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Pinetone: Difference between revisions

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→Pinetone harmonic diminished octatonic: added table of intervals, fixed typos in table of TE modes
Line 2,696: Line 2,696:
!Comments
!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--- Hlanithian diminished]
|[https://xenpaper.com/#%7B0c_251.901c_318.667c_494.559c_637.334c_813.226c_956.002c_1131.893c_1198.660c_1450.561c_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--- Hlanithian diminished]
|AsLMLMLs
|AsLMLMLs
|~ 56/45 6/5 4/3 13/9 8/5 26/15 48/25 2/1
|~ 52/45 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
|251.901 318.667 494.559 637.334 813.226 956.002 1131.893 1198.660
|
|
|-
|-
Line 2,738: Line 2,738:
|root 4:5:6,10:12:15
|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--- Illarnekian diminished]*<sup>†</sup>
|[https://xenpaper.com/#%7B0c_66.766c_242.658c_385.433c_561.325c_704.101c_879.993c_946.759c_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--- Illarnekian diminished]*<sup>†</sup>
|sLMLMLsA
|sLMLMLsA
|~ 25/24 15/13 5/4 11/8 3/2 5/3 45/26 2/1
|~ 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
|66.766 242.658 385.433 561.325 704.101 879.993 946.759 1198.660
|root 4:5:6
|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"
|+Intervals of the Pinetone harmonic diminished
!Interval class
!sizes
!Diminished[8] name
!Porcupine[8] name
!Pinetone octatonic name
!JI ratios approximated*
!size in cents (TE)
!Occurence
|-
!1-step
|s
M
 
L
 
A
|minor step
minor step
 
major step
 
major step
|minor step
major step
 
major step
 
augmented step
|minor step
minor-major step
 
major step
 
augmented step
|25/24, 33/32, 27/26
27/25, 12/11, 13/12
 
10/9, 11/10
 
144/125, 64/55, 52/45
|66.766
142.775
 
175.892
 
251.901
|2
2
 
3
 
1
|-
!2-step
|L + s
L + M = A + s
|perfect 2-step
perfect 2-step
|minor 2-step
major 2-step
|minor 2-step
major 2-step
|15/13 (7/6 or 8/7)
6/5
|242.658
318.667
|2
6
|-
!3-step
|L + M + s = A + 2s
L + 2M
 
2L + M = A + L + s
|minor 3-step
minor 3-step
 
major 3-step
|minor 3-step
major 3-step
 
major 3-step
|minor 3-step
minor-major 3-step
 
major 3-step
|5/4
13/10 (9/7 or 21/16)
 
4/3
|385.433
461.433
 
494.559
|3
1
 
4
|-
!4-step
|2L + M + s = A + L + 2s
2L + 2M = A + L + M + s
|perfect 4-step
perfect 4-step
|minor 4-step
major 4-step
|minor 4-step
major 4-step
|25/18, 11/8, 18/13
36/25, 16/11, 13/9
|561.325
637.334
|4
4
|-
!5-step
|2L + 2M + s = A + L + M + 2s
A + 2L + 2s
 
3L + 2M = A + 2L + M + s
|minor 5-step
major 5-step
 
major 5-step
|minor 5-step
minor 5-step
 
major 5-step
|minor 5-step
major-minor 5-step
 
major 5-step
|3/2
20/13 (14/9 or 32/16)
 
8/5
|704.101
737.217
 
813.227
|4
1
 
3
|-
!6-step
|3L + 2M + s = A + 2L + M + 2s
A + 2L + 2M + s
|perfect 6-step
perfect 6-step
|minor 6-step
major 6-step
|minor 6-step
major 6-step
|5/3
26/15 (12/7 or 7/4)
|879.992
956.002
|6
2
|-
!7-step
|3L + 2M + 2s
A + 2L + 2M + 2s
 
A + 3L + M + 2s
 
4L + 3M
|minor 7-step
minor 7-step
 
major 7-step
 
major 7-step
|diminished 7-step
minor 7-step
 
minor 7-step
 
major 7-step
|diminished 7-step
minor 7-step
 
major-minor 7-step
 
major 7-step
|125/72, 55/32, 45/26
9/5, 20/11
 
50/27, 11/6, 24/13
 
48/25, 64/33, 52/27
|946.759
1022.768
 
1055.884
 
1131.983
|1
3
 
2


2
|}<nowiki>*</nowiki> Non-bracketed JI ratios are those approximated in 2.3.5.11.13 ptolemismic Pinetone; bracketed JI ratios are shows in pairs: the first interval in each pair is the 2.3.7 interval approximated by additionally tempering out 91/90 or 126/125 (Starling); the second is the 2.3.7 interval approximated by additionally tempering out 105/104 or 245/243 (Supermagic).
{| class="wikitable"
{| class="wikitable"
|+3-step stacked triads of the Pinetone harmonic diminished
|+3-step stacked triads of the Pinetone harmonic diminished
Retrieved from "https://en.xen.wiki/w/Pinetone"