Pinetone: Difference between revisions
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====Pinetone diminished chromatic==== | ====Pinetone diminished chromatic==== | ||
We can extend the Pinetone diminished into an alternative 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 | We can extend the Pinetone diminished into an alternative 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 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 Pinetone diminished chromatic. | We could call these scales, which temper to Diminished[12], the left and right-handed Pinetone diminished chromatic. | ||
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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 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 | 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 45/26 11/6 2/1 and 25/24 10/9 52/45 5/4 4/3 11/8 3/2 8/5 5/3 9/5 48/25 2/1. | ||
Tuned to TE 2.3.5.11.13 Ptolemismic, the Pinetone diminished chromatic scales in those modes in cents are: | |||
66.766 175.892 318.667 385.433 494.559 637.334 704.101 813.227 879.992 956.002 1055.884 1198.660 and 66.766 175.892 251.901 385.433 494.559 561.325 704.101 813.227 879.992 946.759 1131.983 1198.660. | |||
The Pinetone diminished chromatic scales provide the same amount of major and minor triads as the Pinetone chromatic (4 of each), however, they are more evenly spread in the diminished chromatic scales. | |||
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 | 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 | ||