Direct approximation: Difference between revisions
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Noted that the term "direct mapping" can refer to patent intervals themselves via analogy with the relationship between "nearest edomapping" and "patent vals". |
Trying to make sure the text reads well, though I don't know if I'm succeeding at this or not. |
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A '''patent interval''' in a given [[EDO]] is the number of EDO steps needed to reach the best approximation of a given interval – usually, but not necessarily just – in that EDO. The method for calculating patent intervals | A '''patent interval''' in a given [[EDO]] is the number of EDO steps needed to reach the best approximation of a given interval – usually, but not necessarily just – in that EDO. The method for calculating patent intervals is referred to as '''direct mapping''', and it involves [[rounding]] the product of the [[Wikipedia: binary logarithm|binary logarithm]] (''log2'') of the interval ratio (''r'') and the EDO number (''nEdo''). | ||
round(log2(r)*nEdo) | round(log2(r)*nEdo) | ||
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A [[patent val]] is the best mapping of a representative set of intervals (taken to be [[generator]]s for a [[JI subgroup]]) in a given EDO; for the ''p''-[[prime limit]] this set consists of [[prime interval]]s. | A [[patent val]] is the best mapping of a representative set of intervals (taken to be [[generator]]s for a [[JI subgroup]]) in a given EDO; for the ''p''-[[prime limit]] this set consists of [[prime interval]]s. Just as the patent val itself can be referred to as the "nearest edomapping", so a patent interval can be referred to as a "direct mapping". | ||
[[Category:Terms]] | [[Category:Terms]] | ||
[[Category:Method]] | [[Category:Method]] | ||
[[Category:Val]] | [[Category:Val]] | ||
Revision as of 16:48, 19 January 2021
A patent interval in a given EDO is the number of EDO steps needed to reach the best approximation of a given interval – usually, but not necessarily just – in that EDO. The method for calculating patent intervals is referred to as direct mapping, and it involves rounding the product of the binary logarithm (log2) of the interval ratio (r) and the EDO number (nEdo).
round(log2(r)*nEdo)
- Some Examples
| \ | 12edo | 17edo | 19edo | 26edo |
|---|---|---|---|---|
| 3/2 | 7 | 10 | 11 | 15 |
| 5/4 | 4 | 5 | 6 | 8 |
| 6/5 | 3 | 4 | 5 | 7 |
| 7/4 | 10 | 14 | 15 | 21 |
A patent val is the best mapping of a representative set of intervals (taken to be generators for a JI subgroup) in a given EDO; for the p-prime limit this set consists of prime intervals. Just as the patent val itself can be referred to as the "nearest edomapping", so a patent interval can be referred to as a "direct mapping".