Sensamagic–gariboh equivalence continuum: Difference between revisions
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All temperaments in the continuum satisfy {{nowrap|(245/243)<sup>''n''</sup> ~ 3125/3087}}. Varying ''n'' results in different temperaments listed in the table below. It converges to [[Bohlen–Pierce–Stearns]] as ''n'' approaches infinity and [[Sirius]] as ''n'' approaches 0. If we allow non-integer and infinite ''n'', the continuum describes the set of all 3.5.7 subgroup temperaments supported by [[13edt]] (due to it being the unique equal temperament that tempers both commas and thus tempers all combinations of them); additionally, if prime 2 is added, this describes the set of all rank-3 7-limit temperaments supported by [[bohpier]]. Due to 13edt, the equal-tempered [[Bohlen–Pierce scale]], being a very good representation of the subgroup, and many named temperaments within the subgroup being supported by 13edt, this continuum is structurally important. The just value of ''n'' is 1.4926…, and temperaments near this tend to be the most accurate ones. | All temperaments in the continuum satisfy {{nowrap|(245/243)<sup>''n''</sup> ~ 3125/3087}}. Varying ''n'' results in different temperaments listed in the table below. It converges to [[Bohlen–Pierce–Stearns]] as ''n'' approaches infinity and [[Sirius]] as ''n'' approaches 0. If we allow non-integer and infinite ''n'', the continuum describes the set of all 3.5.7 subgroup temperaments supported by [[13edt]] (due to it being the unique equal temperament that tempers both commas and thus tempers all combinations of them); additionally, if prime 2 is added, this describes the set of all rank-3 7-limit temperaments supported by [[bohpier]]. Due to 13edt, the equal-tempered [[Bohlen–Pierce scale]], being a very good representation of the subgroup, and many named temperaments within the subgroup being supported by 13edt, this continuum is structurally important. The just value of ''n'' is 1.4926…, and temperaments near this tend to be the most accurate ones. | ||
An alternative definition of this equivalence continuum satisfies (245/243)<sup>''k''</sup> ~ 7<sup>13</sup>/3<sup>23</sup> = 96889010407/94143178827, which sets ''k'' = 5 | An alternative definition of this equivalence continuum satisfies {{nowrap|(245/243)<sup>''k''</sup> ~ 7<sup>13</sup>/3<sup>23</sup>}} = 96889010407/94143178827, which sets {{nowrap|''k'' {{=}} 5 − ''n''}}. This definition might be advantageous as ''k'' gives the order of harmonic 5 in the corresponding comma, and equals the number of generators to obtain a harmonic 7 in the MOS scale. | ||
{| class="wikitable center-1 center-2" | {| class="wikitable center-1 center-2" | ||
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| 2 | | 2 | ||
| 3 | | 3 | ||
| Procyon | | [[Procyon]] | ||
| 823543/820125 | | 823543/820125 | ||
| {{monzo| -8 -3 7 }} | | {{monzo| -8 -3 7 }} | ||
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{| class="wikitable center-1 center-2" | {| class="wikitable center-1 center-2" | ||
|+ style="font-size: 105%;" Temperaments in the continuum | |+ style="font-size: 105%;" | Temperaments in the continuum | ||
|- | |- | ||
! rowspan="2" | ''m'' | ! rowspan="2" | ''m'' | ||
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|- | |- | ||
| 1 | | 1 | ||
| [[ | | [[Bohlen–Pierce–Stearns]] | ||
| [[245/243]] | | [[245/243]] | ||
| {{monzo| -5 1 2 }} | | {{monzo| -5 1 2 }} | ||
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== Projective tuning space == | == Projective tuning space == | ||
Below is a depiction of the temperaments of this continuum in 3.5.7 [[projective tuning space]]. | Below is a depiction of the temperaments of this continuum in 3.5.7 [[projective tuning space]]. | ||
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</div>}} | </div>}} | ||
[[Category: | [[Category:Bohlen–Pierce]] | ||
[[Category:Equivalence continua]] | [[Category:Equivalence continua]] | ||