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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== 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

|}

@@ 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 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
 |}