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The [[Delta-N|delta]] of a ratio is simply the numerator minus the denominator. All [[Superparticular ratio|superparticular]] ratios are delta-1. Both 5/3 and 7/5 are delta-2.
The [[Delta-N|delta]] of a ratio is simply the numerator minus the denominator. All [[Superparticular ratio|superparticular]] ratios are delta-1. Both 5/3 and 7/5 are delta-2.


Every ratio occurs only once in the Stern-Brocot tree. Every ratio has two ancestors and two children. Both ancestors will have a smaller [[Limit|integer limit]], and one will always be smaller than the other. Thus there is a '''simpler''' ancestor and a '''more''' '''complex''' ancestor.
Every ratio occurs only once in the [[wikipedia:Stern–Brocot_tree|Stern-Brocot tree]]. Every ratio has two ancestors and two children. Both ancestors will have a smaller [[Limit|integer limit]], and one will always be smaller than the other. Thus there is a '''simpler''' ancestor and a '''more''' '''complex''' ancestor.


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Line 36: Line 36:


The correct factor to multiply by is always less than half of the delta, and is always coprime with the delta.
The correct factor to multiply by is always less than half of the delta, and is always coprime with the delta.
{| class="wikitable"
{| class="wikitable center-all"
|+
|+
!delta
!delta
Line 172: Line 172:
* If the numerator mod 10 is 1 or 9, bump it.
* If the numerator mod 10 is 1 or 9, bump it.
* If the numerator mod 10 is 3 or 7, triple the ratio before bumping. (23/13 --> 69/39 --> 70/40 --> 7/4 and 16/9)
* If the numerator mod 10 is 3 or 7, triple the ratio before bumping. (23/13 --> 69/39 --> 70/40 --> 7/4 and 16/9)
== Example: comparing various edos to 41-edo ==
{| class="wikitable center-all"
|+
!edo
!1
!2
!3
!4
!5
!6
!7
!8
!9
!10
!11
!12
!13b
!14
!15
!16
!17
!18b
!19
!20
|-
! rowspan="2" |nearest
misses
|0\1
|1\2
|1\3
|1\4
|1\5
M2
|1\6
|1\7
P2
|1\8
|2\9
M3
|1\10
^m2
|4\11
v4
|5\12
P4
|6\13
P4
|1\14
v2
|4\15
^m3
|7\16
P4
|5\17
~3
|7\18
v4
|6\19
M3
|1\20
^m2
|-
|1\41
^1
|20\41
d5
|14\41
M3
|10\41
m3
|8\41
^M2
|7\41
M2
|6\41
vM2
|5\41
~3
|9\41
vm3
|4\41
^m2
|15\41
^M3
|17\41
P4
|19\41
~4
|3\41
m2
|11\41
^m3
|18\41
^4
|12\41
~3
|16\41
v4
|13\41
vM3
|2\41
vm2
|-
! rowspan="2" |farthest
misses
|0\1
|1\2
|1\3
|2\4
|2\5
P4
|3\6
|3\7
P4
|4\8
|1\9
M2
|5\10
|2\11
M3
|6\12
|3\13
^M3
|7\14
|2\15
vM2
|8\16
|6\17
M3
|9\18
|3\19
M2
|10\20
|-
|20\41
d5
|10\41
m3
|7\41
M2
|5\41
~2
|4\41
^m2
|17\41
P4
|3\41
m2
|18\41
^4
|16\41
v4
|2\41
vm2
|13\41
vM3
|12\41
~3
|11\41
^m3
|19\41
~4
|15\41
^M3
|9\41
vm3
|6\41
vM2
|8\41
^M2
|14\41
M3
|1\41
^1
|-
!edo
!40
!39
!38
!37
!36
!35
!34
!33
!32
!31
!30
!29
!28
!27
!26
!25
!24
!23
!22
!21
|-
! rowspan="2" |nearest
misses
|1\40
^1
|19\39
^d5
|13\38
^M3
|9\37
^m3
|7\36
^M2
|6\35
^2
|5\34
vM2
|4\33
m2
|7\32
m3
|3\31
m2
|11\30
vM3
|12\29
P4
|13\28
^4
|2\27
^m2
|7\26
m3
|11\25
^4
|7\24
~3
|9\23
A4
|7\22
vM3
|1\21
^1
|-
|1\41
^1
|20\41
d5
|14\41
M3
|10\41
m3
|8\41
^M2
|7\41
M2
|6\41
vM2
|5\41
~2
|9\41
vm3
|4\41
^m2
|15\41
^M3
|17\41
P4
|19\41
~4
|3\41
m2
|11\41
^m3
|18\41
^4
|12\41
~3
|16\41
v4
|13\41
vM3
|2\41
vm2
|-
! rowspan="2" |farthest
misses
|20\40
|10\39
^m3
|19\38
|14\37
M3
|18\36
|3\35
vv2
|17\34
|2\33
dd2
|16\32
|14\31
^4
|15\30
|6\29
^M2
|14\28
|1\27
m2
|13\26
|7\25
^^m3
|12\24
|7\23
m3
|11\22
|10\21
^4
|-
|20\41
d5
|10\41
m3
|7\41
M2
|5\41
~2
|4\41
^m2
|17\41
P4
|3\41
m2
|18\41
^4
|16\41
v4
|2\41
vm2
|13\41
vM3
|12\41
~3
|11\41
^m3
|19\41
~4
|15\41
^M3
|9\41
vm3
|6\41
vM2
|8\41
^M2
|14\41
M3
|1\41
^1
|}
* Note the symmetry of the 41-edo intervals, which results from the symmetry of the Stern-Brocot tree.
* The "b" in 13b and 18b only affects the interval names. For example, 6\13 in 13b-edo nomenclature is a P4, but in 13-edo it would be an ^4.
* The unnamed farthest misses for edos other than 41 are the half-octave.


== Further notes ==
== Further notes ==
The delta method was invented by [[Kite Giedraitis]] in 2022. The ratio approximations for a/b and c/d rely on the formula log [(a+c)/(b+d)] ≈ [a/(a+c)] * log [a/b] + [c/(a+c)] * log [c/d], where ad - bc = ±1. There may be a better formula.
The delta method was invented by [[Kite Giedraitis]] in 2022. The ratio approximations for a/b and c/d rely on the formula log [(a+c)/(b+d)] ≈ [a/(a+c)] * log [a/b] + [c/(a+c)] * log [c/d], where ad - bc = ±1. There may be a better formula.
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