Hemipyth: Difference between revisions

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A '''hemipyth''' (or '''"hemipythagorean"''') interval is an [[interval]] in the <math>\sqrt{2}\,.\sqrt{3}</math> [[subgroup]] i.e. intervals that can be constructed by multiplying half-integer powers of 2 and 3. Hemipythagorean is the pure tuning of [[Ploidacot/Diploid dicot|diploid dicot]].

A '''hemipyth''' (or '''"hemipythagorean"''') interval is an [[interval]] in the <math>\sqrt{2}\,.\sqrt{3}-</math>[[subgroup]]; i.e. intervals that can be constructed by multiplying half-integer powers of primes [[2/1|2]] and [[3/1|3]]. Hemipythagorean is the pure tuning of [[Ploidacot/Diploid dicot|diploid dicot]].

Notable hemipyth intervals include the neutral third <math>\sqrt{\frac{3}{2}} = \frac{\sqrt{3}}{\sqrt{2}}</math>, semioctave <math>\sqrt{2}</math>, and the semifourth <math>\sqrt{\frac{4}{3}} = \frac{2}{\sqrt{3}}</math>.

Notable hemipyth intervals include the [[neutral third]] <math>\sqrt{\frac{3}{2}} = \frac{\sqrt{3}}{\sqrt{2}}</math>, [[semioctave]] <math>\sqrt{2}</math>, and the [[semifourth]] <math>\sqrt{\frac{4}{3}} = \frac{2}{\sqrt{3}}</math>.

Many temperaments naturally produce intervals that split ~{{sfrac|3|2}}, ~2, or ~{{sfrac|4|3}} exactly in half and can thus be interpreted as neutral thirds, semioctaves, or semifourths within the temperament.

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|+ 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 ==

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== MOS patterns ==

{{Idiosyncratic terms|The mos names for hemipyth[14], hemipyth[24], and hemipyth[34] are proposals described on [[TAMNAMS_Extension #Naming mos descendants]].}}

By default hemipyth[n] refers to a [[MOS]] pattern of size n inside the octave i.e. there are always two √2 periods per octave. The generator is √3 unless otherwise stated.

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|-

! hemipyth[n]

!TAMNAMS name!! MOS pattern !! hardness (untempered)

! TAMNAMS name !! MOS pattern !! hardness (untempered)

|-

| hemipyth[4]

|biwood|| [[2L 2s]] || 1.4094

| biwood || [[2L 2s]] || 1.4094

|-

| hemipyth[6]

|citric|| [[4L 2s]] || 2.4424

| citric || [[4L 2s]] || 2.4424

|-

| hemipyth[10]

|lime|| [[4L 6s]] || 1.4424

| lime || [[4L 6s]] || 1.4424

|-

| hemipyth[14]

|m-chro lime|| [[10L 4s]] || 2.260

| m-chro lime || [[10L 4s]] || 2.260

|-

| hemipyth[24]

|f-enhar lime|| [[10L 14s]] || 1.260

| f-enhar lime || [[10L 14s]] || 1.260

|-

|hemipyth[34]

| hemipyth[34]

|paso-lime

| paso-lime

|[[24L 10s]]

| [[24L 10s]]

|3.8459

| 3.8459

|-

|hemipyth[58]

| hemipyth[58]

|

|[[24L 34s]]

| [[24L 34s]]

|2.8459

| 2.8459

|-

|hemipyth[82]

| hemipyth[82]

|

|[[24L 58s]]

| [[24L 58s]]

|1.8459

| 1.8459

|-

|hemipyth[106]

| hemipyth[106]

|

|[[82L 24s]]

| [[82L 24s]]

|1.1822

| 1.1822

|}

== Music ==

~~{{todo|inline-1| Make more music }}~~

[[File:The_Hymn_of_Pergele.mp3]]

@@ Line 1: / Line 1: @@
 {{texchars}}
-A '''hemipyth''' (or '''"hemipythagorean"''') interval is an [[interval]] in the <math>\sqrt{2}\,.\sqrt{3}</math> [[subgroup]] i.e. intervals that can be constructed by multiplying half-integer powers of 2 and 3. Hemipythagorean is the pure tuning of [[Ploidacot/Diploid dicot|diploid dicot]].
+A '''hemipyth''' (or '''"hemipythagorean"''') interval is an [[interval]] in the <math>\sqrt{2}\,.\sqrt{3}-</math>[[subgroup]]; i.e. intervals that can be constructed by multiplying half-integer powers of primes [[2/1|2]] and [[3/1|3]]. Hemipythagorean is the pure tuning of [[Ploidacot/Diploid dicot|diploid dicot]].
-Notable hemipyth intervals include the neutral third <math>\sqrt{\frac{3}{2}} = \frac{\sqrt{3}}{\sqrt{2}}</math>, semioctave <math>\sqrt{2}</math>, and the semifourth <math>\sqrt{\frac{4}{3}} = \frac{2}{\sqrt{3}}</math>.
+Notable hemipyth intervals include the [[neutral third]] <math>\sqrt{\frac{3}{2}} = \frac{\sqrt{3}}{\sqrt{2}}</math>, [[semioctave]] <math>\sqrt{2}</math>, and the [[semifourth]] <math>\sqrt{\frac{4}{3}} = \frac{2}{\sqrt{3}}</math>.
 Many temperaments naturally produce intervals that split ~{{sfrac|3|2}}, ~2, or ~{{sfrac|4|3}} exactly in half and can thus be interpreted as neutral thirds, semioctaves, or semifourths within the temperament.
@@ 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 ==
@@ Line 273: / Line 273: @@
 == MOS patterns ==
+{{Idiosyncratic terms|The mos names for hemipyth[14], hemipyth[24], and hemipyth[34] are proposals described on [[TAMNAMS_Extension #Naming mos descendants]].}}
 By default hemipyth[n] refers to a [[MOS]] pattern of size n inside the octave i.e. there are always two √2 periods per octave. The generator is √3 unless otherwise stated.
@@ Line 279: / Line 280: @@
 |-
 ! hemipyth[n]
-!TAMNAMS name!! MOS pattern !! hardness (untempered)
+! TAMNAMS name !! MOS pattern !! hardness (untempered)
 |-
 | hemipyth[4]
-|biwood|| [[2L 2s]] || 1.4094
+| biwood || [[2L 2s]] || 1.4094
 |-
 | hemipyth[6]
-|citric|| [[4L 2s]] || 2.4424
+| citric || [[4L 2s]] || 2.4424
 |-
 | hemipyth[10]
-|lime|| [[4L 6s]] || 1.4424
+| lime || [[4L 6s]] || 1.4424
 |-
 | hemipyth[14]
-|m-chro lime|| [[10L 4s]] || 2.260
+| m-chro lime || [[10L 4s]] || 2.260
 |-
 | hemipyth[24]
-|f-enhar lime|| [[10L 14s]] || 1.260
+| f-enhar lime || [[10L 14s]] || 1.260
 |-
-|hemipyth[34]
+| hemipyth[34]
-|paso-lime
+| paso-lime
-|[[24L 10s]]
+| [[24L 10s]]
-|3.8459
+| 3.8459
 |-
-|hemipyth[58]
+| hemipyth[58]
 |
-|[[24L 34s]]
+| [[24L 34s]]
-|2.8459
+| 2.8459
 |-
-|hemipyth[82]
+| hemipyth[82]
 |
-|[[24L 58s]]
+| [[24L 58s]]
-|1.8459
+| 1.8459
 |-
-|hemipyth[106]
+| hemipyth[106]
 |
-|[[82L 24s]]
+| [[82L 24s]]
-|1.1822
+| 1.1822
 |}
 == Music ==
-{{todo|inline-1| Make more music }}
 [[File:The_Hymn_of_Pergele.mp3]]