Diaschismic–gothmic equivalence continuum: Difference between revisions
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The '''diaschismic–gothmic equivalence continuum''' (or '''diaschismic–tetracot equivalence continuum''') is a [[equivalence continuum|continuum]] of [[5-limit]] [[regular temperament|temperaments]] describing the set of all [[5-limit]] temperaments [[support]]ed by [[34edo]]. | The '''diaschismic–gothmic equivalence continuum''' (or '''diaschismic–tetracot equivalence continuum''') is a [[equivalence continuum|continuum]] of [[5-limit]] [[regular temperament|temperaments]] describing the set of all [[5-limit]] temperaments [[support]]ed by [[34edo]]. | ||
All temperaments in the continuum satisfy {{nowrap|(2048/2025)<sup>''n''</sup> ~ {{monzo| 27 -17 }}}}, equating a number of [[2048/2025|diaschismas (2048/2025)]] with the [[gothic comma|gothic comma (134217728/129140163)]]. At {{nowrap|''n'' {{=}} 2}} (which we align with {{nowrap|''r'' {{=}} 0}}) we get tetracot, which is an important offset for a number of reasons discussed in [[#Significance of tetracot]]. Varying ''n'' results in different temperaments listed in the table below. It converges to [[diaschismic]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all 5-limit temperaments supported by 34edo 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 3.41464…, and temperaments having ''n'' near this value tend to be the most accurate ones. | All temperaments in the continuum satisfy {{nowrap| (2048/2025)<sup>''n''</sup> ~ {{monzo| 27 -17 }} }}, equating a number of [[2048/2025|diaschismas (2048/2025)]] with the [[gothic comma|gothic comma (134217728/129140163)]]. At {{nowrap| ''n'' {{=}} 2 }} (which we align with {{nowrap| ''r'' {{=}} 0 }}) we get tetracot, which is an important offset for a number of reasons discussed in [[#Significance of tetracot]]. Varying ''n'' results in different temperaments listed in the table below. It converges to [[diaschismic]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all 5-limit temperaments supported by 34edo 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 3.41464…, and temperaments having ''n'' near this value tend to be the most accurate ones. | ||
The [[17-comma|Pythagorean gothma]] a.k.a. gothic comma is the characteristic [[3-limit]] comma tempered out in 34edo. Describing the continuum this way has notable advantages – in particular, due to being determined in terms of the 3-limit comma and the comma with the next lowest power of 5, twice the numerator of the value of ''n'' represents the number of generator steps required to reach the interval class of [[3/1|harmonic 3]]. For example: | The [[17-comma|Pythagorean gothma]] a.k.a. gothic comma is the characteristic [[3-limit]] comma tempered out in 34edo. Describing the continuum this way has notable advantages – in particular, due to being determined in terms of the 3-limit comma and the comma with the next lowest power of 5, twice the numerator of the value of ''n'' represents the number of generator steps required to reach the interval class of [[3/1|harmonic 3]]. For example: | ||
* [[Immunity]] ({{nowrap|''n'' {{=}} 1}}) splits its twelfth in two; | * [[Immunity]] ({{nowrap| ''n'' {{=}} 1 }}) splits its twelfth in two; | ||
* [[Tetracot]] ({{nowrap|''n'' {{=}} 2}}) splits its fifth in four; | * [[Tetracot]] ({{nowrap| ''n'' {{=}} 2 }}) splits its fifth in four; | ||
* [[ | * [[Kleismic]] ({{nowrap| ''n'' {{=}} 3 }}) splits its twelfth in six; | ||
* Etc. | * Etc. | ||
The factor of 2 between ''n'' and the split of the interval class of 3 has to do with the fact that 34et has two [[ring number|rings]] of 17et's. | The factor of 2 between ''n'' and the split of the interval class of 3 has to do with the fact that 34et has two [[ring number|rings]] of 17et's. | ||
Another reasonable way of defining this continuum equates a number of diaschismas with the [[20000/19683|tetracot comma (20000/19683)]], so that {{nowrap|(2048/2025)<sup>''r''</sup> ~ 20000/19683}}. As a result, {{nowrap|''r'' {{=}} ''n'' − 2}}, and this labeling may also be called the ''diaschismic-tetracot equivalence continuum''. The just value of ''r'' is 1.4146…, and temperaments near this tend to be the most accurate. | Another reasonable way of defining this continuum equates a number of diaschismas with the [[20000/19683|tetracot comma (20000/19683)]], so that {{nowrap| (2048/2025)<sup>''r''</sup> ~ 20000/19683 }}. As a result, {{nowrap| ''r'' {{=}} ''n'' − 2 }}, and this labeling may also be called the ''diaschismic-tetracot equivalence continuum''. The just value of ''r'' is 1.4146…, and temperaments near this tend to be the most accurate. | ||
{| class="wikitable center-1 center-2" | {| class="wikitable center-1 center-2" | ||
| Line 31: | Line 31: | ||
| −1.5 | | −1.5 | ||
| 1/2 | | 1/2 | ||
| 22c & | | 22c & 34 | ||
| (30 digits) | | (30 digits) | ||
| {{monzo| 43 -30 2 }} | | {{monzo| 43 -30 2 }} | ||
| Line 43: | Line 43: | ||
| −0.5 | | −0.5 | ||
| 3/2 | | 3/2 | ||
| 34 & | | 34 & 36c | ||
| (22 digits) | | (22 digits) | ||
| {{monzo| 21 -22 6 }} | | {{monzo| 21 -22 6 }} | ||
| Line 61: | Line 61: | ||
| 1 | | 1 | ||
| 3 | | 3 | ||
| [[ | | [[Kleismic]] | ||
| [[15625/15552]] | | [[15625/15552]] | ||
| {{monzo| -6 -5 6 }} | | {{monzo| -6 -5 6 }} | ||
| Line 91: | Line 91: | ||
| 3.5 | | 3.5 | ||
| 11/2 | | 11/2 | ||
| 34 & | | 34 & 166c | ||
| (42 digits) | | (42 digits) | ||
| {{monzo| 67 -10 -22 }} | | {{monzo| 67 -10 -22 }} | ||
| Line 109: | Line 109: | ||
| ∞ | | ∞ | ||
| ∞ | | ∞ | ||
| [[ | | [[Diaschismic]] | ||
| [[2048/2025]] | | [[2048/2025]] | ||
| {{monzo| 11 -4 -2 }} | | {{monzo| 11 -4 -2 }} | ||
|} | |} | ||
We may invert the continuum by setting ''m'' such that {{nowrap|1/''n'' + 1/''m'' {{=}} 1}}. The just value of ''m'' is 1.41414…, and temperaments near this tend to be the most accurate ones. The resulting continuum equates a number of [[immunity comma]]s to the [[gothic comma]], but as the immunity comma is both larger and more complex than the diaschisma, this continuum does not contain as many useful temperaments at simple points which aren't already found by (half-)integer points on the diaschismic-gothmic and kleismic-tetracot continua. | We may invert the continuum by setting ''m'' such that {{nowrap| 1/''n'' + 1/''m'' {{=}} 1 }}. The just value of ''m'' is 1.41414…, and temperaments near this tend to be the most accurate ones. The resulting continuum equates a number of [[immunity comma]]s to the [[gothic comma]], but as the immunity comma is both larger and more complex than the diaschisma, this continuum does not contain as many useful temperaments at simple points which aren't already found by (half-)integer points on the diaschismic-gothmic and kleismic-tetracot continua. | ||
It is worth briefly noting that on this continuum: {{nowrap|''m'' {{=}} 0}} yields [[gothic]], {{nowrap|''m'' {{=}} 1}} yields [[diaschismic]] | It is worth briefly noting that on this continuum: {{nowrap| ''m'' {{=}} 0 }} yields [[gothic]], {{nowrap|''m'' {{=}} 1}} yields [[diaschismic]], {{nowrap| ''m'' {{=}} 2 }} yields [[tetracot]], {{nowrap| ''m'' {{=}} 3 }} yields the {{nowrap| 34 & 36c }} temperament occurring at {{nowrap| ''n'' {{=}} −1/2 }}, and the simplest non-integer convergent that approximates the [[JIP]], {{nowrap| ''m'' {{=}} 3/2 }}, yields [[kleismic]]. A unique (but not very good) temperament in this continuum is {{nowrap| ''m'' {{=}} 1/2 }}, yielding the {{nowrap| 29c & 34 }} temperament which may also be described as the {{nowrap| 34 & 107 }} temperament, which is essentially complementary (w.r.t. [[34edo|34et]]) to the simpler [[immunity]]. | ||
We may also examine temperaments that are structurally nontrivial in that they correspond to non-half-integer fractional ''n'' and ''m'', presented here for potential insight into meanings of their fractional values of ''n'' and ''m'' as they relate to the pergen structures of the temperaments. | We may also examine temperaments that are structurally nontrivial in that they correspond to non-half-integer fractional ''n'' and ''m'', presented here for potential insight into meanings of their fractional values of ''n'' and ''m'' as they relate to the pergen structures of the temperaments. | ||
| Line 139: | Line 139: | ||
Tetracot appears as the unique simplest minimal positive integer ''n'' which achieves: | Tetracot appears as the unique simplest minimal positive integer ''n'' which achieves: | ||
1. The simplest comma (compare the monzos, ratios or expressions of gothic ({{nowrap|''n'' {{=}} 0}}) and immunity ({{nowrap|''n'' {{=}} 1}})). | 1. The simplest comma (compare the monzos, ratios or expressions of gothic ({{nowrap| ''n'' {{=}} 0 }}) and immunity ({{nowrap| ''n'' {{=}} 1 }})). | ||
2. The simplest temperament mapping (compare the mappings of gothic (which has a whopping ''17'' periods per octave, but lacks the accuracy of something like [[chlorine]]) and immunity which takes slightly more generators to reach the same intervals of tetracot, so initially seems comparable, but whose generator's 5-limit interpretation is questionably damaged and complex compared to tetracot). | 2. The simplest temperament mapping (compare the mappings of gothic (which has a whopping ''17'' periods per octave, but lacks the accuracy of something like [[chlorine]]) and immunity which takes slightly more generators to reach the same intervals of tetracot, so initially seems comparable, but whose generator's 5-limit interpretation is questionably damaged and complex compared to tetracot). | ||
| Line 148: | Line 148: | ||
== Kleismic–tetracot continuum == | == Kleismic–tetracot continuum == | ||
We may also describe the set of all [[5-limit]] [[regular temperament|temperaments]] supported by [[34edo|34et]] by expressing the continuum (15625/15552)<sup>''k''</sup> ~ 20000/19683, for a value of ''k'' defined such that {{nowrap|1/''r'' + 1/''k'' {{=}} 1}} – corresponding to an inversion of the diaschismic-tetracot continuum with respect to tetracot. Varying ''k'' (for number of <u>k</u>leismas) results in different temperaments listed in the table below. It converges to | We may also describe the set of all [[5-limit]] [[regular temperament|temperaments]] supported by [[34edo|34et]] by expressing the continuum (15625/15552)<sup>''k''</sup> ~ 20000/19683, for a value of ''k'' defined such that {{nowrap| 1/''r'' + 1/''k'' {{=}} 1 }} – corresponding to an inversion of the diaschismic-tetracot continuum with respect to tetracot. Varying ''k'' (for number of <u>k</u>leismas) results in different temperaments listed in the table below. It converges to kleismic as ''k'' approaches infinity, and is motivated by the fact that many important temperaments of 34edo follow a chain of commas connected by kleismas as discovered by [[User:Lériendil|Lériendil]]. The just value of ''k'' is 3.4117…, and temperaments near this tend to be the most accurate. This also suggests that the kleisma is, loosely speaking, a type of "super-comma" or "meta-comma" for the 5-limit, in its ability to equate so many commas simultaneously into a general purpose comma. | ||
{| class="wikitable center-1 center-2" | {| class="wikitable center-1 center-2" | ||
|+ style="font-size: 105%;" | Temperaments with half-integer ''k'' in the<br | |+ style="font-size: 105%;" | Temperaments with half-integer ''k'' in the<br>kleismic–tetracot continuum | ||
|- | |- | ||
! rowspan="2" | ''k'' | ! rowspan="2" | ''k'' | ||
| Line 163: | Line 163: | ||
| −2 | | −2 | ||
| 8/3 | | 8/3 | ||
| 34 & | | 34 & 113 | ||
| (24 digits) | | (24 digits) | ||
| {{monzo| -7 -19 16 }} | | {{monzo| -7 -19 16 }} | ||
| Line 187: | Line 187: | ||
| 1 | | 1 | ||
| ∞ | | ∞ | ||
| [[ | | [[Diaschismic]] | ||
| [[2048/2025]] | | [[2048/2025]] | ||
| {{monzo| 11 -4 -2 }} | | {{monzo| 11 -4 -2 }} | ||
| Line 240: | Line 240: | ||
| ∞ | | ∞ | ||
| 3 | | 3 | ||
| [[ | | [[Kleismic]] | ||
| [[15625/15552]] | | [[15625/15552]] | ||
| {{monzo| -6 -5 6 }} | | {{monzo| -6 -5 6 }} | ||