Pinetone: Difference between revisions

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====Pinetone diminished heptatonic====

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 lower note of each of 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 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.

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 [[Chirality|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.

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 ====Pinetone diminished heptatonic====
-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 lower note of each of 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 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.
+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 [[Chirality|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.
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