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.

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.

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 {{nowrap|(81/80)<sup>''n''</sup> ~ 64/63}}. Varying ''n'' results in different temperament families listed in the table below. It converges to [[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 the 7-limit dominant temperament (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

| [[Archytas]]

| [[64/63]]

| {{~~monzo~~| 6 -2 0 -1 }}

| {{Monzo| 6 -2 0 -1 }}

|-

| 1/2

| 63 & 68 & 80

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

| [[327680/321489]]

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

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

|-

| 1

| [[~~Hemifamity family|~~Hemifamity]]

| [[Hemifamity]]

| [[5120/5103]]

| {{~~monzo~~| 1 5 1 -4 }}

| {{Monzo| 1 5 1 -4 }}

|-

| 5/4

| 894 & 441 & 1106

|

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

| {{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

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| 118 & 125 & 130

| [[2109289329/2097152000]]

| {{~~monzo~~| -24 16 -3 2 }}

| {{Monzo| -24 16 -3 2 }}

|-

| 2

| 72 & 77 & 79

| [[413343/409600]]

| {{~~monzo~~| -14 10 -2 1 }}

| {{Monzo| -14 10 -2 1 }}

|-

| ∞

| [[~~Didymus rank three family|~~Didymus]]

| [[Didymus]]

| [[81/80]]

| {{~~monzo~~| -4 4 -1 0 }}

| {{Monzo| -4 4 -1 0 }}

|}

[[Category:Equivalence continua]]

@@ 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.
+{{Mathematical interest}}
-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.
+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 {{nowrap|(81/80)<sup>''n''</sup> ~ 64/63}}. Varying ''n'' results in different temperament families listed in the table below. It converges to [[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 the 7-limit dominant temperament (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 14: @@
 ! Ratio
 ! Monzo
+|-
+| −1
+| [[Mint]]
+| [[36/35]]
+| {{Monzo| 2 2 -1 -1 }}
 |-
 | 0
 | [[Archytas]]
 | [[64/63]]
-| {{monzo| 6 -2 0 -1 }}
+| {{Monzo| 6 -2 0 -1 }}
 |-
 | 1/2
 | 63 & 68 & 80
-| [[321489/327680]]
+| [[327680/321489]]
-| {{monzo| 16 -8 1 -2 }}
+| {{Monzo| 16 -8 1 -2 }}
 |-
 | 1
-| [[Hemifamity family|Hemifamity]]
+| [[Hemifamity]]
 | [[5120/5103]]
-| {{monzo| 1 5 1 -4 }}
+| {{Monzo| 1 5 1 -4 }}
 |-
 | 5/4
 | 894 & 441 & 1106
 |
-| {{monzo| 44 -28 5 -4 }}
+| {{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 46: / Line 53: @@
 | 118 & 125 & 130
 | [[2109289329/2097152000]]
-| {{monzo| -24 16 -3 2 }}
+| {{Monzo| -24 16 -3 2 }}
 |-
 | 2
 | 72 & 77 & 79
 | [[413343/409600]]
-| {{monzo| -14 10 -2 1 }}
+| {{Monzo| -14 10 -2 1 }}
 |-
 | ∞
-| [[Didymus rank three family|Didymus]]
+| [[Didymus]]
 | [[81/80]]
-| {{monzo| -4 4 -1 0 }}
+| {{Monzo| -4 4 -1 0 }}
 |}
 [[Category:Equivalence continua]]