Hemipyth: Difference between revisions

Line 127:

Luckily we don't need to introduce any more generalizations to the notation to indicate <math>\sqrt{\frac{4}{3}}</math>. It's a neutral {{sfrac|2|1|2}} or a α{{demiflat2}} (alp semiflat) w.r.t middle C.

Nicknames are still assigned to make it easier to talk about the [[5L 4s]] scale generated by <math>\sim~~\vsp~~\sqrt{\frac{4}{3}}</math> against the octave.

Nicknames are still assigned to make it easier to talk about the [[5L 4s]] scale generated by <math>\sim\sqrt{\frac{4}{3}}</math> against the octave.

{| class="wikitable"

Line 179:

The neutral third receives only half of the tuning damage of the fifth so it has a strong character even if the fifth isn't tuned very pure. The irrational nature of <math>\sqrt{\frac{3}{2}}</math> also makes it more tolerant of imprecise tuning.

The same goes for the semifourth. A poorly tuned ~{{sfrac|4|3}} still results in a decent <math>\sim~~\vsp~~\sqrt{\frac{4}{3}}</math> (assuming it's featured in the tuning in the first place).

The same goes for the semifourth. A poorly tuned ~{{sfrac|4|3}} still results in a decent <math>\sim\sqrt{\frac{4}{3}}</math> (assuming it's featured in the tuning in the first place).

=== Signposts ===

Line 187:

== Temperament interpretations ==

Under [[ploidacot]] classification diploid temperaments feature <math>\sim~~\vsp~~\sqrt{2}</math>, dicot temperaments have <math>\sim~~\vsp~~\sqrt{\frac{3}{2}}</math> and alpha-dicot temperaments feature <math>\sim~~\vsp~~\sqrt{\frac{4}{3}}</math> (by virtue of having a <math>\sim~~\vsp~~\sqrt{3}</math>).

Under [[ploidacot]] classification diploid temperaments feature <math>\sim\sqrt{2}</math>, dicot temperaments have <math>\sim\sqrt{\frac{3}{2}}</math> and alpha-dicot temperaments feature <math>\sim\sqrt{\frac{4}{3}}</math> (by virtue of having a <math>\sim\sqrt{3}</math>).

Full hemipyth support is indicated by at least "diploid dicot". Examples include:

Line 193:

|+ style="font-size: 105%;" | Higher-prime interpretations of hemipyth intervals

|-

! Temperament !! <math>\sim~~\vsp~~\sqrt{2}</math> !! <math>\sim~~\vsp~~\sqrt{\frac{3}{2}}</math> !! <math>\sim~~\vsp~~\sqrt{\frac{4}{3}}</math> !! contorted !! rank-2

! Temperament !! <math>\sim\sqrt{2}</math> !! <math>\sim\sqrt{\frac{3}{2}}</math> !! <math>\sim\sqrt{\frac{4}{3}}</math> !! contorted !! rank-2

|-

| [[decimal]] || ~{{sfrac|7|5}} || ~{{sfrac|5|4}} || ~{{sfrac|7|6}} || no || yes

Line 218:

|}

Above contorted tunings don't have a <math>\sim~~\vsp~~\sqrt{2}</math> period with a <math>\sim~~\vsp~~\sqrt{3}</math> generator, but introduce further splits. Higher than rank-2 temperaments introduce further structure that goes beyond basic hemipyth.

Above contorted tunings don't have a <math>\sim\sqrt{2}</math> period with a <math>\sim\sqrt{3}</math> generator, but introduce further splits. Higher than rank-2 temperaments introduce further structure that goes beyond basic hemipyth.

Some possible interpretations for <math>\sim~~\vsp~~\sqrt{2}</math> are:

Some possible interpretations for <math>\sim\sqrt{2}</math> are:

{| class="wikitable"

|+ style="font-size: 105%;" | Higher-prime interpretations of <math>\sqrt{2}</math>

|-

! Temperament !! <math>\sim~~\vsp~~\sqrt{2}</math> !! contorted !! rank-2

! Temperament !! <math>\sim\sqrt{2}</math> !! contorted !! rank-2

|-

| [[jubilic]] || ~{{sfrac|7|5}} || no || yes (2.5.7)

Line 236:

|}

Some possible interpretations for <math>\sim~~\vsp~~\sqrt{3}</math> are:

Some possible interpretations for <math>\sim\sqrt{3}</math> are:

{| class="wikitable"

|+ style="font-size: 105%;" | Higher-prime interpretations of <math>\sqrt{3}</math>

|-

! Temperament !! <math>\sim~~\vsp~~\sqrt{3}</math> !! contorted !! rank-2

! Temperament !! <math>\sim\sqrt{3}</math> !! contorted !! rank-2

|-

| [[semaphore]] || ~{{sfrac|7|4}} || no || yes (2.3.7)

Line 248:

|}

Some possible interpretations for <math>\sim~~\vsp~~\frac{3}{2}</math> are:

Some possible interpretations for <math>\sim\frac{3}{2}</math> are:

{| class="wikitable"

|+ style="font-size: 105%;" | Higher-prime interpretations of <math>\sqrt{\frac{3}{2}}</math>

|-

! Temperament !! <math>\sqrt{~\frac{3}{2}}</math> !! contorted !! rank-2

! Temperament !! <math>\sim\sqrt{\frac{3}{2}}</math> !! contorted !! rank-2

|-

| [[dicot]] || ~{{sfrac|5|4}} || no || yes (2.3.5)

@@ Line 127: / Line 127: @@
 Luckily we don't need to introduce any more generalizations to the notation to indicate <math>\sqrt{\frac{4}{3}}</math>. It's a neutral {{sfrac|2|1|2}} or a α{{demiflat2}} (alp semiflat) w.r.t middle C.
-Nicknames are still assigned to make it easier to talk about the [[5L&nbsp;4s]] scale generated by <math>\sim\vsp\sqrt{\frac{4}{3}}</math> against the octave.
+Nicknames are still assigned to make it easier to talk about the [[5L&nbsp;4s]] scale generated by <math>\sim\sqrt{\frac{4}{3}}</math> against the octave.
 {| class="wikitable"
@@ Line 179: / Line 179: @@
 The neutral third receives only half of the tuning damage of the fifth so it has a strong character even if the fifth isn't tuned very pure. The irrational nature of <math>\sqrt{\frac{3}{2}}</math> also makes it more tolerant of imprecise tuning.
-The same goes for the semifourth. A poorly tuned ~{{sfrac|4|3}} still results in a decent <math>\sim\vsp\sqrt{\frac{4}{3}}</math> (assuming it's featured in the tuning in the first place).
+The same goes for the semifourth. A poorly tuned ~{{sfrac|4|3}} still results in a decent <math>\sim\sqrt{\frac{4}{3}}</math> (assuming it's featured in the tuning in the first place).
 === Signposts ===
@@ Line 187: / Line 187: @@
 == Temperament interpretations ==
-Under [[ploidacot]] classification diploid temperaments feature <math>\sim\vsp\sqrt{2}</math>, dicot temperaments have <math>\sim\vsp\sqrt{\frac{3}{2}}</math> and alpha-dicot temperaments feature <math>\sim\vsp\sqrt{\frac{4}{3}}</math> (by virtue of having a <math>\sim\vsp\sqrt{3}</math>).
+Under [[ploidacot]] classification diploid temperaments feature <math>\sim\sqrt{2}</math>, dicot temperaments have <math>\sim\sqrt{\frac{3}{2}}</math> and alpha-dicot temperaments feature <math>\sim\sqrt{\frac{4}{3}}</math> (by virtue of having a <math>\sim\sqrt{3}</math>).
 Full hemipyth support is indicated by at least "diploid dicot". Examples include:
@@ Line 193: / Line 193: @@
 |+ style="font-size: 105%;" | Higher-prime interpretations of hemipyth intervals
 |-
-! Temperament !! <math>\sim\vsp\sqrt{2}</math> !! <math>\sim\vsp\sqrt{\frac{3}{2}}</math> !! <math>\sim\vsp\sqrt{\frac{4}{3}}</math> !! contorted !! rank-2
+! Temperament !! <math>\sim\sqrt{2}</math> !! <math>\sim\sqrt{\frac{3}{2}}</math> !! <math>\sim\sqrt{\frac{4}{3}}</math> !! contorted !! rank-2
 |-
 | [[decimal]] || ~{{sfrac|7|5}} || ~{{sfrac|5|4}} || ~{{sfrac|7|6}} || no || yes
@@ Line 218: / Line 218: @@
 |}
-Above contorted tunings don't have a <math>\sim\vsp\sqrt{2}</math> period with a <math>\sim\vsp\sqrt{3}</math> generator, but introduce further splits. Higher than rank-2 temperaments introduce further structure that goes beyond basic hemipyth.
+Above contorted tunings don't have a <math>\sim\sqrt{2}</math> period with a <math>\sim\sqrt{3}</math> generator, but introduce further splits. Higher than rank-2 temperaments introduce further structure that goes beyond basic hemipyth.
-Some possible interpretations for <math>\sim\vsp\sqrt{2}</math> are:
+Some possible interpretations for <math>\sim\sqrt{2}</math> are:
 {| class="wikitable"
 |+ style="font-size: 105%;" | Higher-prime interpretations of <math>\sqrt{2}</math>
 |-
-! Temperament !! <math>\sim\vsp\sqrt{2}</math> !! contorted !! rank-2
+! Temperament !! <math>\sim\sqrt{2}</math> !! contorted !! rank-2
 |-
 | [[jubilic]] || ~{{sfrac|7|5}} || no || yes (2.5.7)
@@ Line 236: / Line 236: @@
 |}
-Some possible interpretations for <math>\sim\vsp\sqrt{3}</math> are:
+Some possible interpretations for <math>\sim\sqrt{3}</math> are:
 {| class="wikitable"
 |+ style="font-size: 105%;" | Higher-prime interpretations of <math>\sqrt{3}</math>
 |-
-! Temperament !! <math>\sim\vsp\sqrt{3}</math> !! contorted !! rank-2
+! Temperament !! <math>\sim\sqrt{3}</math> !! contorted !! rank-2
 |-
 | [[semaphore]] || ~{{sfrac|7|4}} || no || yes (2.3.7)
@@ Line 248: / Line 248: @@
 |}
-Some possible interpretations for <math>\sim\vsp\frac{3}{2}</math> are:
+Some possible interpretations for <math>\sim\frac{3}{2}</math> are:
 {| class="wikitable"
 |+ style="font-size: 105%;" | Higher-prime interpretations of <math>\sqrt{\frac{3}{2}}</math>
 |-
-! Temperament !! <math>\sqrt{~\frac{3}{2}}</math> !! contorted !! rank-2
+! Temperament !! <math>\sim\sqrt{\frac{3}{2}}</math> !! contorted !! rank-2
 |-
 | [[dicot]] || ~{{sfrac|5|4}} || no || yes (2.3.5)