Schismic–Mercator equivalence continuum: Difference between revisions

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The '''schismic–Mercator equivalence continuum''' is a continuum of 5-limit temperaments which equate a number of [[32805/32768|schismas (32805/32768)]] with [[Mercator's comma|Mercator's comma ({{monzo|-84 53}})]]. This continuum is theoretically interesting in that ~~these~~ are all 5-limit microtemperaments.

The '''schismic–Mercator equivalence continuum''' is a continuum of 5-limit temperaments which equate a number of [[32805/32768|schismas (32805/32768)]] with [[Mercator's comma|Mercator's comma ({{monzo|-84 53}})]]. This continuum is theoretically interesting in that the temperaments associated with its various commas are all 5-limit microtemperaments.

All temperaments in the continuum satisfy (32805/32768)<sup>''n''</sup> ~ {{monzo|-84 53}}. Varying ''n'' results in different temperaments listed in the table below. It converges to [[schismic]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all [[5-limit]] temperaments supported by [[53edo]] (due to it being the unique equal temperament that tempers both commas and thus tempers all combinations of them). The just value of ''n'' is approximately 1.8503390493…, and temperaments having ''n'' near this value tend to be the most accurate ones.

All temperaments in the continuum satisfy {{nowrap|(32805/32768)<sup>''n''</sup> ~ {{monzo|-84 53}}}}. Varying ''n'' results in different temperaments listed in the table below. It converges to [[schismic]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all [[5-limit]] temperaments supported by [[53edo]] (due to it being the unique equal temperament that tempers both commas and thus tempers all combinations of them). The just value of ''n'' is approximately 1.8503390493…, and temperaments having ''n'' near this value tend to be the most accurate ones.

For a similar but perhaps more intuitive and practical concept, see [[Syntonic–chromatic equivalence continuum]].

{| class="wikitable center-1"

|+ Temperaments with integer ''n''

|+ style="font-size: 105%;" | Temperaments with integer ''n''

|-

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

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|}

We may invert the continuum by setting ''m'' such that 1/''m'' + 1/''n'' = 1. This may be called the ''counterschismic-Mercator equivalence continuum'', which is essentially the same thing. The just value of ''m'' is 2.17600…

We may invert the continuum by setting ''m'' such that {{nowrap|{{sfrac|1|''m''}} + {{sfrac|1|''n''}} {{=}} 1}}. This may be called the ''counterschismic–Mercator equivalence continuum'', which is essentially the same thing. The just value of ''m'' is 2.17600…

{| class="wikitable center-1"

|+ Temperaments with integer ''m''

|+ style="font-size: 105%;" | Temperaments with integer ''m''

|-

! rowspan="2" | ''m''

@@ Line 1: / Line 1: @@
-The '''schismic&ndash;Mercator equivalence continuum''' is a continuum of 5-limit temperaments which equate a number of [[32805/32768|schismas (32805/32768)]] with [[Mercator's comma|Mercator's comma ({{monzo|-84 53}})]]. This continuum is theoretically interesting in that these are all 5-limit microtemperaments.
+The '''schismic&ndash;Mercator equivalence continuum''' is a continuum of 5-limit temperaments which equate a number of [[32805/32768|schismas (32805/32768)]] with [[Mercator's comma|Mercator's comma ({{monzo|-84 53}})]]. This continuum is theoretically interesting in that the temperaments associated with its various commas are all 5-limit microtemperaments.
-All temperaments in the continuum satisfy (32805/32768)<sup>''n''</sup> ~ {{monzo|-84 53}}. Varying ''n'' results in different temperaments listed in the table below. It converges to [[schismic]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all [[5-limit]] temperaments supported by [[53edo]] (due to it being the unique equal temperament that tempers both commas and thus tempers all combinations of them). The just value of ''n'' is approximately 1.8503390493…, and temperaments having ''n'' near this value tend to be the most accurate ones.
+All temperaments in the continuum satisfy {{nowrap|(32805/32768)<sup>''n''</sup> ~ {{monzo|-84 53}}}}. Varying ''n'' results in different temperaments listed in the table below. It converges to [[schismic]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all [[5-limit]] temperaments supported by [[53edo]] (due to it being the unique equal temperament that tempers both commas and thus tempers all combinations of them). The just value of ''n'' is approximately 1.8503390493…, and temperaments having ''n'' near this value tend to be the most accurate ones.
 For a similar but perhaps more intuitive and practical concept, see [[Syntonic&ndash;chromatic equivalence continuum]].
 {| class="wikitable center-1"
-|+ Temperaments with integer ''n''
+|+ style="font-size: 105%;" | Temperaments with integer ''n''
 |-
 ! rowspan="2" | ''n''
@@ Line 76: / Line 76: @@
 |}
-We may invert the continuum by setting ''m'' such that 1/''m'' + 1/''n'' = 1. This may be called the ''counterschismic-Mercator equivalence continuum'', which is essentially the same thing. The just value of ''m'' is 2.17600…
+We may invert the continuum by setting ''m'' such that {{nowrap|{{sfrac|1|''m''}} + {{sfrac|1|''n''}} {{=}} 1}}. This may be called the ''counterschismic&ndash;Mercator equivalence continuum'', which is essentially the same thing. The just value of ''m'' is 2.17600…
 {| class="wikitable center-1"
-|+ Temperaments with integer ''m''
+|+ style="font-size: 105%;" | Temperaments with integer ''m''
 |-
 ! rowspan="2" | ''m''