Pinetone: Difference between revisions

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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 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).

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 27/25 144/25 6/5 5/4 4/3 25/18 3/2 8/5 5/3 9/5 48/25 3/1 respectively.

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: 12/11 9/8 6/5 5/4 13/10 11/8 3/2 25/16 5/3 9/5 15/8 2/1 and 12/11 15/13 6/5 5/4 4/3 11/8 3/2 8/5 5/3 9/5 25/13 2/1.

== Summary for xen-math nerds ==

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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 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).
+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.,
+/25 9/8 6/5 5/4 125/96 25/18 3/2 25/16 5/3 9/5 15/8 2/1 and 27/25 144/25 6/5 5/4 4/3 25/18 3/2 8/5 5/3 9/5 48/25 3/1 respectively.
+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: 12/11 9/8 6/5 5/4 13/10 11/8 3/2 25/16 5/3 9/5 15/8 2/1 and 12/11 15/13 6/5 5/4 4/3 11/8 3/2 8/5 5/3 9/5 25/13 2/1.
 == Summary for xen-math nerds ==