Pinetone: Difference between revisions

!Occurence

|-

|s

|minor 2nd

|minor 2nd

|small 2nd, minor 2nd

|27/25, 12/11

|146.635

|2

|-

|m + s

|minor 3rd

|minor 3rd

|small 3rd, minor 3rd

|6/5, 40/33

|4

|-

|2m + s

|perfect 4th

|minor 4th

|small 4th, minor 4th

|4/3, 33/25

|494.745

|4

|-

|2m + 2s

|diminished 5th

|minor 5th

|small 5th, minor 5th

|36/25, 16/11

|1

|-

|3m + 2s

|minor 6th

|minor 6th

|small 6th, minor 6th

|8/5, 40/33

|2

|-

|4m + 2s

|minor 7th

|16/9, 44/25

|989.490

|1

|}

{| class="wikitable"

!D

|D-F-A

|minor

|minor

!G

|G-B-D

|major

|major

!B

|B-D-F

|diminished

|diminished

|C-E-G

|major

|major diminished

|27:33:40

!Root note

!Triad notes

!Porcupine tetrad

!JI chord approximated by tetrad

|-

|D-F-A-C

|minor 7

|minor 7

|minor 7

|-

!F

|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

|11:15:20

|-

|3-step major minor

|sus ♭2 major minor

|-

!G

|sus

|sus major minor

|11:15:20

!B

|B-E-A

|3-step minor major

|sus minor major

|sus ♯4

|3-step minor major

|6:8:11

|}

|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)

|P4, P4, M3, P4, P4

|M4, m4, M3, m4, m4

|11:15:20:25:33:44

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

!Pentad root position

!Pentad 1st inversion

!Pentad fourth inversion

|-

|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-D-F-B-E

|E-A-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.  

Using the familiar Bosanquet 12-note keyboard mapping (the preset for 12edo), we set the Pinetone diatonic scale to the white keys, starting on D. We than add, on F♯/G♭, the Pinetone penatonic as a set of 5 chromatic keys. There are two options for the chromatic keys, either all sharps or all flats. All sharps makes the Pinetone harmonic minor available, and all flats makes the Pinetone harmonic major available. These scales will be discussed below. In either case, in the just tuning, the chromatic keys give the scale 9/8 5/4 3/2 5/3 2/1, starting from F♯/G♭, tuned to 100/81 (F♯) or 162/125 (G♭) from D. This scale has step pattern msLsL, with step signature and step mapping 2L 1m 2s = (6/5, 9/8, 10/9). The same scale is also available as G-A-B-D-E.   

We are familiar with this scale as the just pentatonic. If we temper m and s together, we get Meantone[5]: ssLsL. If we temper m and L together instead we get a scale called Father[5], tempering out the diatonic semitone 16/15. This mode of Father[5] has step pattern LsLsL. Keep the connection to Father[5] in the back of your minds for now, we'll come back to it. The Pinetone pentatonic is also a subset of the Pinetone diatonic, since Meantone[5] is a subset of Meantone[7]. It is available as G-A-B-D-E. The scale F-G-A-C-D is a mode of the inverse of G-A-B-D-E. The Pinetone pentatonic is [[Chirality|''chiral'']] (i.e., it is not symmetric, unlike the Pinetone diatonic and procutone chromatic scales, which are ''achiral''). There is a pair of Pinetone pentatonic scales, the right-handed Pinetone pentatonic, 9/8 5/4 3/2 5/3 2/1 in JI, it's mirror inverse the left-handed Pinetone pentatonic, 10/9 5/4 3/2 5/3 2/1 in JI, which tempers to ssLsL in Meantone, but to sLLsL in Father. Since 9/8 5/4 3/2 5/3 2/1 is familiar to us as the 5-limit major pentatonic, we could also call the right-handed Pinetone pentatonic the Pinetone major pentatonic; similarly, 6/5 4/3 3/2 9/5 2/1, a mode of the left-handed Pinetone pentatonic is familiar to us as the 5-limit minor pentatonic, we could also call the left-handed Pinetone pentatonic the Pinetone minor pentatonic. D-F-G-A-C gives the Pinetone minor pentatonic, while G-A-B-D-E, F♯-G♯-A♯-C♯-D♯, and G♭-A♭-B♭-D♭-E♭ are Pinetone major pentatonics.  

Adding the right handed Pinetone pentatonic (on F♯/G♭) to the just Pinetone diatonic, a 12-note mirror-symmetric scale with step signature and step mapping of 7L 1m 4s = (27/25, 25/24, 250/243) = (133.2376c, 70.6724c, 49.1661c), i.e., 7 large steps of what was the small step of the just Pinetone diatonic, 1 medium step of the chromatic semitone 25/24, the distance between 6/5 and 5/4, and 4 small steps of 250/243, the porcupine comma, that separates 10/9 from 27/25. For the all sharps scale, we set mode -3 on D (for all flats we set mode 3 on D): 250/243 10/9 6/5 100/81 4/3 25/18 3/2 125/81 5/3 9/5 50/27 2/1, with step pattern sLLsLmLsLLsL.  

The ptolemismic Pinetone chromatic scale is distinctly xenharmonic, and yet is related to the familiar chromatic scale.

Mode -3 has 3/2 perfect fifths available above D, D♯, E, F, F♯, G, G♯, and C♯.

{| class="wikitable"

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

!Triad class

!Triad in meantone

|minor

dim min 4

10:13:15

|318.667, 704.101

|minor

major

major

|10:12:15

|aug maj 2

major

4:5:6

|242.658, 704.101

385.433, 704.101

|-

|(3, 4)

(4, 3)

|22:27:33

4:5:6

385.433, 704.101

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

4:5:6

385.433, 704.101

|-

|G♯

(4, 3)

|minor

|diminished

diminished min 4

dim min 4 diminished

|15:18:22

|minor

major

diminished

|minor diminished

|minor (sub) min 6

min 4 sub minor 6

dim min 4 (sub) dim min 6

|25:30:39

50:65:78

461.443, 780.120

|-

|(3, 4)

(4, 3)

|minor

dim min 4

10:13:15

|318.667, 704.101

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 The Pinetone System. This leads to the Pinetone octatonic scales: D E F G G♯/A♭ A B C. In just intonation: 10/9 6/5 4/3 25/18 3/2 5/3 9/5 2/1 with G♯, or 10/9 6/5 4/3 36/25 3/2 5/3 9/5 2/1 with A♭. This scale has 4 large steps of 10/9, 3 medium steps of 27/25, and 1 small step of 25/24. It is not mirror-symmetric, or equivalentely, it is ''[[Chirality|chiral]]'' so it cannot be uniquely defined with a step signature like Meantone[7], Porcupine[7], Porcupine[8], Meantone[12], and the Pinetone diatonic (the Zarlino/Ptolemy just major scale is also not mirror symmetric). Scales that can be uniquely defined by a step signature are called ''step-nested scales''. More on that later. The Pinetone octatonic with G♯ is called the Pinetone major-harmonic octatonic, and the Pinetone octatonic with A♭ is called the Pinetone minor-harmonic octatonic. These names will make sense to the reader after further reading on these scales and the chords they contain, and on the Pintone diminished octatonic introduced below. The mirror inverse of any mode of the Pinetone major-harmonic octatonic is a mode of the Pinetone minor-harmonic octatonic (see [[chirality]]). This is true similarly of the familiar harmonic minor and harmonic major scales.  

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

{| class="wikitable"

!Mode number

!Step pattern

!UDP

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

!(with A♭ = H)

|-

|3

|<nowiki>6|1</nowiki>

|Dark quartal

|-

|1

|<nowiki>4|3</nowiki>

|Middle major

|3:4:5

|-

|LLLsLLLL

|<nowiki>3|4</nowiki>

|3:4:5

|-

|LLsLLLLL

|<nowiki>2|5</nowiki>

|12:15:20

|-

|LsLLLLLL

|<nowiki>1|6</nowiki>

|12:15:20

|-

|sLLLLLLL

|<nowiki>0|7</nowiki>

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

|G-B-E

|[8] minor

|12:15:20

|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>

|160:200:243

|-

|sLsLLsLL

|<nowiki>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

!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)

!Occurence

major step

major step

351.784

1

major 5-step

major-minor 5-step

4L + 3M

major 7-step

48/25, 64/33, 52/27

1055.884

|LMLLMLsM

|A-D-G

|[8] augmented

|[8] augmented

|LLMLsMLM

|C-F-A

|[8] major

|3:4:5

|-

|LMLsMLML

|D-G-B

|3:4:5

|-

|MLsMLMLL

|[8] major diminished

|[8] minor

|8:10:13

|-

|LsMLMLLM

|F-A-D

|[8] perfect

|[8] minor

|12:15:20

|-

|sMLMLLML

|G-B-E

|[8] perfect

|[8] minor

|12:15:20

|+3-step stacked triads of the 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

|LMLMsLML

|D-G-B

|[8] major

|[8] major

|3:4:5

|-

|MLMsLMLL

|[8] minor diminished

|[8] major

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

|-

|LMsLMLLM

|F-A-D

|[8] minor

|[8] minor

|12:15:20

|-

|sLMLLMLM

|A♭-C-F

|[8] minor

|[8] minor

Unlike the Pinetone diatonic, and chromatic scales, the Pinetone harmonic octatonics are chiral, and is therefore not step-nested scales. As we can see, they more complex than the Pinetone diatonic. The Pinetone pentatonic and diatonic scales is also wakalix / PWF, and it can be seen that the Pinetone harmonic octatonics are more complex than the Pinetone pentatonic as well. It is left as an exercise for the reader to determine the complexity of the Pinetone chromatic, and compare that to the Pinetone harmonic octatonics.

Modifying the right or left-handed Pinetone octatonic by switching the order of adjacent pairs of large and medium steps, i.e., by modifying steps of the scale by the L-M chroma - the difference between the large and medium steps - leads to similar Porcupine[8] detempers. Since L and M temper together under Porcupine tempering, any resulting scale tempers to Porcupine[8] just as before, but the scale it tempers to under Diminished and Father temperaments are modified.  

|-

|10/9 125/108 625/486 25/18 125/81 5/3 50/27 2/1

|<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

|-

4/3

418.550

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

36/25, 16/11, 13/9

637.334

4

|-

|2L + 2M + s

3L + M + s

813.227

2

2

|-

|3L + 2M + s

3L + 3M

!Mode (rotational order)

!Step pattern

!Diminished[8] name

!Porcupine[8] name

|Bright quartal diminished

|LMLMLMLs

|[8] major

|[8] augmented

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

|-

|Dark quartal diminished

|MLMLMLsL

|[8] minor

|[8] augmented

|[8] minor augmented

|-

|Bright major diminished

|Middle major diminished

|MLMLsLML

|[8] minor

|[8] major

|[8] major

|[8] major

|-

|Bright minor diminished

|MLsLMLML

|[8] minor

|[8] minor

|12:15:20

|LsLMLMLM

|F-A♯-D, G♭-B-E♭

|[8] minor

|[8] major 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.

The following 13 notes are used in total for these scales: E♭, G♭, A♭ D, E, F, G, A, B, C, G♯, A♯, C♯  

We can extend the Pinetone diminished into an alternative chromatic scale: Starting with the bright minor diminished scale, MLsLMLML, we add a small step into the bottom or top of every large step, leading to the scales LsMssMLsMLsM and LMssMsLMsLMs respectively, modes of mirror-inversions of one another. In 5-limit just intonation this pair of scales comprises 3 large steps of 27/25, 4 medium steps of 16/15, and 5 small steps of 25/24, i.e., 27/25 9/8 6/5 5/4 125/96 25/18 3/2 25/16 5/3 9/5 15/8 2/1 and 27/25 144/125 6/5 5/4 4/3 25/18 3/2 8/5 5/3 9/5 48/25 2/1 respectively. In other modes, they can be expressed as 25/24 10/9 6/5 5/4 4/3 36/25 3/2 8/5 5/3 125/72 50/27 2/1, and 25/24 10/9 125/108 5/4 4/3 4/3 25/18 3/2 8/5 5/3 9/5 48/25 2/1, i.e., sMLsMLsMssML and sMsLMsLMsLMs respectively.

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

The Pinetone system is 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.  

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

Additionally, we have another set of [[Porcupine]][7] modes contained in the Pinetone harmonic octatonics: Replacing the G with the G♯ changes the mode of the Porcupine[7] scale represented, and replaces diatonic with harmonic minor modes for the [[Meantone]][7] scale represented, now a MODMOS.  

We get the following 7 modes of Pinetone harmonic minor 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:

Which has modes:

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

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

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

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

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.  

|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]*^†

|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

|242.658

318.667

6

|-

!3-step

L + 2M

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

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

perfect 4-step

|minor 4-step

|3:4:5

|-

|MLMLsAsL

|[8] minor

|3:4:5

|-

|MLsAsLML

|[8] minor

|3:4:5

|-

|sAsLMLML

|[8] minor

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

|-

|sLMLMLsA

|[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. Alternatively, if the consonances of the triads are to be maximised, the scale could be tempered to 2.3.5.7 245/243 i.e., Sensamagic temperament.

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