Direct approximation: Difference between revisions
add names to intervals, links to EDO pages, and some explanation about the "patent fifth" |
m Xenwolf moved page Patent interval to Direct approximation: as per suggestion of many users |
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Revision as of 22:03, 23 February 2022
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)
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.
Examples of Patent Intervals
Of these intervals, the fifth plays an important role for characterizing EDO systems (as it defines the size of M2, m2, A1). Also, a simple test can show if circle-of-fifths notation can be applied to a given EDO system, because for this the sizes of fifth and octave must be relatively prime.
| Interval, ratio | 12edo | 17edo | 19edo | 26edo |
|---|---|---|---|---|
| Just perfect fifth, 3/2 | 7 | 10 | 11 | 15 |
| Just classic major third, 5/4 | 4 | 5 | 6 | 8 |
| Just classic minor third, 6/5 | 3 | 4 | 5 | 7 |
| Harmonic seventh, 7/4 | 10 | 14 | 15 | 21 |