Diaschismic–gothmic equivalence continuum: Difference between revisions
There are still many temps of non-half-integer fractional values of "n" and "m" |
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The ''' | 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 (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 ''n'' = 2 (which we align with ''r'' = 0) we get | 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]] (''n'' = 1) splits its twelfth in two; | * [[Immunity]] ({{nowrap| ''n'' {{=}} 1 }}) splits its twelfth in two; | ||
* [[Tetracot]] (''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 (2048/2025)<sup>''r''</sup> ~ 20000/19683. As a result, ''r'' = ''n'' | 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" | ||
|+ Temperaments with half-integer ''n'' and ''r'' | |+ style="font-size: 105%;" | Temperaments with half-integer ''n'' and ''r'' | ||
|- | |- | ||
! rowspan="2" | ''r'' | ! rowspan="2" | ''r'' | ||
| Line 23: | Line 23: | ||
! Monzo | ! Monzo | ||
|- | |- | ||
| | | −2 | ||
| 0 | | 0 | ||
| [[Gothic]] | | [[Gothic]] | ||
| Line 29: | Line 29: | ||
| {{monzo| 27 -17 }} | | {{monzo| 27 -17 }} | ||
|- | |- | ||
| | | −1.5 | ||
| 1/2 | | 1/2 | ||
| 22c & 34 | | 22c & 34 | ||
| Line 35: | Line 35: | ||
| {{monzo| 43 -30 2 }} | | {{monzo| 43 -30 2 }} | ||
|- | |- | ||
| | | −1 | ||
| 1 | | 1 | ||
| [[Immunity]] | | [[Immunity]] | ||
| Line 41: | Line 41: | ||
| {{monzo| 16 -13 2 }} | | {{monzo| 16 -13 2 }} | ||
|- | |- | ||
| | | −0.5 | ||
| 3/2 | | 3/2 | ||
| 34 & 36c | | 34 & 36c | ||
| Line 61: | Line 61: | ||
| 1 | | 1 | ||
| 3 | | 3 | ||
| [[ | | [[Kleismic]] | ||
| [[15625/15552]] | | [[15625/15552]] | ||
| {{monzo| -6 -5 6 }} | | {{monzo| -6 -5 6 }} | ||
| Line 79: | Line 79: | ||
| 2.5 | | 2.5 | ||
| 9/2 | | 9/2 | ||
| 34 & | | [https://sintel.pythonanywhere.com/result?subgroup=2.3.5.13.17.23&reduce=on&weights=tenney&target=&edos=34+%26+74&submit_edo=submit&commas= Hemienneatonic] | ||
| (28 digits) | | (28 digits) | ||
| {{monzo| 45 -2 18 }} | | {{monzo| 45 -2 -18 }} | ||
|- | |- | ||
| 3 | | 3 | ||
| 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 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 [[immunity comma]] is both larger and more complex than the diaschisma | 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]], {{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. | |||
{| class="wikitable" | {| class="wikitable" | ||
|+ Temperaments with non-half-integer fractional ''n'' and ''m'' | |+ style="font-size: 105%;" | Temperaments with non-half-integer fractional ''n'' and ''m'' | ||
|- | |- | ||
! ''n'' !! ''m'' !! Temperament !! Comma | ! ''n'' !! ''m'' !! Temperament !! Comma | ||
| Line 130: | Line 134: | ||
|} | |} | ||
Note that all of these correspond to half-integer points of either ''n'' or ''k'' (defined below), hence part of the usefulness of the inversion discussed in the [[#Kleismic-tetracot continuum]] subsection. | |||
== Significance of tetracot == | |||
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 }})). | |||
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). | |||
3. A characteristic damage of [[34edo]] which is not trivial; gothic is trivial in that it is just a subgroup restriction, and immunity, though not trivial and comparable in complexity to tetracot, is not characteristic, because it asks for a flat tuning of 5, so that it is arguably more or equally as characteristic of [[63edo]]'s or [[97edo]]'s representation of the 5-limit, but the fact that it has so many 3's in it when 34edo does not do so well in consistency of 3's to long distance should be a hint that we can do better. Compare with tetracot, which admits comparatively more and lower damage patent tunings and is clearly a type of imperfect simplification corresponding to a structural awkwardness in [[5-limit]] [[JI]] – the [[20000/19683|minimal diesis]] – so it has characteristic damage on 10/9 (flatwards) and 9/8 (sharpwards) by exaggerating the difference; this is to say, if you look at 34et's tuning of the 5-limit, its damage is strongly characteristic of tetracot. Exaggerating this difference in this way ''is'' also characteristic of immunity, but its general tunings are at odds with those of tetracot's so that they only merge in 34edo, which is arguably a more unusual tuning for immunity than it is for tetracot, where it is clearly characteristic. | |||
4. As aforementioned, a convenient point to invert the scale to define the '''kleismic–tetracot continuum''' nicely, discussed below. | |||
== 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 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" | ||
|+ Temperaments with half-integer ''k'' in the | |+ style="font-size: 105%;" | Temperaments with half-integer ''k'' in the<br>kleismic–tetracot continuum | ||
|- | |- | ||
! rowspan="2" | ''k'' | ! rowspan="2" | ''k'' | ||
| Line 143: | Line 161: | ||
! Monzo | ! Monzo | ||
|- | |- | ||
| | | −2 | ||
| 8/3 | | 8/3 | ||
| 34 & 113 | | 34 & 113 | ||
| Line 149: | Line 167: | ||
| {{monzo| -7 -19 16 }} | | {{monzo| -7 -19 16 }} | ||
|- | |- | ||
| | | −1 | ||
| 5/2 | | 5/2 | ||
| [[Fifive]] | | [[Fifive]] | ||
| Line 169: | Line 187: | ||
| 1 | | 1 | ||
| ∞ | | ∞ | ||
| [[ | | [[Diaschismic]] | ||
| [[2048/2025]] | | [[2048/2025]] | ||
| {{monzo| 11 -4 -2 }} | | {{monzo| 11 -4 -2 }} | ||
| Line 222: | Line 240: | ||
| ∞ | | ∞ | ||
| 3 | | 3 | ||
| [[ | | [[Kleismic]] | ||
| [[15625/15552]] | | [[15625/15552]] | ||
| {{monzo| -6 -5 6 }} | | {{monzo| -6 -5 6 }} | ||
| Line 228: | Line 246: | ||
{| class="wikitable" | {| class="wikitable" | ||
|+ Temperaments with fractional ''n'', ''r'' and ''k'' | |+ style="font-size: 105%;" | Temperaments with fractional ''n'', ''r'', and ''k'' | ||
|- | |- | ||
! ''n'' !! ''r'' !! ''k'' !! Temperament !! Comma | ! ''n'' !! ''r'' !! ''k'' !! Temperament !! Comma | ||
| Line 236: | Line 254: | ||
| 11/3 = 3.{{overline|6}} || 5/3 = 1.{{overline|6}} || 5/2 = 2.5 || [[Majvam]] || {{monzo| 40 7 -22 }} | | 11/3 = 3.{{overline|6}} || 5/3 = 1.{{overline|6}} || 5/2 = 2.5 || [[Majvam]] || {{monzo| 40 7 -22 }} | ||
|- | |- | ||
| 9/2 = 4.5 || 5/2 = 2.5 || 5/3 = 1.{{overline|6}} || [ | | 9/2 = 4.5 || 5/2 = 2.5 || 5/3 = 1.{{overline|6}} || [http://sintel.pythonanywhere.com/result?subgroup=2.3.5.13.17.23&reduce=on&weights=tenney&target=&edos=34+%26+74&submit_edo=submit&commas= Hemienneatonic] || {{monzo| 45 -2 -18 }} | ||
|} | |} | ||
== Goldis == | |||
: ''For extensions, see [[Keegic temperaments #Aurora]] and [[Breedsmic temperaments #Hemigoldis]].'' | |||
Goldis tempers out the [[goldis comma]]. Despite being a quarter-tone in size, due to its complexity, the damage is spread out, so that simple intervals of the [[5-limit]] tend to be tuned reasonably. Possible edo tunings include [[21edo]], [[34edo]], [[55edo]], and [[89edo]]. [[34edo]] is an especially good and tone-efficient tuning (also evidenced by being the largest "golden edo" appearing in the [[optimal ET sequence]]). | |||
As the generator does not admit a useful interpretation in the [[5-limit]], a number of extensions are possible. One possibility is to notice that the generator is close to [[49/32]], resulting in [[hemigoldis]], which splits the generator in half. | |||
[[Subgroup]]: 2.3.5 | |||
[[Comma list]]: 549755813888/533935546875 | |||
{{Mapping|legend=1| 1 -3 5 | 0 12 -7 }} | |||
: mapping generators: ~2, ~84375/65536 | |||
[[Optimal tuning]]s: | |||
* [[WE]]: ~2 = 1199.1846{{c}}, ~84375/65536 = 458.3229{{c}} | |||
: [[error map]]: {{val| -0.815 +0.366 +1.349 }} | |||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~84375/65536 = 458.6193{{c}} | |||
: error map: {{val| 0.000 +1.477 +3.351 }} | |||
{{Optimal ET sequence|legend=1| 13, 21, 34, 123, 157, 191c }} | |||
[[Badness]] (Sintel): 16.0 | |||
[[Category:34edo]] | [[Category:34edo]] | ||
[[Category:Equivalence continua]] | [[Category:Equivalence continua]] | ||