Marvel–syntonic equivalence continuum: Difference between revisions

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The '''marvel-syntonic equivalence continuum''' is a continuum of 7-limit rank-3 temperament families which equate a number of [[225/224|marvel commas (225/224)]] with an [[81/80|syntonic comma (81/80)]]. This continuum is theoretically interesting in that these are all 7-limit rank-3 temperament families supported by [[septimal meantone]] 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 [[~~marvel family|~~marvel]] 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 septimal meantone (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 2.~~788851128489581...~~, and temperaments having ''n'' near this value will be more accurate.

The '''marvel–syntonic equivalence continuum''' is a [[equivalence continuum|continuum]] of 7-limit rank-3 temperament families which equate a number of [[225/224|marvel commas (225/224)]] with a [[81/80|syntonic comma (81/80)]]. This continuum is theoretically interesting in that these are all 7-limit rank-3 temperament families supported by [[septimal meantone]] temperament.

All temperaments in the continuum satisfy {{nowrap|(225/224)<sup>''n''</sup> ~ 81/80}}. Varying ''n'' results in different temperament families listed in the table below. It converges to [[marvel]] 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 septimal meantone (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 2.788851, and temperaments having ''n'' near this value will be more 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 13:

Line 15:

! Monzo

|-

| [[~~Meantone family|Meantone~~]]

| 0

| [[Didymus]]

| [[81/80]]

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

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

|-

| 1

| [[~~Starling family|~~Starling]]

| [[Starling]]

| [[126/125]]

| {{~~monzo~~|1 2 -3 1⟩}}

| {{Monzo| 1 2 -3 1 }}

|-

| 2

| [[~~Hemimean clan|~~Hemimean]]

| [[Hemimean]]

| [[3136/3125]]

| {{~~monzo~~|6 0 -5 2}}

| {{Monzo| 6 0 -5 2 }}

|-

| 2.75

| {{Nowrap| 1205 & 612 & 1236 }}

| <abbr title="1087046872920473206784/1086294651031494140625">[very long]</abbr>

| {{Monzo| 39 -6 -26 11 }}

|-

| 3

| [[Metric ~~microtemperaments|Meter~~]]

| [[Metric]]

| [[703125/702464]]

| {{~~monzo~~|-11 2 7 -3}}

| {{Monzo| -11 2 7 -3 }}

|-

| ∞

| [[~~Marvel family|~~Marvel]]

| [[Marvel]]

| [[225/224]]

| {{~~monzo~~|-5 2 2 -1}}

| {{Monzo| -5 2 2 -1 }}

|}

[[Category:Equivalence continua]]

@@ Line 1: / Line 1: @@
-The '''marvel-syntonic equivalence continuum''' is a continuum of 7-limit rank-3 temperament families which equate a number of [[225/224|marvel commas (225/224)]] with an [[81/80|syntonic comma (81/80)]]. This continuum is theoretically interesting in that these are all 7-limit rank-3 temperament families supported by [[septimal meantone]] 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 [[marvel family|marvel]] 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 septimal meantone (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 2.788851128489581..., and temperaments having ''n'' near this value will be more accurate.
+The '''marvel–syntonic equivalence continuum''' is a [[equivalence continuum|continuum]] of 7-limit rank-3 temperament families which equate a number of [[225/224|marvel commas (225/224)]] with a [[81/80|syntonic comma (81/80)]]. This continuum is theoretically interesting in that these are all 7-limit rank-3 temperament families supported by [[septimal meantone]] temperament.
+All temperaments in the continuum satisfy {{nowrap|(225/224)<sup>''n''</sup> ~ 81/80}}. Varying ''n'' results in different temperament families listed in the table below. It converges to [[marvel]] 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 septimal meantone (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 2.788851, and temperaments having ''n'' near this value will be more 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 13: / Line 15: @@
 ! Monzo
 |-
-| [[Meantone family|Meantone]]
+| 0
+| [[Didymus]]
 | [[81/80]]
-| {{monzo| -4 4 -1 0}}
+| {{Monzo| -4 4 -1 0 }}
 |-
 | 1
-| [[Starling family|Starling]]
+| [[Starling]]
 | [[126/125]]
-| {{monzo|1 2 -3 1⟩}}
+| {{Monzo| 1 2 -3 1 }}
 |-
 | 2
-| [[Hemimean clan|Hemimean]]
+| [[Hemimean]]
 | [[3136/3125]]
-| {{monzo|6 0 -5 2}}
+| {{Monzo| 6 0 -5 2 }}
+|-
+| 2.75
+| {{Nowrap| 1205 & 612 & 1236 }}
+| <abbr title="1087046872920473206784/1086294651031494140625">[very long]</abbr>
+| {{Monzo| 39 -6 -26 11 }}
 |-
 | 3
-| [[Metric microtemperaments|Meter]]
+| [[Metric]]
 | [[703125/702464]]
-| {{monzo|-11 2 7 -3}}
+| {{Monzo| -11 2 7 -3 }}
 |-
 | ∞
-| [[Marvel family|Marvel]]
+| [[Marvel]]
 | [[225/224]]
-| {{monzo|-5 2 2 -1}}
+| {{Monzo| -5 2 2 -1 }}
 |}
 [[Category:Equivalence continua]]