Schismic–Pythagorean equivalence continuum: Difference between revisions
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All temperaments in the continuum satisfy {{nowrap|(32805/32768)<sup>''n''</sup> ~ {{monzo| -19 12 }}}}. 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 12edo due to it being the unique equal temperament that [[tempering out|tempers out]] both commas and thus tempers out all combinations of them. The just value of ''n'' is approximately 12.0078623975…, and temperaments having ''n'' near this value tend to be the most accurate ones – indeed, the fact that this number is so close to 12 reflects how small [[Kirnberger's atom]] (the difference between 12 schismas and the Pythagorean comma) is. | All temperaments in the continuum satisfy {{nowrap|(32805/32768)<sup>''n''</sup> ~ {{monzo| -19 12 }}}}. 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 12edo due to it being the unique equal temperament that [[tempering out|tempers out]] both commas and thus tempers out all combinations of them. The just value of ''n'' is approximately 12.0078623975…, and temperaments having ''n'' near this value tend to be the most accurate ones – indeed, the fact that this number is so close to 12 reflects how small [[Kirnberger's atom]] (the difference between 12 schismas and the Pythagorean comma) is. | ||
The [[Pythagorean comma]] is the characteristic 3-limit comma tempered out in 12edo, and has many advantages as a target. In each case, ''n'' equals the order of [[5/1|harmonic 5]] in the corresponding comma, and equals the number of steps to obtain the interval class of [[3/1|harmonic 3]] in the generator chain. For an ''n'' that is not coprime with 12, however, the corresponding temperament splits the [[octave]] into gcd (''n'', | The [[Pythagorean comma]] is the characteristic 3-limit comma tempered out in 12edo, and has many advantages as a target. In each case, ''n'' equals the order of [[5/1|harmonic 5]] in the corresponding comma, and equals the number of steps to obtain the interval class of [[3/1|harmonic 3]] in the generator chain. For an ''n'' that is not coprime with 12, however, the corresponding temperament splits the [[octave]] into {{nowrap| gcd(''n'', 12) }} parts, and splits the interval class of 3 into {{nowrap| ''n''/gcd(''n'', 12) }}. For example: | ||
* [[Meantone]] ({{nowrap|''n'' {{=}} 1}}) is generated by a fifth with an unsplit octave; | * [[Meantone]] ({{nowrap| ''n'' {{=}} 1 }}) is generated by a fifth with an unsplit octave; | ||
* [[Diaschismic]] ({{nowrap|''n'' {{=}} 2}}) splits the octave in two, as 2 divides 12; | * [[Diaschismic]] ({{nowrap| ''n'' {{=}} 2 }}) splits the octave in two, as 2 divides 12; | ||
* [[Misty]] ({{nowrap|''n'' {{=}} 3}}) splits the octave in three, as 3 divides 12; | * [[Misty]] ({{nowrap| ''n'' {{=}} 3 }}) splits the octave in three, as 3 divides 12; | ||
* [[Undim]] ({{nowrap|''n'' {{=}} 4}}) splits the octave in four, as 4 divides 12; | * [[Undim]] ({{nowrap| ''n'' {{=}} 4 }}) splits the octave in four, as 4 divides 12; | ||
* [[Quindromeda]] ({{nowrap|''n'' {{=}} 5}}) does not split the octave but splits the fourth in five, as 5 is coprime with 12. | * [[Quindromeda]] ({{nowrap| ''n'' {{=}} 5 }}) does not split the octave but splits the fourth in five, as 5 is coprime with 12. | ||
Alternatively, because the the 5-limit otonal detemperament of 12edo is a 4×3 rectangle (known as the [[duodene]]), we may be interested in expressing the continuum in terms of the boundary commas of this detemper, that is, as {{nowrap| ([[81/80]])<sup>''k''</sup> ~ ([[128/125]]) }}. This corresponds to these commas' structural significance via 128/125 being entirely in the 2.5 subgroup while 81/80 explains 5 in the simplest way relative to the 3-limit. This choice of coordinates has the advantage of finding all temperaments discussed in a relatively intuitive and simple way so that less accurate but structurally simpler temperaments are found at integer points while microtemperaments converging to atomic are found as successive mediants towards the JIP. Specifically, its JIP is at 1.90915584… which is approximated very closely by the microtemperament atomic at {{nowrap| 21/11 {{=}} 1.90909… }} so that the main continuum can be seen as taking successive mediants towards 2/1 (schismic) starting from 1/1 (diaschismic). It is noted for its nontrivial relation to the other better-motivated (in terms of mapping) coordinates discussed. | Alternatively, because the the 5-limit otonal detemperament of 12edo is a 4×3 rectangle (known as the [[duodene]]), we may be interested in expressing the continuum in terms of the boundary commas of this detemper, that is, as {{nowrap| ([[81/80]])<sup>''k''</sup> ~ ([[128/125]]) }}. This corresponds to these commas' structural significance via 128/125 being entirely in the 2.5 subgroup while 81/80 explains 5 in the simplest way relative to the 3-limit. This choice of coordinates has the advantage of finding all temperaments discussed in a relatively intuitive and simple way so that less accurate but structurally simpler temperaments are found at integer points while microtemperaments converging to atomic are found as successive mediants towards the JIP. Specifically, its JIP is at 1.90915584… which is approximated very closely by the microtemperament atomic at {{nowrap| 21/11 {{=}} 1.90909… }} so that the main continuum can be seen as taking successive mediants towards 2/1 (schismic) starting from 1/1 (diaschismic). It is noted for its nontrivial relation to the other better-motivated (in terms of mapping) coordinates discussed. | ||
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| {{monzo| -176 92 13 }} | | {{monzo| -176 92 13 }} | ||
|- | |- | ||
| … | |||
| … | | … | ||
| … | | … | ||
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| {{monzo| -5 12 -6 }} | | {{monzo| -5 12 -6 }} | ||
|- | |- | ||
| … | |||
| … | | … | ||
| … | | … | ||
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[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[ | * [[WE]]: ~2 = 1200.8769{{c}}, ~3/2 = 699.5409{{c}} | ||
: [[error map]]: {{val| +0.876 -1.537 +0.203 }} | |||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 699.0789{{c}} | * [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 699.0789{{c}} | ||
: error map: {{val| 0.000 -2.876 -1.051 }} | |||
<!-- * [[CTE]]: ~2 = 1200.0000{{c}}, ~3/2 = 699.1680{{c}} --> | |||
{{Optimal ET sequence|legend=1| 12, …, 79, 91, 103 }} | {{Optimal ET sequence|legend=1| 12, …, 79, 91, 103 }} | ||
[[Badness]] ( | [[Badness]] (Sintel): 6.92 | ||
== Sextile == | == Sextile == | ||
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[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[ | * [[WE]]: ~4096/3645 = 199.9836{{c}}, ~3/2 = 702.1782{{c}} (~4428675/4194304 = 97.7564{{c}}) | ||
: [[error map]]: {{val| -0.098 +0.125 +0.045 }} | |||
* [[CWE]]: ~4096/3645 = 200.0000{{c}}, ~3/2 = 702.2434{{c}} (~4428675/4194304 = 97.7566{{c}}) | * [[CWE]]: ~4096/3645 = 200.0000{{c}}, ~3/2 = 702.2434{{c}} (~4428675/4194304 = 97.7566{{c}}) | ||
: error map: {{val| 0.000 +0.288 +0.226 }} | |||
<!-- * [[CTE]]: ~4096/3645 = 200.0000{{c}}, ~3/2 = 702.2627{{c}} (~4428675/4194304 = 97.7373{{c}}) --> | |||
{{Optimal ET sequence|legend=1| 12, …, 222, 234, 246, 258, 270, 1068, 1338, 1608, 1878, 4026bc }} | {{Optimal ET sequence|legend=1| 12, …, 222, 234, 246, 258, 270, 1068, 1338, 1608, 1878, 4026bc }} | ||
[[Badness]] ( | [[Badness]] (Sintel): 13.0 | ||
== Heptacot == | == Heptacot == | ||
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[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[ | * [[WE]]: ~2 = 1199.9328{{c}}, ~{{monzo| -37 19 3 }} = 100.3012{{c}} | ||
: [[error map]]: {{val| -0.067 +0.086 +0.029 }} | |||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~{{monzo| -37 19 3 }} = 100.3076{{c}} | * [[CWE]]: ~2 = 1200.0000{{c}}, ~{{monzo| -37 19 3 }} = 100.3076{{c}} | ||
: error map: {{val| 0.000 +0.198 +0.153 }} | |||
<!-- * [[CTE]]: ~2 = 1200.0000{{c}}, ~{{monzo| -37 19 3 }} = 100.3094{{c}} --> | |||
{{Optimal ET sequence|legend=1| 12, …, 287, 299, 311, 323, 981, 1304, 5539bc, 6843bbcc }} | {{Optimal ET sequence|legend=1| 12, …, 287, 299, 311, 323, 981, 1304, 5539bc, 6843bbcc }} | ||
[[Badness]] ( | [[Badness]] (Sintel): 16.0 | ||
[[Category:12edo]] | [[Category:12edo]] | ||
[[Category:Equivalence continua]] | [[Category:Equivalence continua]] | ||