Syntonic–chromatic equivalence continuum: Difference between revisions
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If we let {{nowrap|''k'' {{=}} ''n'' − 2}} so that {{nowrap|''k'' {{=}} 0}} means {{nowrap|''n'' {{=}} 2}}, {{nowrap|''k'' {{=}} −1}} means {{nowrap|''n'' {{=}} 1}}, etc. then the continuum corresponds to {{nowrap|(81/80)<sup>''k''</sup> {{=}} 25/24}}. Some prefer this way of conceptualising it because: | If we let {{nowrap|''k'' {{=}} ''n'' − 2}} so that {{nowrap|''k'' {{=}} 0}} means {{nowrap|''n'' {{=}} 2}}, {{nowrap|''k'' {{=}} −1}} means {{nowrap|''n'' {{=}} 1}}, etc. then the continuum corresponds to {{nowrap|(81/80)<sup>''k''</sup> {{=}} 25/24}}. Some prefer this way of conceptualising it because: | ||
* 25/24 is the classic chromatic semitone, notable in the 5-limit as the difference between 5/4 and 6/5, so this shifted continuum could also logically be termed the "syntonic-chromatic equivalence continuum". This means that at {{nowrap|''k'' {{=}} 0}}, 5/4 and 6/5 are mapped to the same interval while 81/80 becomes independent of 25/24 (meaning 81/80 may or may not be tempered) because the relation becomes {{nowrap|(81/80)<sup>0</sup> ~ 1/1 ~ 25/24}}. | * 25/24 is the classic chromatic semitone, notable in the 5-limit as the difference between 5/4 and 6/5, so this shifted continuum could also logically be termed the "syntonic-chromatic equivalence continuum". This means that at {{nowrap|''k'' {{=}} 0}}, 5/4 and 6/5 are mapped to the same interval while 81/80 becomes independent of 25/24 (meaning 81/80 may or may not be tempered) because the relation becomes {{nowrap|(81/80)<sup>0</sup> ~ 1/1 ~ 25/24}}. | ||
* {{nowrap|''k'' {{=}} 1}} and upwards up to a point represent temperaments with the potential for reasonably good accuracy as equating at least one 81/80 with 25/24 seems like a good lower bound for a temperament intended to model JI. A good upper bound might be gravity {{nowrap|(''k'' {{=}} 4)}}, with the only exception being meantone {{nowrap|(''n'' {{=}} ''k'' {{=}} ∞)}}. Temperaments corresponding to {{nowrap|''k'' {{=}} 0, −1, −2 | * {{nowrap|''k'' {{=}} 1}} and upwards up to a point represent temperaments with the potential for reasonably good accuracy as equating at least one 81/80 with 25/24 seems like a good lower bound for a temperament intended to model JI. A good upper bound might be gravity {{nowrap|(''k'' {{=}} 4)}}, with the only exception being meantone {{nowrap|(''n'' {{=}} ''k'' {{=}} ∞)}}. Temperaments corresponding to {{nowrap|''k'' {{=}} 0, −1, −2, …}} are comparatively low-accuracy to the point of developing various intriguing structures and consequences. | ||
* 25/24 is the simplest ratio to be tempered in the continuum, the second simplest being 81/80 at unsigned infinity, which together are the two smallest 5-limit [[List of superparticular intervals|superparticular intervals]] and the only superparticular intervals in the continuum. | * 25/24 is the simplest ratio to be tempered in the continuum, the second simplest being 81/80 at unsigned infinity, which together are the two smallest 5-limit [[List of superparticular intervals|superparticular intervals]] and the only superparticular intervals in the continuum. | ||
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We may invert the continuum by setting ''m'' such that {{nowrap|{{frac|1|''m''}} + {{frac|1|''n''}} {{=}} 1}}. This may be called the ''mavila/pelogic-chromatic equivalence continuum'', which is essentially the same thing. The just value of ''m'' is 1.2333… The [[135/128| | We may invert the continuum by setting ''m'' such that {{nowrap|{{frac|1|''m''}} + {{frac|1|''n''}} {{=}} 1}}. This may be called the ''mavila/pelogic-chromatic equivalence continuum'', which is essentially the same thing. The just value of ''m'' is 1.2333… The [[135/128|mavila comma]] is both larger and more complex than the syntonic comma. As such, this continuum does not contain as many useful temperaments, but still interesting nonetheless. | ||
{| class="wikitable center-1" | {| class="wikitable center-1" | ||
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{{Mapping|legend=1| 1 2 2 | 0 -3 2 }} | {{Mapping|legend=1| 1 2 2 | 0 -3 2 }} | ||
: | : mapping generators: ~2, ~16/15 | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
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{{Mapping|legend=1| 7 0 -17 | 0 1 3 }} | {{Mapping|legend=1| 7 0 -17 | 0 1 3 }} | ||
: | : mapping generators: ~800/729, ~3 | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
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{{Mapping|legend=1| 1 8 18 | 0 -9 -22 }} | {{Mapping|legend=1| 1 8 18 | 0 -9 -22 }} | ||
: | : mapping generators: ~2, ~400/243 | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
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{{Mapping|legend=1| 1 0 -15 | 0 1 11 }} | {{Mapping|legend=1| 1 0 -15 | 0 1 11 }} | ||
: | : mapping generators: ~2, ~3 | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
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{{Mapping|legend=1| 1 0 18 | 0 1 -10 }} | {{Mapping|legend=1| 1 0 18 | 0 1 -10 }} | ||
: | : mapping generators: ~2, ~3 | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
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{{Mapping|legend=1| 1 1 -1 | 0 2 15 }} | {{Mapping|legend=1| 1 1 -1 | 0 2 15 }} | ||
: | : mapping generators: ~2, ~2560/2187 | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
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{{Mapping|legend=1| 1 2 5 | 0 -3 -19 }} | {{Mapping|legend=1| 1 2 5 | 0 -3 -19 }} | ||
: | : mapping generators: ~2, ~729/640 | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||