Diamond function: Difference between revisions
Remove sentece about 17/16, as it occurs in 17-odd limit so that doesn't make any sense. Also remove remark about multisets, which has no clear relevance or application. Add crystal ball link. |
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The scale steps of the tonality diamond are superparticular ratios, but they are not very evenly distributed. Filling in the gaps, as [[Harry Partch]] did with the 11-limit diamond to create a constant structure for his famous Genesis scale, is one way to go about constructing a just intonation scale. A constant structure is where each occurrence of a ratio will always have the same number of scale steps. While this is not completely possible with the [[11-limit]] diamond, Partch was able to do so except in two places. This makes his 43 tone scale related to a 41 tone constant structure with two alternates. | The scale steps of the tonality diamond are superparticular ratios, but they are not very evenly distributed. Filling in the gaps, as [[Harry Partch]] did with the 11-limit diamond to create a constant structure for his famous Genesis scale, is one way to go about constructing a just intonation scale. A constant structure is where each occurrence of a ratio will always have the same number of scale steps. While this is not completely possible with the [[11-limit]] diamond, Partch was able to do so except in two places. This makes his 43 tone scale related to a 41 tone constant structure with two alternates. | ||
The diamond construction can be iterated, giving diamond(diamond(S)), diamond(diamond(diamond(S))), and so forth. These scales are known as [[ | The diamond construction can be iterated, giving diamond(diamond(S)), diamond(diamond(diamond(S))), and so forth. These scales are known as [[crystal ball]]s. As scales, the results are too large for many applications, but iterating the tonality diamonds, or taking its [[Scale products and scale powers|scale powers]], provides a convenient means of obtaining ''p''-limit intervals, or intervals in a desired [[Just intonation subgroups|JI subgroup]], in abundance. | ||
[[Category:Diamond]] | [[Category:Diamond]] | ||