Pinetone: Difference between revisions
m →Pinetone diminished heptatonic: typo |
|||
| Line 23: | Line 23: | ||
* [[Pinetone#Pinetone harmonic minor and harmonic major|Harmonic minor]]: 5455475, 6566585 (subset of major and minor-harmonic octatonic) | * [[Pinetone#Pinetone harmonic minor and harmonic major|Harmonic minor]]: 5455475, 6566585 (subset of major and minor-harmonic octatonic) | ||
* [[Pinetone#Pinetone harmonic minor and harmonic major|Harmonic major]]: 5545474, 6656585 (subset of major and minor-harmonic octatonic) | * [[Pinetone#Pinetone harmonic minor and harmonic major|Harmonic major]]: 5545474, 6656585 (subset of major and minor-harmonic octatonic) | ||
* [[Pinetone#Pinetone diminished heptatonic|Diminished heptatonic]]: | * [[Pinetone#Pinetone diminished heptatonic|Diminished heptatonic]]: 4754545 and 4574545, 5865656 and 5685656 (subset of diminished) | ||
* [[Pinetone#Pinetone octatonic scales|Major-harmonic octatonic]]: 54524545, 65625656 (subset of chromatic, diminished chromatic) | * [[Pinetone#Pinetone octatonic scales|Major-harmonic octatonic]]: 54524545, 65625656 (subset of chromatic, diminished chromatic) | ||
* [[Pinetone#Pinetone octatonic scales|Minor-harmonic octatonic]]: 54254554, 65265665 (subset of chromatic, diminished chromatic) | * [[Pinetone#Pinetone octatonic scales|Minor-harmonic octatonic]]: 54254554, 65265665 (subset of chromatic, diminished chromatic) | ||
| Line 2,530: | Line 2,530: | ||
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. | 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 ssssLss, Porcupine[7] | Tempering the small and medium steps together gives us the scale ssssLss, Porcupine[7]; tempering L=m implies tempering out 25/24, which leads to Dicot[7] as sLsLssL; 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. The scale is chiral, with mirror-inverse msmLsms, approximating 12/11 6/5 11/8 3/2 5/3 9/5 2/1, or in other modes, Lmsmsms and smsmsmL, the brightest and darkest modes of the pair of scales, approximating 15/13 33/26 11/8 20/13 5/3 11/6 2/1 and 12/11 6/5 13/10 13/9 39/25 26/15 2/1. The scales, like the Pinetone diatonic, are also trivalent. There's only one major triad and one minor triad available in the scale. In their simplest modes with 3/2 above the root, the pairs of scales in 5-limit JI are 27/25 5/4 25/18 3/2 5/3 9/5 2/1 and 27/25 6/5 25/18 3/2 5/3 2/1, and as Ptolemismic tempered scales their simplest pre-images are 12/11 5/4 11/8 3/2 5/3 9/5 2/1 and 12/11 6/5 11/8 3/2 5/3 9/5 2/1, with step patterns sLmsmsm and smLsmsm respectively. | ||
{| class="wikitable" | {| class="wikitable" | ||