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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 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'').
{{interwiki
| en = Direct approximation
| ja = 直接近似
}}
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:


round(log2(r)*nEdo)
<math>\operatorname {round} (n\log_2(i))</math>


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.
for ratio ''i'' in ''n''-edo.  


==== Examples of Patent Intervals ====
== Examples ==
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.


{| class="wikitable center-all"
{| class="wikitable center-all"
Line 12: Line 15:
! [[12edo]] || [[17edo]] || [[19edo]] || [[26edo]]
! [[12edo]] || [[17edo]] || [[19edo]] || [[26edo]]
|-
|-
! Just perfect fifth, [[3/2]]
! Perfect fifth, [[3/2]]
|  7  ||  10  ||  11  || 15
|  7  ||  10  ||  11  || 15
|-
|-
! Just classic major third, [[5/4]]
! Just major third, [[5/4]]
|  4  ||  5  ||  6  || 8
|  4  ||  5  ||  6  || 8
|-
|-
! Just classic minor third, [[6/5]]
! Just minor third, [[6/5]]
|  3  ||  4  ||  5  || 7
|  3  ||  4  ||  5  || 7
|-
|-
Line 25: Line 28:
|}
|}


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.
== Problems ==
Unless one sticks to one or two notes at a time, direct approximation is not always practical in harmony. For example, it is impossible to construct a [[4:5:6|just major triad]] using the direct approximations of 3/2, 5/4, and 6/5 in 17edo since the step numbers do not add up (5 steps + 4 steps ≠ 10 steps). The closest 3/2 and 5/4 imply the second closest 6/5; the closest 3/2 and 6/5 imply the second closest 5/4; and the closest 5/4 and 6/5 imply the second closest 3/2. We see one of the direct approximations must be given up. This is called [[consistency|inconsistency]], and chords like this exists in every edo.
In [[regular temperament theory]], intervals are mapped through [[val]]s. Although more complex, it recognizes the fact that intervals like 3/2, 5/4, and 6/5 are related, as the number of steps of one interval is determined once the other two have been determined. The three situations in the above example correspond to using vals {{val| 17 27 39 }}, {{val| 17 27 40 }}, and {{val| 17 26 39 }}, respectively.
[[Category:Interval]]
[[Category:Terms]]
[[Category:Terms]]
[[Category:Method]]
[[Category:Method]]
[[Category:Val]]
[[Category:Todo]]
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