Syntonic–Archytas equivalence continuum: Difference between revisions

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The '''~~syntonic-Archytas~~ equivalence continuum''' is a continuum of 7-limit rank-3 temperament families which equate a number of [[81/80|syntonic commas (81/80)]] with an [[64/63|Archytas comma (64/63)]]. This continuum is theoretically interesting in that these are all 7-limit rank-3 temperament families supported by [[Meantone family#Dominant|dominant]] temperament.

The '''syntonic–Archytas equivalence continuum''' is a continuum of 7-limit rank-3 temperament families which equate a number of [[81/80|syntonic commas (81/80)]] with an [[64/63|Archytas comma (64/63)]]. This continuum is theoretically interesting in that these are all 7-limit rank-3 temperament families supported by [[Meantone family#Dominant|dominant]] temperament.

All temperaments in the continuum satisfy (81/80)<sup>''n''</sup> ~ 64/63. Varying ''n'' results in different temperament families listed in the table below. It converges to [[Didymus rank three family|didymus]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all [[7-limit]] temperament families supported by squares (due to it being the unique rank-2 temperament that tempers both commas and thus tempers all combinations of them). The just value of ''n'' is approximately 1.~~267726433120519…~~, and temperaments having ''n'' near this value will be ~~more~~ accurate.

All temperaments in the continuum satisfy {{nowrap|(81/80)<sup>''n''</sup> ~ 64/63}}. Varying ''n'' results in different temperament families listed in the table below. It converges to [[Didymus rank three family|didymus]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all [[7-limit]] temperament families supported by squares (due to it being the unique rank-2 temperament that tempers both commas and thus tempers all combinations of them). The just value of ''n'' is approximately 1.267726, and temperaments having ''n'' near this value will be the most accurate.

{| class="wikitable center-1 center-2"

|+ Temperament families in the continuum

|+ style="font-size: 105%;" | Temperament families in the continuum

|-

! rowspan="2" | ''n''

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! Ratio

! Monzo

|-

| −1

| [[Mint]]

| [[36/35]]

| {{monzo| 2 2 -1 -1 }}

|-

| 0

Line 20:

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| 1/2

| 63 & 68 & 80

| [[321489~~/327680~~]]

| [[327680/321489]]

| {{monzo| 16 -8 1 -2 }}

|-

Line 33:

Line 38:

| {{monzo| 44 -28 5 -4 }}

|-

|19/15

| 19/15

|5 & 12 & 836

| 5 & 12 & 836

|

|[166 -106 19 -15⟩

| [166 -106 19 -15⟩

|-

| 4/3

@@ Line 1: / Line 1: @@
-The '''syntonic-Archytas equivalence continuum''' is a continuum of 7-limit rank-3 temperament families which equate a number of [[81/80|syntonic commas (81/80)]] with an [[64/63|Archytas comma (64/63)]]. This continuum is theoretically interesting in that these are all 7-limit rank-3 temperament families supported by [[Meantone family#Dominant|dominant]] temperament.
+The '''syntonic–Archytas equivalence continuum''' is a continuum of 7-limit rank-3 temperament families which equate a number of [[81/80|syntonic commas (81/80)]] with an [[64/63|Archytas comma (64/63)]]. This continuum is theoretically interesting in that these are all 7-limit rank-3 temperament families supported by [[Meantone family#Dominant|dominant]] temperament.
-All temperaments in the continuum satisfy (81/80)<sup>''n''</sup> ~ 64/63. Varying ''n'' results in different temperament families listed in the table below. It converges to [[Didymus rank three family|didymus]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all [[7-limit]] temperament families supported by squares (due to it being the unique rank-2 temperament that tempers both commas and thus tempers all combinations of them). The just value of ''n'' is approximately 1.267726433120519…, and temperaments having ''n'' near this value will be more accurate.
+All temperaments in the continuum satisfy {{nowrap|(81/80)<sup>''n''</sup> ~ 64/63}}. Varying ''n'' results in different temperament families listed in the table below. It converges to [[Didymus rank three family|didymus]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all [[7-limit]] temperament families supported by squares (due to it being the unique rank-2 temperament that tempers both commas and thus tempers all combinations of them). The just value of ''n'' is approximately 1.267726, and temperaments having ''n'' near this value will be the most accurate.
 {| class="wikitable center-1 center-2"
-|+ Temperament families in the continuum
+|+ style="font-size: 105%;" | Temperament families in the continuum
 |-
 ! rowspan="2" | ''n''
@@ Line 12: / Line 12: @@
 ! Ratio
 ! Monzo
+|-
+| −1
+| [[Mint]]
+| [[36/35]]
+| {{monzo| 2 2 -1 -1 }}
 |-
 | 0
@@ Line 20: / Line 25: @@
 | 1/2
 | 63 & 68 & 80
-| [[321489/327680]]
+| [[327680/321489]]
 | {{monzo| 16 -8 1 -2 }}
 |-
@@ Line 33: / Line 38: @@
 | {{monzo| 44 -28 5 -4 }}
 |-
-|19/15
+| 19/15
-|5 & 12 & 836
+| 5 & 12 & 836
 |
-|[166 -106 19 -15⟩
+| [166 -106 19 -15⟩
 |-
 | 4/3