan="2" |Medium

|-

| rowspan="4" |Small

| rowspan="2" |Medium

|-

|-

| rowspan="2" |Small

| rowspan="2" |Small

|-

|-

|}

!D

|D-F-A

|minor

|minor

!G

|G-B-D

|major

|major

|diminished

|minor diminished

|-

!B

|B-D-F

| diminished

|diminished

|25:30:36

|C-E-G

|major

|}

!Root note

!Triad notes

!Porcupine tetrad

!JI chord approximated by tetrad

|-

|D-F-A-C

|minor 7

|-

!E

|minor 7

|minor 7

| 10:12:15:18

|-

|major minor 7

|major major-minor 7

|-

!G

|half diminished 7

|half diminished 7

|25:30:36:45

|-

|C-E-G-B

|major 7

|major half diminished 7

|27:33:40:50

|}

Also of interest are the quartal triads of the Pinetone diatonic. We describe these as stacked 3-step intervals (fourths) of the scale, with major and minor designating the large and small 3-step intervals (fourths) respectively. This may seem an odd way to describe quartal chords, but it is consistent with the naming scheme I introduce for the Porcupine[7] 3-step (quartal) triads along side the quartal triads of the Pinetone diatonic.

!Triad notes

!Meantone triad

!JI chord approximated by triad

|-

|-

!E

|sus

| 3-step major minor

|11:15:20

|-

|3-step major minor

|sus ♭2 major minor

|-

!G

|sus

| 3-step major minor

!B

|B-E-A

|3-step minor major

|sus minor major

|M4, m4, m4, M4, (m3)

|mM4, m4, m4, mM4, (Mm3)

|27/20, 4/3, 4/3, 65/48, (16/13)

|-

|F-B-E-A-D

|A4, P4, P4, P4, (m3)

|27/20, 4/3, 25/18, 33/25, (40/33)

|-

|A-D-G-C-F

|P4, P4, P4, P4, (M3)

|m4, m4, M4, m4, (M3)

|25:33:44:60:80

|33/25, 4/3, 15/11, 4/3, (5/4)

|D-G-C-E-A-D

|P4, P4, M3, P4, P4

|15:20:27:33:45:60

|4/3, 27/20, 11/9, 15/11, 4/3

|E-A-D-F-B-E

|P4, P4, m3, A4, P4

|15/11, 4/3, 6/5, 11/8, 4/3

|-

|F-B-E-G-C-F

|A4, P4, m3, P4, P4

|18:25:33:40:54:72

|25/18, 33/25, 40/33, 27/20, 4/3

|G-C-F-A-D-G

|P4, P4, M3, P4, P4

| mM4, m4, M3, m4, m4

|15/11, 4/3, 5/4, 33/25, 4/3

|-

|A-D-G-B-E-A

|P4, P4, M3, P4, P4

|27:36:48:60:80:108

|4/3, 4/3, 5/4, 4/3, 27/20

|B-E-A-C-F-B

|P4, P4, m3, P4, A4

|4/3, 15/11, 6/5, 4/3, 11/8

|-

!C

|P4, A4, m3, P4, P4

|27:36:50:60:80:108

|4/3, 25/18, 6/5, 4/3, 27/20

!Pentad root position

!Pentad 1st inversion

|-

|[https://xenpaper.com/#%7B1%2F1_174.055c_320.690c_494.745c_704.524c_878.579c_1025.214c_1199.269c'%7D%5B0_3_6%5D------._%5B0_3_6_9_12%5D------._%5B0_3_6_9_11%5D------._%5B0_3_6_8_11_14%5D------._%5B0_3_5_8_11%5D------._%5B0_2_5_8_11%5D------. Dorian symmetric minor]

|D-G-C-E-A-D

|D-G-B-E-A*

|-

|[https://xenpaper.com/#%7B1%2F1_146.635c_320.690c_530.469c_704.524c_851.159c_1025.214c_1199.269c'%7D%5B0_3_6%5D------._%5B0_3_6_9_12%5D------._%5B0_3_6_9_11%5D------._%5B0_3_6_8_11_14%5D------._%5B0_3_5_8_11%5D------._%5B0_2_5_8_11%5D------. Phrygian bright minor]

|E-A-D

|E-A-D-G-C

|E-A-C-F-B

|E-G-C-F-B

|G-C-F-A-D-G

|G-C-E-A-D

|-

|[https://xenpaper.com/#%7B1%2F1_174.055c_320.690c_494.745c_668.800c_815.435c_989.490c_1199.269c'%7D%5B0_3_6%5D------._%5B0_3_6_9_12%5D------._%5B0_3_6_9_11%5D------._%5B0_3_6_8_11_14%5D------._%5B0_3_5_8_11%5D------._%5B0_2_5_8_11%5D------. Aeolian magical seventh]

|A-D-G-C-F

|A-D-G-C-E

|A-D-F-B-E

|A-C-F-B-E

|-

|[https://xenpaper.com/#%7B1%2F1_174.055c_348.110c_494.745c_668.800c_878.579c_1052.633c_1199.269c'%7D%5B0_3_6%5D------._%5B0_3_6_9_12%5D------._%5B0_3_6_9_11%5D------._%5B0_3_6_8_11_14%5D------._%5B0_3_5_8_11%5D------._%5B0_2_5_8_11%5D------. Ionian bright diminished]

|C-F-B-E-A

|C-F-B-E-G

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

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

{| class="wikitable"

|+(3, 4) and (4, 3) triads of the Ptolemismic Pinetone chromatic mode -3

!Triad class

!Triad in meantone

!Triad in porcupine

!Pinetone triad name

!Triads in cents

|-

! rowspan="2" |D

|(3, 4)

|minor

|minor

|318.667, 704.101

|-

|major

|minor

|minor

|minor

|318.667, 704.101

|-

|dim 4

|min 4

! rowspan="2" |E

|(3, 4)

|minor

|minor

|10:12:15

|318.667, 704.101

|-

|major

|major

! rowspan="2" |F

|(3, 4)

|maj 2

|aug maj 2

|242.658, 704.101

|-

|major

|major

|385.433, 704.101

|-

|(3, 4)

|minor

|minor-major

|22:27:33

|-

|major

|major

! rowspan="2" |G

|(3, 4)

|maj 2

|aug maj 2

|(14:16:21 or 6:7:9)

|-

|major

|major

|-

! rowspan="2" |G♯

|minor

|diminished

|318.667, 670.451

|-

|dim 4

|dim minor 4

|dim min 4 diminished

|461.443, 670.451

|-

! rowspan="2" |A

|(3, 4)

|318.667, 670.451

|-

|major

|diminished

|major diminished

|351.784, 670.451

|-

|minor dim 6

|minor (sub) min 6

|25:30:39

|-

|dim 4 dim 6

|min 4 sub minor 6

|461.443, 780.120

|-

|(3, 4)

|minor

|diminished

|minor diminished

|318.667, 670.451

|-

|major

|diminished

|major diminished

|351.784, 670.451

|-

! rowspan="2" |C

|(3, 4)

|dim min 2

|aug min 2 diminished

|209.008, 670.451

|-

|major

|diminished

|major diminished

|351.784, 670.451

|-

|minor

|minor

|318.667, 704.101

|-

|dim 4

|min 4

We can see that the large step of Tetracot[7] is the medium step of the Pinetone diatonic, and the small step of Tetracot[7] is the small step of the Pinetone diatonic. The large step of the Pinetone diatonic is the augmented second of tetracot[7].

As with the Pinetone diatonic, tuning the Pinetone chromatic to 19edo collapses it to the Meantone[12] (Flattone[12]) chromatic scale. Tuning it to 15edo, 22edo, or 29edo collapses it to Porcupine[8]. Step signatures, mappings and sizes for tunings to 27edo, 34edo, and 41edo are as follows:

7L 1m 4s = (27/25~12/11~13/12, 25/24~33/32~27/26, 81/80~250/243~55/54~121/120~40/39) = (142.6653, 66.6782, 33.3391), which we note is very similar to 2.3.5.11.13 Ptolemismic.  

The Porcupine comma is the small step of the Pinetone chromatic, so tempering the Pinetone chromatic scale to Porcupine leads from 7L 1m 4s = (27/25, 25/24, 250/243) to 7L 1s = (10/9~27/25, 25/24~81/80), which is Porcupine[8]! The Porcupine[7] scale has its large step between G and A, so the eighth note of Porcupine[8] is either G♯ or A♭, adding another small step of Porcupine[7] below A (for G♯) or above G (A♭). Mode -3 or mode 3 of the Pinetone chromatic scale, respectively, are set to D so that this is preserved in Pinetone. This leads to the Pinetone octatonic scales: D E F G G♯/A♭ A B C. In just intonation: 10/9 6/5 4/3 25/18 3/2 5/3 9/5 2/1 with G♯, or 10/9 6/5 4/3 36/25 3/2 5/3 9/5 2/1 with A♭. This scale has 4 large steps of 10/9, 3 medium steps of 27/25, and 1 small step of 25/24. It is not mirror-symmetric, or equivalentely, it is ''[[Chirality|chiral]]'' so it cannot be uniquely defined with a step signature like Meantone[7], Porcupine[7], Porcupine[8], Meantone[12], and the Pinetone diatonic (the Zarlino/Ptolemy just major scale is also not mirror symmetric). Scales that can be uniquely defined by a step signature are called ''step-nested scales''. More on that later. The Pinetone octatonic with G♯ is called the Pinetone major-harmonic octatonic, and the Pinetone octatonic with A♭ is called the Pinetone minor-harmonic octatonic. These names will make sense to the reader after further reading on these scales and the chords they contain, and on the Pintone diminished octatonic introduced below. The mirror inverse of any mode of the Pinetone major-harmonic octatonic is a mode of the Pinetone minor-harmonic octatonic (see [[chirality]]). This is true similarly of the familiar harmonic minor and harmonic major scales.  

!Step pattern

!UDP

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

!(with A♭ = H)

|-

|3

|<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

|[8] augmented

|9:12:16

|-

|1

|<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

|[8] major

|3:4:5

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

|D-G-B

|[8] major

|3:4:5

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

|F-A-D

|[8] minor

|12:15:20

|3:4:5, 9:12:16

|-

|LLsLsLLs

|<nowiki>6|1</nowiki>

|3:4:5, 9:12:16

|-

|LsLLsLLs

|<nowiki>5|2</nowiki>

|3:4:5, 9:12:16

|-

|LsLLsLsL

|<nowiki>4|3</nowiki>

|3:4:5, 9:12:16

|-

|LsLsLLsL

|<nowiki>3|4</nowiki>

|3:4:5, 9:12:16

|-

|sLLsLLsL

|<nowiki>2|5</nowiki>

|3:4:5, 9:12:16

|-

|sLLsLsLL

|<nowiki>1|6</nowiki>

|Hlanithian (lə-NITH-iən)

|[8] major diminished

|-

| -4

|Sarnathian (sar-NA(H)TH-iən), can be shortened to "Sarn"

|[8] minor diminished

|}

For our modes of the left handed and right handed porcupine octatonic scales we prefix the functional mode names for Porcupine[8], with the [[oneirotonic]] mode names associated with Father[8]. Like in the tables of modes of the Pinetone diatonic, the modes are listed in order of brightest, with the brightest mode at the top, and the darkest mode at the bottom.

|Major with 10:12:15 on root

|-

|LsMLMLLM

|<nowiki>LsLLLLLL 1|6</nowiki>

|Minor with 4:5:6 on root

|-

|MLsMLMLL

|<nowiki>LLsLLLLL 2|5</nowiki>

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

|-

|sMLMLLML

|<nowiki>sLLLLLLL 0|7</nowiki>

|

|-

|MLLMLMsL

|<nowiki>LLLLLLsL 6|1</nowiki>

|-

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

|<nowiki>LLLLsLLL 4|3</nowiki>

|Middle major

|Celephaïsian

|[https://xenpaper.com/#%7B1%2F1_10%2F9_6%2F5_5%2F4_25%2F18_3%2F2_5%2F3_50%2F27_2%2F1_20%2F9_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--- Celephaïsian bright minor]*^†^†

|-

|27/25 6/5 162/125 27/20 3/2 81/50 9/5 2/1

|10:12:15 on root

|-

|sLMLLMLM

|<nowiki>sLLLLLLL 0|7</nowiki>

!Mode (height order)

!Step pattern

!Mode in cents

!Comments

|}

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

{| class="wikitable"

!Porcupine[8] name

!Pinetone octatonic name

!size in cents (TE)

|-

! rowspan="3" |1-step

|s

| minor step

|25/24, 33/32, 27/26

|66.766

|9/8

|209.542

|-

|L + s

|minor 5-step

|minor 5-step

|3/2

|704.101

! rowspan="3" |7-step

|3L + 3M + s

|minor 7-step

|minor 7-step

|9/5, 20/11

|4

|-

|G♯-C-F

|[8] minor diminished

|[8] minor augmented

|30:39:52 (7:9:12, 16:21:28)

|LMLLMLsM

|A-D-G

|[8] augmented

|[8] augmented

|LLMLsMLM

|C-F-A

|[8] major

|3:4:5

|-

|Ultharian dark major

|D-G-B

|[8] perfect

|Hlanithian bright minor

|MLsMLMLL

|[8] major diminished

|[8] diminished minor

|8:10:13

|[8] perfect

|[8] minor

|12:15:20

|-

|Mnarian dark minor

|G-B-E

|[8] perfect

|[8] minor

|12:15:20

|+3-step stacked triads of the Ptolemismic Pinetone minor-harmonic octatonic (G-A♭ gamut)

!Mode (rotational order)

!3-step stacked triad on root

!Oneirotonic name

|Ultharian bright quartal

|LMLLMLMs

|[8] perfect

|[8] augmented

|B-E-A♭

|[8] major diminished

|[8] major augmented

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

|LMLMsLML

|D-G-B

|[8] major

|[8] major

|-

|Sarnathian dark major

|[8] minor diminished

|[8] diminished major

|55:72:90 (16:21:26)

|LMsLMLLM

|F-A-D

|[8] minor

|[8] minor

|-

|Dylathian dark minor

|A♭-C-F

|[8] minor

|[8] minor

|12:15:20

|-

|MsLMLLML

|G-B-E

|10:12:15 on root

|-

|<nowiki>LsLLLLLL 1|6</nowiki>

|Middle minor

|[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]

|LMLMLMLs

|175.892 318.667 494.559 637.334 813.226 956.002 1131.893 1198.660

|

|[https://xenpaper.com/#%7B0c_142.775c_318.667c_385.433c_561.325c_704.101c_879.993c_1022.768c_1198.660c_1341.435c_1441.318c_1517.327c_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 diminished]*^†^†

|MLsLMLML

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

|-

!JI ratios approximated*

!size in cents (TE)

|-

! rowspan="3" |1-step

|s

|minor step

|minor step

|minor 3-step

|5/4

|2

|-

|minor 4-step

|25/18, 11/8, 18/13

|4

|-

|4

|-

|2L + 2M + s

|minor 5-step

|3/2

|704.101

|-

|3L + M + s

|2

|-

|3L + 2M + s

|perfect 6-step

{| class="wikitable"

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

!Step pattern

!Diminished[8] name

!Porcupine[8] name

|-

|Bright quartal diminished

|G♯-C♯-F, A-D-G♭

|[8] major

|[8] augmented

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

|-

|Dark quartal diminished

|A♯-D-G, B-E♭-A♭

|[8] minor

|[8] augmented

|[8] minor augmented

|-

|Bright major diminished

|B-E-G♯, C-F-A

|[8] major

|-

|Middle major diminished

|[8] minor

|[8] major

|-

|Dark major diminished

|D-G-B, E♭-A♭-C

|[8] major

|[8] major

|[8] major

|-

|Bright minor diminished

|[8] minor

|[8] minor

|12:15:20

|-

|Middle minor diminished

|F-A♯-D, G♭-B-E♭

|[8] minor

|[8] major minor

|-

|Dark minor diminished

|G-B-E, A♭-C-F

|[8] minor

|[8] minor

|[8] minor

|}

<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.

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

Starting with the 5-limit diminished tetrad 6/5 36/25 5/3 2/1, putting 10/9 above each note takes us to the Pinetone diminished octatonic 10/9 6/5 4/3 36/25 8/5 5/3 50/27 2/1. The 5-limit diminished tetrad has 3 large steps of 6/5, and one small step of 125/108. If we only put 10/9 above the large steps, we get the scale 10/9 6/5 4/3 36/25 5/3 50/27 2/1, with step pattern msmsLms, comprising 3 small steps of 27/25, 3 medium steps of 10/9, and one large step of 125/108. Tempering the small and medium steps together gives us the scale sssssLss, Porcupine[7]. Tempering L=m gives us LsLsLLs, Dicot[7], and tempering s=L gives us sLsLLsL as Sixix[7]. Tempering out 100/99 leads to simplest JI pre-image 10/9 6/5 4/3 16/11 5/3 11/6 2/1, and additionally tempering out 144/143 to 10/9 6/5 4/3 13/9 5/3 11/6 2/1. The Pinetone diminished heptatonic's large step tempers to 55/48 under 2.3.5.11 Ptolemismic, and to 15/13 under 2.3.5.11.13 Ptolemismic.  

{| class="wikitable"

|+Modes of the just Pinetone harmonic diminished

!Step pattern

!Oneirotonic step pattern

!Porcupine[8] step pattern

!Mode in 5-limit JI

|-

|[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--- Hlanithian diminished]

|LLLLsAsL

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

|-

|[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--- Kadathian diminished]*^††

|sAsLLLLL

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

|-

|[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--- Illarnekian diminished]*^†

|sLLLLLsA

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

|}

{| class="wikitable"

|+Modes of the Ptolemismic Pinetone harmonic diminished

!Step pattern

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

!Comments

|-

|LsAsLMLM

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

|

|-

|[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--- Sarnathian diminished]^†

|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--- Kadathian diminished]*^††

|~ 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

!JI ratios approximated*

!size in cents (TE)

|-

! rowspan="4" |1-step

|s

|minor step

|minor step

|15/13 (7/6 or 8/7)

|242.658

|-

|L + M = A + s

|-

! rowspan="3" |3-step

|minor 3-step

|minor 3-step

! rowspan="3" |5-step

|2L + 2M + s = A + L + M + 2s

|minor 5-step

|minor 5-step

|-

|3L + 2M = A + 2L + M + s

|major 5-step

|major 5-step

{| class="wikitable"

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

!Step pattern

!Diminished[8] name

|-

|Mnarian diminished

|[8] major

|[8] major

|-

|Sarnathian diminished

|[8] minor

|[8] augmented

|-

|Celephaïsian diminished

|[8] major

|[8] major

|-

|Kadathian diminished

|[8] minor

|[8] minor

|-

|Dylathian diminished

|[8] major

|[8] major

|-

|Ultharian diminished

|[8] minor

|[8] minor

|-

|Hlanithian diminished

|[8] major

|[8] augmented

|-

|Illarnekian diminished

|[8] minor

|[8] minor