Direct approximation: Difference between revisions
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changed the definition of "direct mapping" to refer to the process of creating patent intervals- as per the clarification by Xenwolf. |
Noted that the term "direct mapping" can refer to patent intervals themselves via analogy with the relationship between "nearest edomapping" and "patent vals". |
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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. | 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 – referred to as '''direct mapping''', though this same term can also refer to the result of this process – is by [[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) | ||
Revision as of 16:37, 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 – referred to as direct mapping, though this same term can also refer to the result of this process – is by 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.