Hemipyth: Difference between revisions

Line 16:

|+ style="font-size: 105%;" | List of edo mappings with full or partial hemipyth support

|-

! Edo (warts) !! Has <math>\sqrt{2}</math> !! Has <math>\sqrt{\frac{3}{2}}</math> !! Has <math>\sqrt{\frac{4}{3}}</math>

! Edo ([[Wart notation|warts]]) !! Has <math>\sqrt{2}</math> !! Has <math>\sqrt{\frac{3}{2}}</math> !! Has <math>\sqrt{\frac{4}{3}}</math>

|-

| 2 || yes || no || no

| [[2edo|2]] || yes || no || no

|-

| 3 || no || yes || no

| [[3edo|3]] || no || yes || no

|-

| 4 || yes || yes || yes

| [[4edo|4]] || yes || yes || yes

|-

| 5 || no || no || yes

| [[5edo|5]] || no || no || yes

|-

| 6 || yes || yes || yes

| [[6edo|6]] || yes || yes || yes

|-

| 7 || no || yes || no

| [[7edo|7]] || no || yes || no

|-

| 8 || yes || no || no

| [[8edo|8]] || yes || no || no

|-

| 9 || no || no || yes

| [[9edo|9]] || no || no || yes

|-

| 10 || yes || yes || yes

| [[10edo|10]] || yes || yes || yes

|-

| 11 || no || yes || no

| [[11edo|11]] || no || yes || no

|-

| 12 || yes || no || no

| [[12edo|12]] || yes || no || no

|-

| 13 || no || yes || no

| [[13edo|13]] || no || yes || no

|-

| 13b || no || no || yes

|-

| 14 || yes || yes || yes

| [[14edo|14]] || yes || yes || yes

|-

| 15 || no || no || yes

| [[15edo|15]] || no || no || yes

|-

| 16 || yes || no || no

| [[16edo|16]] || yes || no || no

|-

| 17 || no || yes || no

| [[17edo|17]] || no || yes || no

|-

| 18 || yes || no || no

| [[18edo|18]] || yes || no || no

|-

| 18b || yes || yes || yes

|-

| 19 || no || no || yes

| [[19edo|19]] || no || no || yes

|-

| 20* || yes || yes || yes

| [[20edo|20]]* || yes || yes || yes

|-

| 20b || yes || no || no

|-

| 21 || no || yes || no

| [[21edo|21]] || no || yes || no

|-

| 22 || yes || no || no

| [[22edo|22]] || yes || no || no

|-

| 23 || no || no || yes

| [[23edo|23]] || no || no || yes

|-

| 24 || yes || yes || yes

| [[24edo|24]] || yes || yes || yes

|}

<nowiki>*</nowiki> Above the patent val of 20edo results in the same tuning as the patent val of 10edo, so it adds nothing new.

Note how in hemipyth the patent val of 24edo is not tuned the same as 12edo's patent val. In fact 24edo is arguably the smallest edo where all of the important hemipyth intervals are tuned reasonably accurately.

Note how in hemipyth the patent val of 24edo is not tuned the same as 12edo's patent val. In fact, 24edo is arguably the smallest edo where all of the important hemipyth intervals are tuned reasonably accurately.

Other edos with hemipyth-supporting patent vals are 28, 30, 34, 38, 44, 48, 52, 54, 58, etc. 58edo is the first one to reduce the absolute error of the neutral third generator compared to 24edo~~. You need~~ to go all the way to 82edo in order to get an improvement in terms of relative error.

Other edos with hemipyth-supporting patent vals are {{edos|28, 30, 34, 38, 44, 48, 52, 54, 58}}, etc. 58edo is the first one to reduce the absolute error of the neutral third generator compared to 24edo, though one needs to go all the way to [[82edo]] in order to get an improvement in terms of relative error.

== Notation ==

@@ Line 16: / Line 16: @@
 |+ style="font-size: 105%;" | List of edo mappings with full or partial hemipyth support
 |-
-! Edo (warts) !! Has <math>\sqrt{2}</math> !! Has <math>\sqrt{\frac{3}{2}}</math> !! Has <math>\sqrt{\frac{4}{3}}</math>
+! Edo ([[Wart notation|warts]]) !! Has <math>\sqrt{2}</math> !! Has <math>\sqrt{\frac{3}{2}}</math> !! Has <math>\sqrt{\frac{4}{3}}</math>
 |-
-| 2 || yes || no || no
+| [[2edo|2]] || yes || no || no
 |-
-| 3 || no || yes || no
+| [[3edo|3]] || no || yes || no
 |-
-| 4 || yes || yes || yes
+| [[4edo|4]] || yes || yes || yes
 |-
-| 5 || no || no || yes
+| [[5edo|5]] || no || no || yes
 |-
-| 6 || yes || yes || yes
+| [[6edo|6]] || yes || yes || yes
 |-
-| 7 || no || yes || no
+| [[7edo|7]] || no || yes || no
 |-
-| 8 || yes || no || no
+| [[8edo|8]] || yes || no || no
 |-
-| 9 || no || no || yes
+| [[9edo|9]] || no || no || yes
 |-
-| 10 || yes || yes || yes
+| [[10edo|10]] || yes || yes || yes
 |-
-| 11 || no || yes || no
+| [[11edo|11]] || no || yes || no
 |-
-| 12 || yes || no || no
+| [[12edo|12]] || yes || no || no
 |-
-| 13 || no || yes || no
+| [[13edo|13]] || no || yes || no
 |-
 | 13b || no || no || yes
 |-
-| 14 || yes || yes || yes
+| [[14edo|14]] || yes || yes || yes
 |-
-| 15 || no || no || yes
+| [[15edo|15]] || no || no || yes
 |-
-| 16 || yes || no || no
+| [[16edo|16]] || yes || no || no
 |-
-| 17 || no || yes || no
+| [[17edo|17]] || no || yes || no
 |-
-| 18 || yes || no || no
+| [[18edo|18]] || yes || no || no
 |-
 | 18b || yes || yes || yes
 |-
-| 19 || no || no || yes
+| [[19edo|19]] || no || no || yes
 |-
-| 20* || yes || yes || yes
+| [[20edo|20]]* || yes || yes || yes
 |-
 | 20b || yes || no || no
 |-
-| 21 || no || yes || no
+| [[21edo|21]] || no || yes || no
 |-
-| 22 || yes || no || no
+| [[22edo|22]] || yes || no || no
 |-
-| 23 || no || no || yes
+| [[23edo|23]] || no || no || yes
 |-
-| 24 || yes || yes || yes
+| [[24edo|24]] || yes || yes || yes
 |}
 <nowiki>*</nowiki> Above the patent val of 20edo results in the same tuning as the patent val of 10edo, so it adds nothing new.
-Note how in hemipyth the patent val of 24edo is not tuned the same as 12edo's patent val. In fact 24edo is arguably the smallest edo where all of the important hemipyth intervals are tuned reasonably accurately.
+Note how in hemipyth the patent val of 24edo is not tuned the same as 12edo's patent val. In fact, 24edo is arguably the smallest edo where all of the important hemipyth intervals are tuned reasonably accurately.
-Other edos with hemipyth-supporting patent vals are 28, 30, 34, 38, 44, 48, 52, 54, 58, etc. 58edo is the first one to reduce the absolute error of the neutral third generator compared to 24edo. You need to go all the way to 82edo in order to get an improvement in terms of relative error.
+Other edos with hemipyth-supporting patent vals are {{edos|28, 30, 34, 38, 44, 48, 52, 54, 58}}, etc. 58edo is the first one to reduce the absolute error of the neutral third generator compared to 24edo, though one needs to go all the way to [[82edo]] in order to get an improvement in terms of relative error.
 == Notation ==