Direct approximation: Difference between revisions

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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. 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"~~.

for ratio ''i'' in ''n''-edo.

== Examples ==

~~=== Examples of Patent Intervals ===~~

{| class="wikitable center-all"

! \

! Interval, ratio

! 12edo || 17edo || 19edo || 26edo

! [[12edo]] || [[17edo]] || [[19edo]] || [[26edo]]

|-

! 3/2

! Perfect fifth, [[3/2]]

| 7 || 10 || 11 || 15

|-

! 5/4

! Just major third, [[5/4]]

| 4 || 5 || 6 || 8

|-

! 6/5

! Just minor third, [[6/5]]

| 3 || 4 || 5 || 7

|-

! 7/4

! 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.

== 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:Method]]

~~[[Category:Val]]~~

~~[[Category:Todo]]~~

@@ Line 1: / Line 1: @@
-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.  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".
+for ratio ''i'' in ''n''-edo.
+== Examples ==
-=== Examples of Patent Intervals ===
 {| class="wikitable center-all"
-! \
+! Interval, ratio
-! 12edo || 17edo || 19edo || 26edo
+! [[12edo]] || [[17edo]] || [[19edo]] || [[26edo]]
 |-
-! 3/2
+! Perfect fifth, [[3/2]]
 |   7   ||   10  ||   11  || 15
 |-
-! 5/4
+! Just major third, [[5/4]]
 |   4   ||   5   ||   6   || 8
 |-
-! 6/5
+! Just minor third, [[6/5]]
 |   3   ||   4   ||   5   || 7
 |-
-! 7/4
+! 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.
+== 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:Method]]
-[[Category:Val]]
-[[Category:Todo]]