Ennealimmal–enneadecal equivalence continuum: Difference between revisions
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The ''' | The '''ennealimmal–enneadecal equivalence continuum''' is a [[equivalence continuum|continuum]] of 5-limit temperaments which equate a number of [[ennealimma|ennealimmas ({{monzo|1 -27 18}})]] with [[Enneadeca|enneadeca comma ({{monzo|-14 -19 19}})]]. This continuum is theoretically interesting in that these are all 5-limit microtemperaments. | ||
All temperaments in the continuum satisfy (2･3<sup> | All temperaments in the continuum satisfy {{nowrap|(2･3<sup>−27</sup>･5<sup>18</sup>)<sup>''n''</sup> ~ (2<sup>−14</sup>･3<sup>−19</sup>･5<sup>19</sup>)}}. Varying ''n'' results in different temperaments listed in the table below. It converges to [[ennealimmal]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all [[5-limit]] temperaments supported by [[171edo]] (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 3.2669545024..., and temperaments having ''n'' near this value tend to be the most accurate ones. | ||
{| class="wikitable center-1 center-2" | {| class="wikitable center-1 center-2" | ||
|+ Temperaments in the continuum | |+ style="font-size: 105%" | Temperaments in the continuum | ||
|- | |- | ||
! rowspan="2" | ''n'' | ! rowspan="2" | ''n'' | ||
| Line 13: | Line 13: | ||
! Monzo | ! Monzo | ||
|- | |- | ||
| | | −4 | ||
| 171 & 2429 | | 171 & 2429 | ||
| | | | ||
| {{monzo|-10 -127 91}} | | {{monzo|-10 -127 91}} | ||
|- | |- | ||
| | | −3 | ||
| 171 & 1817 | | 171 & 1817 | ||
| | | | ||
| {{monzo|-11 -100 73}} | | {{monzo|-11 -100 73}} | ||
|- | |- | ||
| | | −2 | ||
| [[Ragismic microtemperaments #Semidimi|Semidimi]] | | [[Ragismic microtemperaments #Semidimi|Semidimi]] | ||
| | | | ||
| {{monzo|-12 -73 55}} | | {{monzo|-12 -73 55}} | ||
|- | |- | ||
| | | −1 | ||
| [[Ragismic microtemperaments #Supermajor|Supermajor]] | | [[Ragismic microtemperaments #Supermajor|Supermajor]] | ||
| | | | ||
| Line 75: | Line 75: | ||
Examples of temperaments with fractional values of ''n'': | Examples of temperaments with fractional values of ''n'': | ||
* 171 & 3193 (''n'' = | * 171 & 3193 ({{nowrap|''n'' {{=}} −5/2 {{=}} −2.5}}) | ||
* 171 & 2140 (''n'' = | * 171 & 2140 ({{nowrap|''n'' {{=}} −3/2 {{=}} −1.5}}) | ||
* 171 & 1087 (''n'' = | * 171 & 1087 ({{nowrap|''n'' {{=}} −1/2 {{=}} −0.5}}) | ||
* [[Landscape microtemperaments #Pnict|Pnict]] (''n'' = 1/3 = 0.{{overline|3}}) | * [[Landscape microtemperaments #Pnict|Pnict]] ({{nowrap|''n'' {{=}} 1/3 {{=}} 0.{{overline|3}}}}) | ||
* [[Gammic family|Gammic]] (''n'' = 1/2 = 0.5) | * [[Gammic family|Gammic]] ({{nowrap|''n'' {{=}} 1/2 {{=}} 0.5}}) | ||
* [[Horwell temperaments #Mutt|Mutt]] (''n'' = 2/3 = 0.{{overline|6}}) | * [[Horwell temperaments #Mutt|Mutt]] ({{nowrap|''n'' {{=}} 2/3 {{=}} 0.{{overline|6}}}}) | ||
* [[Landscape microtemperaments #Septichrome|Septichrome]] (''n'' = 4/3 = 1.{{overline|3}}) | * [[Landscape microtemperaments #Septichrome|Septichrome]] ({{nowrap|''n'' {{=}} 4/3 {{=}} 1.{{overline|3}}}}) | ||
* [[Metric microtemperaments #Geb|Geb]] (''n'' = 3/2 = 1.5) | * [[Metric microtemperaments #Geb|Geb]] ({{nowrap|''n'' {{=}} 3/2 {{=}} 1.5}}) | ||
* 171 & 1901 (''n'' = 5/2 = 2.5) | * 171 & 1901 ({{nowrap|''n'' {{=}} 5/2 {{=}} 2.5}}) | ||
* 171 & 4125 (''n'' = 10/3 = 3.{{overline|3}}) | * 171 & 4125 ({{nowrap|''n'' {{=}} 10/3 {{=}} 3.{{overline|3}}}}) | ||
* 171 & 3125 (''n'' = 7/2 = 3.5) | * 171 & 3125 ({{nowrap|''n'' {{=}} 7/2 {{=}} 3.5}}) | ||
[[Category:171edo]] | [[Category:171edo]] | ||
[[Category:Equivalence continua]] | [[Category:Equivalence continua]] | ||
Latest revision as of 19:48, 17 December 2024
The ennealimmal–enneadecal equivalence continuum is a continuum of 5-limit temperaments which equate a number of ennealimmas ([1 -27 18⟩) with enneadeca comma ([-14 -19 19⟩). This continuum is theoretically interesting in that these are all 5-limit microtemperaments.
All temperaments in the continuum satisfy (2･3−27･518)n ~ (2−14･3−19･519). Varying n results in different temperaments listed in the table below. It converges to ennealimmal as n approaches infinity. If we allow non-integer and infinite n, the continuum describes the set of all 5-limit temperaments supported by 171edo (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 3.2669545024..., and temperaments having n near this value tend to be the most accurate ones.
| n | Temperament | Comma | |
|---|---|---|---|
| Ratio | Monzo | ||
| −4 | 171 & 2429 | [-10 -127 91⟩ | |
| −3 | 171 & 1817 | [-11 -100 73⟩ | |
| −2 | Semidimi | [-12 -73 55⟩ | |
| −1 | Supermajor | [-13 -46 37⟩ | |
| 0 | Enneadecal | 19073486328125/19042491875328 | [-14 -19 19⟩ |
| 1 | Schismic | 32805/32768 | [-15 8 1⟩ |
| 2 | Minortone | 50031545098999707/50000000000000000 | [-16 35 -17⟩ |
| 3 | Senior | [-17 62 -35⟩ | |
| 4 | 171 & 1783 | [18 -89 53⟩ | |
| 5 | 171 & 2395 | [19 -116 71⟩ | |
| … | … | … | … |
| ∞ | Ennealimmal | 7629394531250/7625597484987 | [1 -27 18⟩ |
Examples of temperaments with fractional values of n:
- 171 & 3193 (n = −5/2 = −2.5)
- 171 & 2140 (n = −3/2 = −1.5)
- 171 & 1087 (n = −1/2 = −0.5)
- Pnict (n = 1/3 = 0.3)
- Gammic (n = 1/2 = 0.5)
- Mutt (n = 2/3 = 0.6)
- Septichrome (n = 4/3 = 1.3)
- Geb (n = 3/2 = 1.5)
- 171 & 1901 (n = 5/2 = 2.5)
- 171 & 4125 (n = 10/3 = 3.3)
- 171 & 3125 (n = 7/2 = 3.5)