Direct approximation: Difference between revisions
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Cmloegcmluin (talk | contribs) "direct mapping" and "patent interval" → "direct approximation", per discussion (and page move) |
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A '''direct approximation''' of an interval in a given [[ | A '''direct approximation''' of an interval in a given [[edo]] is the number of edosteps that most closely approximates it, found by [[rounding]] to the nearest integer the edo number times the [[log2|binary logarithm]] of the interval: | ||
<math>\operatorname {round} (n\log_2(i))</math> | |||
for ratio ''i'' in ''n''-edo. | |||
== Examples of direct approximations == | == Examples of direct approximations == | ||
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! [[12edo]] || [[17edo]] || [[19edo]] || [[26edo]] | ! [[12edo]] || [[17edo]] || [[19edo]] || [[26edo]] | ||
|- | |- | ||
! | ! Perfect fifth, [[3/2]] | ||
| 7 || 10 || 11 || 15 | | 7 || 10 || 11 || 15 | ||
|- | |- | ||
! Just | ! Just major third, [[5/4]] | ||
| 4 || 5 || 6 || 8 | | 4 || 5 || 6 || 8 | ||
|- | |- | ||
! Just | ! Just minor third, [[6/5]] | ||
| 3 || 4 || 5 || 7 | | 3 || 4 || 5 || 7 | ||
|- | |- | ||
| Line 20: | Line 24: | ||
|} | |} | ||
Of these intervals, the fifth plays an important role for characterizing [[ | 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. | ||
[[Category:Terms]] | [[Category:Terms]] | ||
[[Category:Method]] | [[Category:Method]] | ||
Revision as of 22:19, 24 February 2022
A direct approximation of an interval in a given edo is the number of edosteps that most closely approximates it, found by rounding to the nearest integer the edo number times the binary logarithm of the interval:
[math]\displaystyle{ \operatorname {round} (n\log_2(i)) }[/math]
for ratio i in n-edo.
Examples of direct approximations
| Interval, ratio | 12edo | 17edo | 19edo | 26edo |
|---|---|---|---|---|
| Perfect fifth, 3/2 | 7 | 10 | 11 | 15 |
| Just major third, 5/4 | 4 | 5 | 6 | 8 |
| Just minor third, 6/5 | 3 | 4 | 5 | 7 |
| Harmonic seventh, 7/4 | 10 | 14 | 15 | 21 |
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.