Pinetone: Difference between revisions
/* Porcutone Octatonic scales added links to xenpaper pages set up to play modes |
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We know that this scale tempers to Porcupine[8]; tempering M=s instead leads to LsLsLsLs, i.e., Diminished[8]; and finally tempering L=s leads to LsLLLsLs, a mod of Father[8]. Like the Porcutone chromatic and diatonic scale, this scale is an SN scale, and is therefor achiral. We may name this scale perhaps the Porcupine-Diminished scale, or we may include it in the Porcutone system as the Porcutone diminished scale. It may be more wise to refer to this scale as the Porcupine-Diminished scale to avoid confusion that might result if, while the octatonic scale and the diminished scale are different names for the same scale, the Porcutone octatonic and the Porcutone diminished scale are not. | We know that this scale tempers to Porcupine[8]; tempering M=s instead leads to LsLsLsLs, i.e., Diminished[8]; and finally tempering L=s leads to LsLLLsLs, a mod of Father[8]. Like the Porcutone chromatic and diatonic scale, this scale is an SN scale, and is therefor achiral. We may name this scale perhaps the Porcupine-Diminished scale, or we may include it in the Porcutone system as the Porcutone diminished scale. It may be more wise to refer to this scale as the Porcupine-Diminished scale to avoid confusion that might result if, while the octatonic scale and the diminished scale are different names for the same scale, the Porcutone octatonic and the Porcutone diminished scale are not. | ||
Every other step of any mode of the Porcutone Diminished scale gives an inversion of the 5-limit diminished tetrad; therefore every second step of the Porcutone Diminished scales only comes in two different sizes, as opposed to the four different sizes of every second step of the Porcutone octatonic. Although the Porcutone Diminished is simpler in this way, the Porcutone octatonic provides more major and minor | Every other step of any mode of the Porcutone Diminished scale gives an inversion of the 5-limit diminished tetrad; therefore every second step of the Porcutone Diminished scales only comes in two different sizes, as opposed to the four different sizes of every second step of the Porcutone octatonic. Although the Porcutone Diminished is simpler in this way, the Porcutone octatonic provides more major and minor triads. | ||
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<nowiki>*</nowiki> Non-bracketed JI ratios are those approximated in 2.3.5.11.13 ptolemismic Porcutone; bracketed JI ratios are the 2.3.7 interval approximated by additionally tempering out 105/104 or 245/243 as in Supermagic temperament. | <nowiki>*</nowiki> Non-bracketed JI ratios are those approximated in 2.3.5.11.13 ptolemismic Porcutone; bracketed JI ratios are the 2.3.7 interval approximated by additionally tempering out 105/104 or 245/243 as in Supermagic temperament. | ||
We can extend the Porcutone Diminished into an alterative 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. | The following 13 notes are used in total for these scales: E♭, G♭, A♭ D, E, F, G, A, B, C, G♯, A♯, C♯ | ||
==== Porcutone Diminished chromatic ==== | |||
We can extend the Porcutone Diminished into an alterative 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. | |||
We could call these scales, which temper to Diminished[12], the left and right-handed Porcutone Diminished chromatic. | We could call these scales, which temper to Diminished[12], the left and right-handed Porcutone Diminished chromatic. | ||
Tempering out 100/99 and 144/143 leads to the following simplest pre-images | Tempering M=L alternatively leads to sLLsLLsLssLL and sLsLLsLLsLLs, which are MODMOS of Meantone[12], i.e., C C♯ D E♭ E F G♭ G A♭ A A♯ B and G G♯ A A♯ B C C♯ D E♭ E F G♭. | ||
Tempering out 100/99 and 144/143 leads to the following simplest pre-images 25/24 10/9 6/5 5/4 4/3 13/9 3/2 8/5 5/3 26/15 11/6 2/1 and 25/24 10/9 15/13 5/4 4/3 11/8 3/2 8/5 5/3 9/5 48/25 2/1. | |||
From sMsLMsLMsLMs, putting a small step below the top of each large step (replacing ''L'' with ''sm'', where ''m'' is the small step of the 12-note scale, the medium step of the 15-note scale) leads to | |||
mLm(sm)Lm(sm)Lm(sm)Lm | |||
mLmsmLmsmLmsmLm, which we later introduce as Porcutone-15. | |||
== Summary for xen-math nerds == | == Summary for xen-math nerds == | ||