Diaschismic family: Difference between revisions

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The ~~[[5-limit]] parent [[comma]] for the~~ '''diaschismic family''' of [[regular temperament|temperaments]] ~~is 2048/2025, the~~ [[~~diaschisma]]. The [[period]] is half an [[2/1~~|~~octave~~]]~~, and~~ the ~~[[generator]] is a fifth. Three periods gives 1800 cents~~, ~~and decreasing this by two fifths gives the major third.~~ [[~~34edo~~]] is a good tuning choice, with [[46edo]], [[56edo]], [[58edo]], or [[80edo]] being other possibilities. Both [[12edo]] and [[22edo]] support it, and retuning them to a [[mos]] of diaschismic gives two scale possibilities.

The '''diaschismic family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the diaschisma, [[2048/2025]].

== Diaschismic ==

The [[period]] of diaschismic is half an [[2/1|octave]], and the [[generator]] is a fifth; the [[ploidacot]] is diploid monocot. Three periods gives 1800 cents, and decreasing this by two fifths gives the major third. [[34edo]] is a good tuning choice, with [[46edo]], [[56edo]], [[58edo]], or [[80edo]] being other possibilities. Both [[12edo]] and [[22edo]] support it, and retuning them to a [[mos]] of diaschismic gives two scale possibilities.

This temperament is also known as '''srutal''' in the 5-limit, but that name more strictly speaking refers to the [[#Srutal|34d & 46 extension]] to the [[7-limit]] that adds [[4375/4374]] to the comma list.

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* [[CWE]]: ~45/32 = 600.0000{{c}}, ~3/2 = 704.9585{{c}}

: error map: {{val| 0.000 +3.003 +3.769 }}

~~~~

[[Tuning ranges]]:

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==== 7-limit extensions ====

To get the 7-limit extensions, we add another comma:

* Septimal diaschismic adds [[126/125]], the starling comma, to obtain 7-limit harmony by more complex methods than pajara, but with greater accuracy.

* Pajara derives from [[64/63]] and is a popular and well-known choice.

* Srutal adds [[4375/4374]], the ragisma, which is about as accurate as septimal diaschismic but has a much more complex mapping of 7.

* Keen adds [[875/864]].

* Bidia adds [[3136/3125]], the hemimean comma.

* Echidna adds [[1728/1715]], the orwellisma.

* Shrutar adds [[245/243]], the sensamagic comma.

~~Pajara, diaschismic, srutal and keen~~ keep the same half-octave period and fifth generator, ~~but shrutar has~~ a generator of a ~~quarter-tone (which can be taken as~~ [[36/35]], the ~~septimal quarter-tone) and echidna has~~ a ~~generator of~~ 9/7. ~~Bidia has~~ a ~~quarter-octave period~~ and ~~a fifth~~ generator.

Those all keep the same half-octave period and fifth generator.

Bidia adds [[3136/3125]], the hemimean comma, with a 1/4-octave period. Shrutar adds [[245/243]] and shru adds [[392/375]], with a quartertone generator. Sruti adds [[19683/19600]] and anguirus adds [[49/48]], with a neutral third or hemitwelfth generator. Those split the original generator in two. Echidna adds [[1728/1715]], the orwellisma, with a ~9/7 generator. Echidnic adds [[686/675]], the senga, with a ~8/7 generator. Those split the original generator in three. Finally, quadrasruta adds [[2401/2400]] and splits the original generator in four.

==== Subgroup extensions ====

Since the diaschisma factors into ([[256/255]])2([[289/288]]) in the 17-limit, it extends naturally to the 2.3.5.17 subgroup as ''srutal archagall'', documented right below. The [[S-expression]]-based comma list of this temperament is {[[256/255|S16]], [[289/288|S17]]}.

=== Srutal archagall ===

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* WE: ~45/32 = 599.5585{{c}}, ~3/2 = 704.6188{{c}}

* CWE: ~45/32 = 600.0000{{c}}, ~3/2 = 705.1356{{c}}

~~~~

{{Optimal ET sequence|legend=0| 10, 12, 22, 34, 80, 114, 194bc }}

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A simpler characterization than the one given by the normal comma list is that diaschismic adds [[126/125]] or [[5120/5103]] to the set of commas, and it can also be called {{nowrap| 46 & 58 }}. However described, diaschismic has a 1/2-octave period and a sharp fifth generator like ~~pajara~~, but not so sharp, giving a more accurate but more complex temperament. [[~~58edo~~]] provides an excellent tuning, ~~but an alternative~~ is to ~~make~~ [[7/4]] just by making the fifth 703.897 cents~~, as opposed to 703.448 cents for 58edo~~.

A simpler characterization than the one given by the normal comma list is that septimal diaschismic adds [[126/125]] or [[5120/5103]] to the set of commas, and it can also be called {{nowrap| 46 & 58 }}. However described, septimal diaschismic has a 1/2-octave period and a sharp fifth generator like the 5-limit version, but not so sharp, giving a more accurate but more complex temperament. [[104edo]] provides an excellent tuning, which is close to tuning [[7/4]] just by making the fifth 703.897 cents.

Diaschismic extends naturally to the 17-limit, for which the same tunings may be used, making it one of the most important of the higher-limit rank-2 temperaments. Adding the 11-limit adds the commas 176/175, 896/891 and 441/440. The 13-limit yields 196/195, 351/350, and 364/363; the 17-limit adds 136/135, 221/220, and 442/441. If you want to explore higher-limit harmonies, diaschismic is certainly one excellent way to do it; [[mos]] of 34 notes and even more the 46-note mos will encompass very great deal of it. Of course 46 or 58 equal provide alternatives which in many ways are similar, particularly in the case of 58.

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* [[CWE]]: ~45/32 = 600.0000{{c}}, ~3/2 = 703.7739{{c}}

: error map: {{val| 0.000 +1.819 +6.138 +0.983 }}

~~~~

[[Tuning ranges]]:

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* WE: ~45/32 = 599.4471{{c}}, ~3/2 = 703.0657{{c}}

* CWE: ~45/32 = 600.0000{{c}}, ~3/2 = 703.7996{{c}}

~~~~

Tuning ranges:

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* WE: ~45/32 = 599.4451{{c}}, ~3/2 = 703.0528{{c}}

* CWE: ~45/32 = 600.0000{{c}}, ~3/2 = 703.7813{{c}}

~~~~

Tuning ranges:

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* WE: ~17/12 = 599.6253{{c}}, ~3/2 = 703.3726{{c}}

* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 703.8520{{c}}

~~~~

Tuning ranges:

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* WE: ~17/12 = 599.6272{{c}}, ~3/2 = 703.4326{{c}}

* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 703.9093{{c}}

~~~~

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Pajara is closely associated with 22edo (not to mention [[Paul Erlich]]) but other tunings are possible. The 1/2-octave period serves as both a [[10/7]] and a [[7/5]]. Aside from 22edo, 34 with the val {{val| 34 54 79 96 }} and 56 with the val {{val| 56 89 130 158 }} are interesting alternatives, with more acceptable fifths, and a tetrad which is more clearly a dominant seventh. As such, they are closer to the tuning of 12edo and of common practice Western music in general, while retaining the distictiveness of a sharp fifth.

Pajara is closely associated with 22edo (not to mention [[Paul Erlich]]) but other tunings are possible. The 1/2-octave period serves as both a [[10/7]] and a [[7/5]]. Aside from 22edo, 34 with the val {{val| 34 54 79 96 }} (34d) and 56 with the val {{val| 56 89 130 158 }} (56d) are interesting alternatives, with more acceptable fifths, and a tetrad which is more clearly a dominant seventh. As such, they are closer to the tuning of 12edo and of common practice Western music in general, while retaining the distictiveness of a sharp fifth.

Pajara extends nicely to an 11-limit version, for which the 56edo tuning can be used, but a good alternative is to make the major thirds pure by setting the fifth to be 706.843 cents. Now 99/98, 100/99, 176/175 and 896/891 are being tempered out.

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* [[CWE]]: ~7/5 = 600.0000{{c}}, ~3/2 = 707.3438{{c}}

: error map: {{val| 0.000 +5.389 -1.001 +16.487 }}

~~~~

[[Tuning ranges]]:

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* WE: ~7/5 = 598.8485{{c}}, ~3/2 = 705.5285{{c}}

* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 707.1826{{c}}

~~~~

Tuning ranges:

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* WE: ~7/5 = 599.9732{{c}}, ~3/2 = 708.8873{{c}}

* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 708.9227{{c}}

~~~~

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* WE: ~7/5 = 599.8871{{c}}, ~3/2 = 708.6725{{c}}

* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 708.8176{{c}}

~~~~

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* WE: ~7/5 = 598.7732{{c}}, ~3/2 = 704.6889{{c}}

* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 706.3950{{c}}

~~~~

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* WE: ~7/5 = 599.0204{{c}}, ~3/2 = 705.2572{{c}}

* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 706.5660{{c}}

~~~~

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* WE: ~7/5 = 598.3048{{c}}, ~3/2 = 705.4512{{c}}

* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 707.9238{{c}}

~~~~

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* WE: ~7/5 = 598.6103{{c}}, ~3/2 = 706.3076{{c}}

* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 708.2256{{c}}

~~~~

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* WE: ~7/5 = 599.4055{{c}}, ~3/2 = 708.8747{{c}}

* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 709.5508{{c}}

~~~~

Tuning ranges:

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* WE: ~7/5 = 599.9064{{c}}, ~3/2 = 710.1289{{c}}

* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 710.2325{{c}}

~~~~

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* WE: ~7/5 = 599.8239{{c}}, ~3/2 = 710.0128{{c}}

* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 710.2067{{c}}

~~~~

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* WE: ~7/5 = 598.8257{{c}}, ~3/2 = 709.4266{{c}}

* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 710.8414{{c}}

~~~~

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* WE: ~7/5 = 598.8865{{c}}, ~3/2 = 709.5472{{c}}

* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 710.8704{{c}}

~~~~

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* WE: ~7/5 = 597.4807{{c}}, ~3/2 = 702.5616{{c}}

* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 706.0542{{c}}

~~~~

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* WE: ~7/5 = 597.1952{{c}}, ~3/2 = 704.1350{{c}}

* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 708.1989{{c}}

~~~~

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* WE: ~7/5 = 597.6509{{c}}, ~3/2 = 705.7702{{c}}

* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 708.9719{{c}}

~~~~

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* WE: ~7/5 = 597.6509{{c}}, ~16/11 = 652.7788{{c}}

* CWE: ~7/5 = 600.0000{{c}}, ~16/11 = 653.7119{{c}}

~~~~

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* WE: ~7/5 = 597.6509{{c}}, ~55/32 = 950.8475{{c}}

* CWE: ~7/5 = 600.0000{{c}}, ~55/32 = 953.1172{{c}}

~~~~

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* WE: ~7/5 = 598.6748{{c}}, ~26/15 = 950.9691{{c}}

* CWE: ~7/5 = 600.0000{{c}}, ~26/15 = 953.1052{{c}}

~~~~

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* WE: ~7/5 = 598.8411{{c}}, ~26/15 = 951.3687{{c}}

* CWE: ~7/5 = 600.0000{{c}}, ~26/15 = 953.2169{{c}}

~~~~

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Srutal can be described as the 34d & 46 temperament, where 7/4 is located at 15 generator steps, or the double-augmented fifth (C–Gx). 80edo and 126edo are among the possible tunings. Srutal, shrutar and bidia have similar 19-limit properties, tempering out 190/189, related to rank-3 [[julius]].

Srutal can be described as the {{nowrap| 34d & 46 }} temperament, where 7/4 is located at 15 generator steps, or the double-augmented fifth (C–Gx). As such, it weakly extends [[leapfrog]]. 80edo and [[126edo]] are among the possible tunings. Srutal, shrutar and bidia have similar 19-limit properties, tempering out 190/189, related to rank-3 [[julius]].

[[Subgroup]]: 2.3.5.7

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* [[CWE]]: ~45/32 = 600.0000{{c}}, ~3/2 = 704.7646{{c}}

: error map: {{val| 0.000 +2.810 +4.157 +2.643 }}

~~~~

[[Tuning ranges]]:

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* WE: ~45/32 = 599.4413{{c}}, ~3/2 = 704.1999{{c}}

* CWE: ~45/32 = 600.0000{{c}}, ~3/2 = 704.8017{{c}}

~~~~

Tuning ranges:

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* WE: ~45/32 = 599.5490{{c}}, ~3/2 = 704.3516{{c}}

* CWE: ~45/32 = 600.0000{{c}}, ~3/2 = 704.8347{{c}}

~~~~

Tuning ranges:

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* WE: ~17/12 = 599.6459{{c}}, ~3/2 = 704.4237{{c}}

* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 704.8083{{c}}

~~~~

Tuning ranges:

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* WE: ~17/12 = 599.6371{{c}}, ~3/2 = 704.4790{{c}}

* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 704.8745{{c}}

~~~~

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* WE: ~17/12 = 599.6690{{c}}, ~3/2 = 704.5098{{c}}

* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 704.8713{{c}}

~~~~

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* WE: ~17/12 = 599.6664{{c}}, ~3/2 = 704.5138{{c}}

* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 704.8807{{c}}

~~~~

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* WE: ~17/12 = 599.8115{{c}}, ~3/2 = 704.5958{{c}}

* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 704.8086{{c}}

~~~~

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== Keen ==

Keen adds 875/864 as well as 2240/2187 to the set of commas. It may also be described as the {{nowrap| 22 & 56 }} temperament. [[78edo]] is a good tuning choice, and remains a good one in the 11-limit, where the temperament is really more interesting, adding 100/99 and 385/384 to the list of commas.

Keen adds 875/864 as well as 2240/2187 to the set of commas. It may also be described as the {{nowrap| 22 & 34 }} temperament. [[78edo]] is a good tuning choice, and remains a good one in the 11-limit, where the temperament is really more interesting, adding 100/99 and 385/384 to the list of commas.

[[Subgroup]]: 2.3.5.7

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* [[CWE]]: ~45/32 = 600.0000{{c}}, ~3/2 = 707.5294{{c}}

: error map: {{val| 0.000 +5.574 -1.373 -1.061 }}

~~~~

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* WE: ~45/32 = 599.6286{{c}}, ~3/2 = 707.1712{{c}}

* CWE: ~45/32 = 600.0000{{c}}, ~3/2 = 707.5984{{c}}

~~~~

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* WE: ~45/32 = 599.3498{{c}}, ~3/2 = 706.4009{{c}}

* CWE: ~45/32 = 600.0000{{c}}, ~3/2 = 707.1309{{c}}

~~~~

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* WE: ~17/12 = 599.4053{{c}}, ~3/2 = 706.4544{{c}}

* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 707.1243{{c}}

~~~~

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* WE: ~45/32 = 599.8547{{c}}, ~3/2 = 707.0858{{c}}

* CWE: ~45/32 = 600.0000{{c}}, ~3/2 = 707.2596{{c}}

~~~~

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* WE: ~17/12 = 599.8338{{c}}, ~3/2 = 707.0558{{c}}

* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 707.2537{{c}}

~~~~

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== Bidia ==

Bidia adds [[3136/3125]] to the commas, splitting the period into 1/4 octave. It may be called the {{nowrap| 12 & 56 }} temperament.

Bidia adds [[3136/3125]] to the commas, splitting the period into 1/4 octave. It may be called the {{nowrap| 12 & 68 }} temperament; its ploidacot is tetraploid monocot.

[[Subgroup]]: 2.3.5.7

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* [[CWE]]: ~25/21 = 300.0000{{c}}, ~3/2 = 705.5070{{c}}

: error map: {{val| 0.000 +3.552 +2.672 +3.639 }}

~~~~

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* WE: ~25/21 = 299.6809{{c}}, ~3/2 = 704.3367{{c}}

* CWE: ~25/21 = 600.0000{{c}}, ~3/2 = 705.2170{{c}}

~~~~

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* WE: ~25/21 = 299.7538{{c}}, ~3/2 = 704.7222{{c}}

* CWE: ~25/21 = 600.0000{{c}}, ~3/2 = 705.3241{{c}}

~~~~

{{Optimal ET sequence|legend=0| 12, 68, 80, 148d, 228bcd, 376bbcddf }}

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* WE: ~25/21 = 299.7883{{c}}, ~3/2 = 704.8365{{c}}

* CWE: ~25/21 = 600.0000{{c}}, ~3/2 = 705.3496{{c}}

~~~~

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* WE: ~19/16 = 299.7967{{c}}, ~3/2 = 704.8609{{c}}

* CWE: ~19/16 = 600.0000{{c}}, ~3/2 = 705.3519{{c}}

~~~~

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* WE: ~19/16 = 299.7961{{c}}, ~3/2 = 704.8577{{c}}

* CWE: ~19/16 = 300.0000{{c}}, ~3/2 = 705.3413{{c}}

~~~~

Badness (Sintel): 1.24

~~== Echidna ==~~

Echidna adds 1728/1715 to the commas and takes 9/7 as a generator. It may be called the {{nowrap| 22 & 58 }} temperament. [[58edo]] or [[80edo]] make for good tunings, or their vals can be added to {{val| 138 219 321 388 }} (138cde). In most of the tunings it has a significantly sharp 7/4 which some prefer.

Echidna becomes more interesting when extended to be an 11-limit temperament by adding 176/175, 540/539 or 896/891 to the commas, where the same tunings can be used as before. It then is able to represent the entire 11-odd-limit diamond to within about six cents of error, within a compass of 24 notes. The 22-note 2mos gives scope for this, and the 36-note mos much more. Better yet, it is related to three important 11-limit edos: 22edo, a trivial tuning, is the smallest consistent in the 11-odd-limit, corresponding to the merge of this temperament with [[hedgehog]]; [[58edo]] is the smallest tuning that is distinctly consistent in the 11-odd-limit and [[80edo]] is the third smallest distinctly consistent in the 11-odd-limit.

The generator can be interpreted as 11/10, the period complement of 9/7, as a stack of 11/10 and 9/7 makes [[99/70]] which is extremely close to 600{{cent}} and is equal to it if we temper out [[9801/9800|S99]]. Three 11/10's then make a 4/3 (tempering out [[4000/3993|S10/S11]] thus making 10/9 and 12/11 equidistant from 11/10), implying a flat tuning of 4/3.

Like most srutal extensions, the 13- and 17-limit interpretations are possible by observing that since we have tempered out [[176/175]], tempering out [[351/350]] and [[352/351]] which sum to 176/175 is very elegant. In the 17-limit we can equate the half-octave with 17/12 and 24/17 and we can take advantage of the sharp fifth by combining echidna with [[srutal archagall]], leading to a particularly beautiful temperament (one that prefers a very slightly less sharp fifth than srutal archagall). This mapping of 13 and 17 is supported by the patent vals of the three main echidna edos of 22, 58 and 80, of which all except 22 are consistent in the [[17-odd-limit]].

~~[[Subgroup]]: 2.3.5.7~~

~~[[Comma list]]: 1728/1715, 2048/2025~~

~~{{Mapping|legend=1| 2 1 9 2 | 0 3 -6 5 }}~~

~~: mapping generators: ~45/32, ~9/7~~

~~[[Optimal tuning]]s:~~

* [[WE]]: ~45/32 = 599.3056{{c}}, ~9/7 = 434.3524{{c}}

~~: [[error map]]: {{val| -1.389 +0.408 +1.322 +1.547 }}~~

* [[CWE]]: ~45/32 = 600.0000{{c}}, ~9/7 = 434.8327{{c}}

~~: error map: {{val| 0.000 +2.543 +4.690 +5.338 }}~~

~~~~

~~{{Optimal ET sequence|legend=1| 22, 58, 80, 138cd, 218cd }}~~

~~[[Badness]] (Sintel): 1.47~~

~~=== 11-limit ===~~

~~Subgroup: 2.3.5.7.11~~

~~Comma list: 176/175, 540/539, 896/891~~

~~Mapping: {{mapping| 2 1 9 2 12 | 0 3 -6 5 -7 }}~~

~~Optimal tunings:~~

* WE: ~45/32 = 599.3085{{c}}, ~9/7 = 434.3511{{c}}

* CWE: ~45/32 = 600.0000{{c}}, ~9/7 = 434.8647{{c}}

~~~~

~~Minimax tuning:~~

* 11-odd-limit: ~9/7 = {{monzo| 5/12 0 0 1/12 -1/12 }}

~~: [{{monzo| 1 0 0 0 0 }}, {{monzo| 7/4 0 0 1/4 -1/4 }}, {{monzo| 2 0 0 -1/2 1/2 }}, {{monzo| 37/12 0 0 5/12 -5/12 }}, {{monzo| 37/12 0 0 -7/12 7/12 }}]~~

~~: unchanged-interval (eigenmonzo) basis: 2.11/7~~

~~{{Optimal ET sequence|legend=0| 22, 58, 80, 138cde, 218cde }}~~

~~Badness (Sintel): 0.859~~

~~=== 13-limit ===~~

~~Subgroup: 2.3.5.7.11.13~~

~~Comma list: 176/175, 351/350, 364/363, 540/539~~

~~Mapping: {{mapping| 2 1 9 2 12 19 | 0 3 -6 5 -7 -16 }}~~

~~Optimal tunings:~~

* WE: ~45/32 = 599.3397{{c}}, ~9/7 = 434.2772{{c}}

* CWE: ~45/32 = 600.0000{{c}}, ~9/7 = 434.7864{{c}}

~~~~

~~{{Optimal ET sequence|legend=0| 22, 36f, 58, 80, 138cde }}~~

~~Badness (Sintel): 0.978~~

~~=== 17-limit ===~~

~~Subgroup: 2.3.5.7.11.13.17~~

~~Comma list: 136/135, 176/175, 221/220, 256/255, 540/539~~

~~Mapping: {{mapping| 2 1 9 2 12 19 6 | 0 3 -6 5 -7 -16 3 }}~~

~~Optimal tunings:~~

* WE: ~45/32 = 599.4645{{c}}, ~9/7 = 434.4282{{c}}

* CWE: ~45/32 = 600.0000{{c}}, ~9/7 = 434.8340{{c}}

~~~~

~~{{Optimal ET sequence|legend=0| 22, 36f, 58, 80, 138cde }}~~

~~Badness (Sintel): 1.03~~

~~== Echidnic ==~~

~~[[Subgroup]]: 2.3.5.7~~

~~[[Comma list]]: 686/675, 1029/1024~~

~~{{Mapping|legend=1| 2 2 7 6 | 0 3 -6 -1 }}~~

~~: mapping generators: ~45/32, ~8/7~~

~~[[Optimal tuning]]s:~~

* [[WE]]: ~45/32 = 599.7208{{c}}, ~8/7 = 234.8330{{c}}

~~: [[error map]]: {{val| -0.558 +1.986 +2.733 -5.334 }}~~

* [[CWE]]: ~45/32 = 600.0000{{c}}, ~8/7 = 234.9539{{c}}

~~: error map: {{val| 0.000 +2.907 +3.963 -3.780 }}~~

~~~~

~~{{Optimal ET sequence|legend=1| 10, 26c, 36, 46 }}~~

~~[[Badness]] (Sintel): 1.83~~

~~=== 11-limit ===~~

~~Subgroup: 2.3.5.7.11~~

~~Comma list: 385/384, 441/440, 686/675~~

~~Mapping: {{mapping| 2 2 7 6 3 | 0 3 -6 -1 10 }}~~

~~Optimal tunings:~~

* WE: ~45/32 = 599.8022{{c}}, ~8/7 = 235.0185{{c}}

* CWE: ~45/32 = 600.0000{{c}}, ~8/7 = 235.0893{{c}}

~~~~

~~{{Optimal ET sequence|legend=0| 10, 36e, 46, 102, 148 }}~~

~~Badness (Sintel): 1.49~~

~~=== 13-limit ===~~

~~Subgroup: 2.3.5.7.11.13~~

~~Comma list: 91/90, 169/168, 385/384, 441/440~~

~~Mapping: {{mapping| 2 2 7 6 3 7 | 0 3 -6 -1 10 1 }}~~

~~Optimal tunings:~~

* WE: ~45/32 = 599.9570{{c}}, ~8/7 = 235.0708{{c}}

* CWE: ~45/32 = 600.0000{{c}}, ~8/7 = 235.0862{{c}}

~~~~

~~{{Optimal ET sequence|legend=0| 10, 36e, 46, 102, 148f }}~~

~~Badness (Sintel): 1.19~~

~~=== 17-limit ===~~

~~Subgroup: 2.3.5.7.11.13.17~~

~~Comma list: 91/90, 136/135, 154/153, 169/168, 256/255~~

~~Mapping: {{mapping| 2 2 7 6 3 7 7 | 0 3 -6 -1 10 1 3 }}~~

~~Optimal tunings:~~

* WE: ~17/12 = 599.9571{{c}}, ~8/7 = 235.0709{{c}}

* CWE: ~17/12 = 600.0000{{c}}, ~8/7 = 235.0860{{c}}

~~~~

~~{{Optimal ET sequence|legend=0| 10, 36e, 46, 102, 148f }}~~

~~Badness (Sintel): 0.983~~

~~; Music~~

* [https://untwelve.org/competition/2011 ''A Stiff Shot of Turpentine''] [https://untwelve.org/static/audio/competition/2011/Kosmorsky-A_Stiff_Shot_of_Turpentine.mp3 play] by [[Peter Kosmorsky]]

* [https://www.youtube.com/watch?v=VsBXIvBZY6A ''56edo Track (Echidnic16 Scale)''] by [[Budjarn Lambeth]] (2025)

== Shrutar ==

Shrutar adds 245/243 to the commas, and also tempers out 6144/6125. It can also be described as {{nowrap| 22 & 46 }}. Its generator can be taken as either 36/35 or 35/24; the latter is interesting since along with 15/14 and 21/20, it connects opposite sides of a hexany. [[68edo]] makes for a good tuning, but another excellent choice is a generator of 14(1/7), making 7's just.

Shrutar adds 245/243 to the commas, and also tempers out [[6144/6125]]. It can also be described as {{nowrap| 22 & 46 }}. Its generator can be taken as either ~36/35 or ~35/24; the latter is interesting since along with 15/14 and 21/20, it connects opposite sides of a hexany. Its ploidacot is diploid alpha-dicot. [[68edo]] makes for a good tuning, but another excellent choice is a generator of 14(1/7), making 7's just.

By adding 121/120 or 176/175 to the commas, shrutar can be extended to the 11-limit, which loses a bit of accuracy, but picks up low-complexity 11-limit harmony, making shrutar quite an interesting 11-limit system. 68, 114 or a 14(1/7) generator can again be used as tunings.

Line 1,045:

Line 845:

* [[CWE]]: ~45/32 = 600.0000{{c}}, ~35/24 = 652.7736{{c}}

: error map: {{val| 0.000 +3.592 +2.592 +0.589 }}

~~~~

Line 1,061:

Line 860:

* WE: ~45/32 = 599.7721{{c}}, ~16/11 = 652.4321{{c}}

* CWE: ~45/32 = 600.0000{{c}}, ~16/11 = 652.6672{{c}}

~~~~

Line 1,077:

Line 875:

* WE: ~45/32 = 599.7699{{c}}, ~16/11 = 652.4035{{c}}

* CWE: ~45/32 = 600.0000{{c}}, ~16/11 = 652.6374{{c}}

~~~~

Line 1,093:

Line 890:

* WE: ~17/12 = 599.7995{{c}}, ~16/11 = 652.4287{{c}}

* CWE: ~17/12 = 600.0000{{c}}, ~16/11 = 652.6334{{c}}

~~~~

Line 1,109:

Line 905:

* WE: ~17/12 = 599.8060{{c}}, ~16/11 = 652.5190{{c}}

* CWE: ~17/12 = 600.0000{{c}}, ~16/11 = 652.7164{{c}}

~~~~

Line 1,125:

Line 920:

* WE: ~17/12 = 599.7879{{c}}, ~16/11 = 652.4776{{c}}

* CWE: ~17/12 = 600.0000{{c}}, ~16/11 = 652.6926{{c}}

~~~~

Badness (Sintel): 1.03

== Shru ==

Shru tempers out 392/375 and slices the compound semitone into two generators of ~10/7. Its ploidacot is diploid alpha-dicot, the same as shrutar.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 392/375, 1323/1280

: mapping generators: ~45/32, ~10/7

[[Optimal tuning]]s:

* [[WE]]: ~45/32 = 600.2519{{c}}, ~10/7 = 650.4083{{c}}

: [[error map]]: {{val| +0.504 -0.887 +14.321 -18.096 }}

* [[CWE]]: ~45/32 = 600.0000{{c}}, ~10/7 = 650.1017{{c}}

: error map: {{val| 0.000 -1.752 +13.279 -19.334 }}

[[Badness]] (Sintel): 3.99

=== 11-limit ===

Subgroup: 2.3.5.7.11

Comma list: 56/55, 77/75, 1323/1280

Mapping: {{mapping| 2 1 9 11 8 | 0 2 -4 -5 -1 }}

Optimal tunings:

* WE: ~17/12 = 600.2356{{c}}, ~10/7 = 650.3856{{c}}

* CWE: ~17/12 = 600.0000{{c}}, ~10/7 = 650.1008{{c}}

Badness (Sintel): 2.10

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

Comma list: 56/55, 77/75, 105/104, 507/500

Mapping: {{mapping| 2 1 9 11 8 15 | 0 2 -4 -5 -1 -7 }}

Optimal tunings:

* WE: ~45/32 = 599.9067{{c}}, ~10/7 = 649.4907{{c}}

* CWE: ~45/32 = 600.0000{{c}}, ~10/7 = 649.5950{{c}}

Badness (Sintel): 2.12

== Sruti ==

Sruti tempers out 19683/19600, setting itself up as a [[hemipyth]] temperament. It has the same semi-octave period as diaschismic, but the generator can be taken as a neutral third or a hemitwelfth. The temperament can be described as {{nowrap| 24 & 34d }}; its ploidacot is diploid dicot. [[58edo]] may be recommended as a tuning.

[[Subgroup]]: 2.3.5.7

Line 1,145:

Line 992:

* [[CWE]]: ~45/32 = 600.0000{{c}}, ~140/81 = 951.8227{{c}}

: error map: {{val| 0.000 +1.690 +6.395 +4.869 }}

~~~~

{{Optimal ET sequence|legend=1| 24, 34d, 58, 150cd, 208ccdd, 266ccdd }}

Line 1,161:

Line 1,007:

* WE: ~45/32 = 599.1951{{c}}, ~121/70 = 950.5864{{c}}

* CWE: ~45/32 = 600.0000{{c}}, ~121/70 = 951.7972{{c}}

~~~~

{{Optimal ET sequence|legend=0| 24, 34d, 58, 150cdee, 208ccddee, 266ccddeee }}

Line 1,177:

Line 1,022:

* WE: ~45/32 = 599.1479{{c}}, ~26/15 = 950.5337{{c}}

* CWE: ~45/32 = 600.0000{{c}}, ~26/15 = 951.8314{{c}}

~~~~

{{Optimal ET sequence|legend=0| 24, 34d, 58, 150cdeef, 208ccddeeff, 266ccddeeefff }}

Line 1,193:

Line 1,037:

* WE: ~17/12 = 599.3003{{c}}, ~26/15 = 950.7465{{c}}

* CWE: ~17/12 = 600.0000{{c}}, ~26/15 = 951.8142{{c}}

~~~~

Line 1,200:

Line 1,043:

== Anguirus ==

As another hemipyth temperament, anguirus tempers out 49/48. It can be described as the {{nowrap| 10 & 24 }} temperament; its ploidacot is diploid dicot, the same as sruti.

[[Subgroup]]: 2.3.5.7

Line 1,213:

Line 1,058:

* [[CWE]]: ~45/32 = 600.0000{{c}}, ~7/4 = 953.0188{{c}}

: error map: {{val| 0.000 +4.083 +1.611 -15.807 }}

~~~~

Line 1,229:

Line 1,073:

* WE: ~45/32 = 599.9250{{c}}, ~7/4 = 952.0646{{c}}

* CWE: ~45/32 = 600.0000{{c}}, ~7/4 = 952.1784{{c}}

~~~~

Line 1,245:

Line 1,088:

* WE: ~45/32 = 599.7575{{c}}, ~7/4 = 951.9241{{c}}

* CWE: ~45/32 = 600.0000{{c}}, ~7/4 = 952.2980{{c}}

~~~~

Line 1,261:

Line 1,103:

* WE: ~17/12 = 599.7925{{c}}, ~7/4 = 952.0004{{c}}

* CWE: ~17/12 = 600.0000{{c}}, ~7/4 = 952.3178{{c}}

~~~~

Line 1,267:

Line 1,108:

Badness (Sintel): 1.10

== ~~Shru~~ ==

== Echidna ==

Echidna adds 1728/1715 to the commas and takes 9/7 as a generator. It may be called the {{nowrap| 22 & 58 }} temperament; its ploidacot is diploid alpha-tricot. [[58edo]] or [[80edo]] make for good tunings, or their vals can be added to {{val| 138 219 321 388 }} (138cde). In most of the tunings it has a significantly sharp 7/4 which some prefer.

Echidna becomes more interesting when extended to be an 11-limit temperament by adding 176/175, 540/539 or 896/891 to the commas, where the same tunings can be used as before. It then is able to represent the entire 11-odd-limit diamond to within about six cents of error, within a compass of 24 notes. The 22-note 2mos gives scope for this, and the 36-note mos much more. Better yet, it is related to three important 11-limit edos: 22edo, a trivial tuning, is the smallest consistent in the 11-odd-limit, corresponding to the merge of this temperament with [[hedgehog]]; [[58edo]] is the smallest tuning that is distinctly consistent in the 11-odd-limit and [[80edo]] is the third smallest distinctly consistent in the 11-odd-limit.

The generator can be interpreted as 11/10, the period complement of 9/7, as a stack of 11/10 and 9/7 makes [[99/70]] which is extremely close to 600{{cent}} and is equal to it if we temper out [[9801/9800|S99]]. Three 11/10's then make a 4/3 (tempering out [[4000/3993|S10/S11]] thus making 10/9 and 12/11 equidistant from 11/10), implying a flat tuning of 4/3.

Like most srutal extensions, the 13- and 17-limit interpretations are possible by observing that since we have tempered out [[176/175]], tempering out [[351/350]] and [[352/351]] which sum to 176/175 is very elegant. In the 17-limit we can equate the half-octave with 17/12 and 24/17 and we can take advantage of the sharp fifth by combining echidna with [[srutal archagall]], leading to a particularly beautiful temperament (one that prefers a very slightly less sharp fifth than srutal archagall). This mapping of 13 and 17 is supported by the patent vals of the three main echidna edos of 22, 58 and 80, of which all except 22 are consistent in the [[17-odd-limit]].

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 1728/1715, 2048/2025

: mapping generators: ~45/32, ~9/7

[[Optimal tuning]]s:

* [[WE]]: ~45/32 = 599.3056{{c}}, ~9/7 = 434.3524{{c}}

: [[error map]]: {{val| -1.389 +0.408 +1.322 +1.547 }}

* [[CWE]]: ~45/32 = 600.0000{{c}}, ~9/7 = 434.8327{{c}}

: error map: {{val| 0.000 +2.543 +4.690 +5.338 }}

[[Badness]] (Sintel): 1.47

=== 11-limit ===

Subgroup: 2.3.5.7.11

Comma list: 176/175, 540/539, 896/891

Mapping: {{mapping| 2 1 9 2 12 | 0 3 -6 5 -7 }}

Optimal tunings:

* WE: ~45/32 = 599.3085{{c}}, ~9/7 = 434.3511{{c}}

* CWE: ~45/32 = 600.0000{{c}}, ~9/7 = 434.8647{{c}}

Minimax tuning:

* 11-odd-limit: ~9/7 = {{monzo| 5/12 0 0 1/12 -1/12 }}

: [{{monzo| 1 0 0 0 0 }}, {{monzo| 7/4 0 0 1/4 -1/4 }}, {{monzo| 2 0 0 -1/2 1/2 }}, {{monzo| 37/12 0 0 5/12 -5/12 }}, {{monzo| 37/12 0 0 -7/12 7/12 }}]

: unchanged-interval (eigenmonzo) basis: 2.11/7

Badness (Sintel): 0.859

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

Comma list: 176/175, 351/350, 364/363, 540/539

Mapping: {{mapping| 2 1 9 2 12 19 | 0 3 -6 5 -7 -16 }}

Optimal tunings:

* WE: ~45/32 = 599.3397{{c}}, ~9/7 = 434.2772{{c}}

* CWE: ~45/32 = 600.0000{{c}}, ~9/7 = 434.7864{{c}}

Badness (Sintel): 0.978

=== 17-limit ===

Subgroup: 2.3.5.7.11.13.17

Comma list: 136/135, 176/175, 221/220, 256/255, 540/539

Mapping: {{mapping| 2 1 9 2 12 19 6 | 0 3 -6 5 -7 -16 3 }}

Optimal tunings:

* WE: ~45/32 = 599.4645{{c}}, ~9/7 = 434.4282{{c}}

* CWE: ~45/32 = 600.0000{{c}}, ~9/7 = 434.8340{{c}}

Badness (Sintel): 1.03

== Echidnic ==

Echidnic tempers out 686/675 and [[1029/1024]]. It has the same semi-octave period as diaschismic, but slices the generator of a fifth into three ~8/7's. It can be described as the {{nowrap| 10 & 46 }} temperament; its ploidacot is diploid tricot.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: ~~392~~/~~375~~, ~~1323~~/~~1280~~

[[Comma list]]: 686/675, 1029/1024

: mapping generators: ~45/32, ~10/7

: mapping generators: ~45/32, ~8/7

[[Optimal tuning]]s:

* [[WE]]: ~45/32 = ~~600~~.~~2519~~{{c}}, ~10/7 = ~~650~~.~~4083~~{{c}}

* [[WE]]: ~45/32 = 599.7208{{c}}, ~8/7 = 234.8330{{c}}

: [[error map]]: {{val| +0.~~504 -0~~.~~887~~ +14.~~321~~ -18.~~096~~ }}

: [[error map]]: {{val| -0.558 +1.986 +2.733 -5.334 }}

* [[CWE]]: ~45/32 = 600.0000{{c}}, ~10/7 = ~~650~~.~~1017~~{{c}}

* [[CWE]]: ~45/32 = 600.0000{{c}}, ~8/7 = 234.9539{{c}}

: error map: {{val| 0.000 -1.~~752~~ +13.~~279~~ -19.~~334~~ }}

: error map: {{val| 0.000 +2.907 +3.963 -3.780 }}

~~~~

[[Badness]] (Sintel): 3.99

[[Badness]] (Sintel): 1.83

=== 11-limit ===

Subgroup: 2.3.5.7.11

Comma list: 56/55, 77/75, ~~1323~~/~~1280~~

Comma list: 385/384, 441/440, 686/675

Mapping: {{mapping| 2 ~~1 9 11 8~~ | 0 ~~2 -4~~ -5 -1 }}

Mapping: {{mapping| 2 2 7 6 3 | 0 3 -6 -1 10 }}

Optimal tunings:

* WE: ~17/12 = ~~600~~.~~2356~~{{c}}, ~10/7 = ~~650~~.~~3856~~{{c}}

* WE: ~45/32 = 599.8022{{c}}, ~8/7 = 235.0185{{c}}

* CWE~~: ~17/12 = 600.0000{{c}}, ~10/7 = 650.1008{{c}}~~

* CWE: ~45/32 = 600.0000{{c}}, ~8/7 = 235.0893{{c}}

~~~~

Badness (Sintel): 2.10

Badness (Sintel): 1.49

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

Comma list: 56/55, 77/75, ~~105~~/~~104~~, ~~507~~/~~500~~

Comma list: 91/90, 169/168, 385/384, 441/440

Mapping: {{mapping| 2 2 7 6 3 7 | 0 3 -6 -1 10 1 }}

Optimal tunings:

* WE: ~45/32 = 599.9570{{c}}, ~8/7 = 235.0708{{c}}

* CWE: ~45/32 = 600.0000{{c}}, ~8/7 = 235.0862{{c}}

Badness (Sintel): 1.19

=== 17-limit ===

Subgroup: 2.3.5.7.11.13.17

Comma list: 91/90, 136/135, 154/153, 169/168, 256/255

Mapping: {{mapping| 2 ~~1 9 11 8 15~~ | 0 2 -~~4 -5~~ -1 -7 }}

Mapping: {{mapping| 2 2 7 6 3 7 7 | 0 3 -6 -1 10 1 3 }}

Optimal tunings:

* WE: ~45/32 = 599.~~9067~~{{c}}, ~10/7 = ~~649~~.~~4907~~{{c}}

* WE: ~17/12 = 599.9571{{c}}, ~8/7 = 235.0709{{c}}

* CWE: ~45/32 = 600.0000{{c}}, ~10/7 = ~~649~~.~~5950~~{{c}}

* CWE: ~17/12 = 600.0000{{c}}, ~8/7 = 235.0860{{c}}

~~~~

~~{{Optimal ET sequence|legend=~~0~~| 2, 24 }}~~

Badness (Sintel): 0.983

~~Badness~~ (~~Sintel~~)~~: 2.12~~

; Music

* [https://untwelve.org/competition/2011 ''A Stiff Shot of Turpentine''] [https://untwelve.org/static/audio/competition/2011/Kosmorsky-A_Stiff_Shot_of_Turpentine.mp3 play] by [[Peter Kosmorsky]]

* [https://www.youtube.com/watch?v=VsBXIvBZY6A ''56edo Track (Echidnic16 Scale)''] by [[Budjarn Lambeth]] (2025)

== Quadrasruta ==

Named by [[Xenllium]] in 2022, quadrasruta tempers out 2401/2400, the breedsma, and extends [[buzzard]]. It may be described as {{nowrap| 58 & 68 }}; its ploidacot is diploid alpha-tetracot. 126edo may be recommended as a tuning.

[[Subgroup]]: 2.3.5.7

Line 1,333:

Line 1,271:

* [[CWE]]: ~45/32 = 600.0000{{c}}, ~21/16 = 476.2394{{c}}

: error map: {{val| 0.000 +3.003 +3.771 +2.456 }}

~~~~

Line 1,349:

Line 1,286:

* WE: ~45/32 = 599.4648{{c}}, ~21/16 = 475.6929{{c}}

* CWE: ~45/32 = 600.0000{{c}}, ~21/16 = 476.1507{{c}}

~~~~

{{Optimal ET sequence|legend=0| 10e, …, 58, 126, 184c, 310bccde }}

Line 1,365:

Line 1,301:

* WE: ~45/32 = 599.3787{{c}}, ~21/16 = 475.6065{{c}}

* CWE: ~45/32 = 600.0000{{c}}, ~21/16 = 476.1345{{c}}

~~~~

Line 1,381:

Line 1,316:

* WE: ~17/12 = 599.5077{{c}}, ~21/16 = 475.7713{{c}}

* CWE: ~17/12 = 600.0000{{c}}, ~21/16 = 476.1814{{c}}

~~~~

Line 1,397:

Line 1,331:

* WE: ~45/32 = 599.2593{{c}}, ~21/16 = 475.4292{{c}}

* CWE: ~45/32 = 600.0000{{c}}, ~21/16 = 476.0088{{c}}

~~~~

{{Optimal ET sequence|legend=0| 10, 48c, 58, 184cee, 242ccdeee }}

Line 1,413:

Line 1,346:

* WE: ~45/32 = 599.2147{{c}}, ~21/16 = 475.4052{{c}}

* CWE: ~45/32 = 600.0000{{c}}, ~21/16 = 476.0253{{c}}

~~~~

{{Optimal ET sequence|legend=0| 10, 48c, 58, 126eef, 184ceeff, 242ccdeeeff }}

Line 1,429:

Line 1,361:

* WE: ~17/12 = 599.3353{{c}}, ~21/16 = 475.5495{{c}}

* CWE: ~17/12 = 600.0000{{c}}, ~21/16 = 476.0691{{c}}

~~~~

@@ Line 1: / Line 1: @@
 {{Technical data page}}
-The [[5-limit]] parent [[comma]] for the '''diaschismic family''' of [[regular temperament|temperaments]] is 2048/2025, the [[diaschisma]]. The [[period]] is half an [[2/1|octave]], and the [[generator]] is a fifth. Three periods gives 1800 cents, and decreasing this by two fifths gives the major third. [[34edo]] is a good tuning choice, with [[46edo]], [[56edo]], [[58edo]], or [[80edo]] being other possibilities. Both [[12edo]] and [[22edo]] support it, and retuning them to a [[mos]] of diaschismic gives two scale possibilities.
+The '''diaschismic family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the diaschisma, [[2048/2025]].
 == Diaschismic ==
 {{Main| Diaschismic }}
+The [[period]] of diaschismic is half an [[2/1|octave]], and the [[generator]] is a fifth; the [[ploidacot]] is diploid monocot. Three periods gives 1800 cents, and decreasing this by two fifths gives the major third. [[34edo]] is a good tuning choice, with [[46edo]], [[56edo]], [[58edo]], or [[80edo]] being other possibilities. Both [[12edo]] and [[22edo]] support it, and retuning them to a [[mos]] of diaschismic gives two scale possibilities.
 This temperament is also known as '''srutal''' in the 5-limit, but that name more strictly speaking refers to the [[#Srutal|34d & 46 extension]] to the [[7-limit]] that adds [[4375/4374]] to the comma list.
@@ Line 20: / Line 22: @@
 * [[CWE]]: ~45/32 = 600.0000{{c}}, ~3/2 = 704.9585{{c}}
 : error map: {{val| 0.000 +3.003 +3.769 }}
-<!-- * [[POTE]]: ~45/32 = 600.000{{c}}, ~3/2 = 704.898{{c}} -->
 [[Tuning ranges]]:
@@ Line 33: / Line 34: @@
 ==== 7-limit extensions ====
 To get the 7-limit extensions, we add another comma:
 * Septimal diaschismic adds [[126/125]], the starling comma, to obtain 7-limit harmony by more complex methods than pajara, but with greater accuracy.
 * Pajara derives from [[64/63]] and is a popular and well-known choice.
 * Srutal adds [[4375/4374]], the ragisma, which is about as accurate as septimal diaschismic but has a much more complex mapping of 7.
 * Keen adds [[875/864]].
-* Bidia adds [[3136/3125]], the hemimean comma.
-* Echidna adds [[1728/1715]], the orwellisma.
-* Shrutar adds [[245/243]], the sensamagic comma.
-Pajara, diaschismic, srutal and keen keep the same half-octave period and fifth generator, but shrutar has a generator of a quarter-tone (which can be taken as [[36/35]], the septimal quarter-tone) and echidna has a generator of 9/7. Bidia has a quarter-octave period and a fifth generator.
+Those all keep the same half-octave period and fifth generator.
+Bidia adds [[3136/3125]], the hemimean comma, with a 1/4-octave period. Shrutar adds [[245/243]] and shru adds [[392/375]], with a quartertone generator. Sruti adds [[19683/19600]] and anguirus adds [[49/48]], with a neutral third or hemitwelfth generator. Those split the original generator in two. Echidna adds [[1728/1715]], the orwellisma, with a ~9/7 generator. Echidnic adds [[686/675]], the senga, with a ~8/7 generator. Those split the original generator in three. Finally, quadrasruta adds [[2401/2400]] and splits the original  generator in four.
 ==== Subgroup extensions ====
 Since the diaschisma factors into ([[256/255]])<sup>2</sup>([[289/288]]) in the 17-limit, it extends naturally to the 2.3.5.17 subgroup as ''srutal archagall'', documented right below. The [[S-expression]]-based comma list of this temperament is {[[256/255|S16]], [[289/288|S17]]}.
 === Srutal archagall ===
@@ Line 61: / Line 60: @@
 * WE: ~45/32 = 599.5585{{c}}, ~3/2 = 704.6188{{c}}
 * CWE: ~45/32 = 600.0000{{c}}, ~3/2 = 705.1356{{c}}
-<!-- * CTE: ~17/12 = 600.0000{{c}}, ~3/2 = 705.1272{{c}} -->
 {{Optimal ET sequence|legend=0| 10, 12, 22, 34, 80, 114, 194bc }}
@@ Line 71: / Line 69: @@
 {{See also| Srutal vs diaschismic }}
-A simpler characterization than the one given by the normal comma list is that diaschismic adds [[126/125]] or [[5120/5103]] to the set of commas, and it can also be called {{nowrap| 46 & 58 }}. However described, diaschismic has a 1/2-octave period and a sharp fifth generator like pajara, but not so sharp, giving a more accurate but more complex temperament. [[58edo]] provides an excellent tuning, but an alternative is to make [[7/4]] just by making the fifth 703.897 cents, as opposed to 703.448 cents for 58edo.
+A simpler characterization than the one given by the normal comma list is that septimal diaschismic adds [[126/125]] or [[5120/5103]] to the set of commas, and it can also be called {{nowrap| 46 & 58 }}. However described, septimal diaschismic has a 1/2-octave period and a sharp fifth generator like the 5-limit version, but not so sharp, giving a more accurate but more complex temperament. [[104edo]] provides an excellent tuning, which is close to tuning [[7/4]] just by making the fifth 703.897 cents.
 Diaschismic extends naturally to the 17-limit, for which the same tunings may be used, making it one of the most important of the higher-limit rank-2 temperaments. Adding the 11-limit adds the commas 176/175, 896/891 and 441/440. The 13-limit yields 196/195, 351/350, and 364/363; the 17-limit adds 136/135, 221/220, and 442/441. If you want to explore higher-limit harmonies, diaschismic is certainly one excellent way to do it; [[mos]] of 34 notes and even more the 46-note mos will encompass very great deal of it. Of course 46 or 58 equal provide alternatives which in many ways are similar, particularly in the case of 58.
@@ Line 86: / Line 84: @@
 * [[CWE]]: ~45/32 = 600.0000{{c}}, ~3/2 = 703.7739{{c}}
 : error map: {{val| 0.000 +1.819 +6.138 +0.983 }}
-<!-- * [[POTE]]: ~45/32 = 600.000{{c}}, ~3/2 = 703.681{{c}} -->
 [[Tuning ranges]]:
@@ Line 106: / Line 103: @@
 * WE: ~45/32 = 599.4471{{c}}, ~3/2 = 703.0657{{c}}
 * CWE: ~45/32 = 600.0000{{c}}, ~3/2 = 703.7996{{c}}
-<!-- * POTE: ~45/32 = 600.000{{c}}, ~3/2 = 703.714{{c}} -->
 Tuning ranges:
@@ Line 126: / Line 122: @@
 * WE: ~45/32 = 599.4451{{c}}, ~3/2 = 703.0528{{c}}
 * CWE: ~45/32 = 600.0000{{c}}, ~3/2 = 703.7813{{c}}
-<!-- * POTE: ~45/32 = 600.000{{c}}, ~3/2 = 703.704{{c}} -->
 Tuning ranges:
@@ Line 147: / Line 142: @@
 * WE: ~17/12 = 599.6253{{c}}, ~3/2 = 703.3726{{c}}
 * CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 703.8520{{c}}
-<!-- * POTE: ~17/12 = 600.000{{c}}, ~3/2 = 703.812{{c}} -->
 Tuning ranges:
@@ Line 169: / Line 163: @@
 * WE: ~17/12 = 599.6272{{c}}, ~3/2 = 703.4326{{c}}
 * CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 703.9093{{c}}
-<!-- * POTE: ~17/12 = 600.000{{c}}, ~3/2 = 703.870{{c}} -->
 {{Optimal ET sequence|legend=0| 12i, 34efi, 46, 58i, 104ci }}
@@ Line 178: / Line 171: @@
 {{Main| Pajara }}
-Pajara is closely associated with 22edo (not to mention [[Paul Erlich]]) but other tunings are possible. The 1/2-octave period serves as both a [[10/7]] and a [[7/5]]. Aside from 22edo, 34 with the val {{val| 34 54 79 96 }} and 56 with the val {{val| 56 89 130 158 }} are interesting alternatives, with more acceptable fifths, and a tetrad which is more clearly a dominant seventh. As such, they are closer to the tuning of 12edo and of common practice Western music in general, while retaining the distictiveness of a sharp fifth.
+Pajara is closely associated with 22edo (not to mention [[Paul Erlich]]) but other tunings are possible. The 1/2-octave period serves as both a [[10/7]] and a [[7/5]]. Aside from 22edo, 34 with the val {{val| 34 54 79 96 }} (34d) and 56 with the val {{val| 56 89 130 158 }} (56d) are interesting alternatives, with more acceptable fifths, and a tetrad which is more clearly a dominant seventh. As such, they are closer to the tuning of 12edo and of common practice Western music in general, while retaining the distictiveness of a sharp fifth.
 Pajara extends nicely to an 11-limit version, for which the 56edo tuning can be used, but a good alternative is to make the major thirds pure by setting the fifth to be 706.843 cents. Now 99/98, 100/99, 176/175 and 896/891 are being tempered out.
@@ Line 193: / Line 186: @@
 * [[CWE]]: ~7/5 = 600.0000{{c}}, ~3/2 = 707.3438{{c}}
 : error map: {{val| 0.000 +5.389 -1.001 +16.487 }}
-<!-- [[POTE]]: ~7/5 = 600.000{{c}}, ~3/2 = 707.048{{c}} -->
 [[Tuning ranges]]:
@@ Line 213: / Line 205: @@
 * WE: ~7/5 = 598.8485{{c}}, ~3/2 = 705.5285{{c}}
 * CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 707.1826{{c}}
-<!-- * POTE: ~7/5 = 600.000{{c}}, ~3/2 = 706.885{{c}} -->
 Tuning ranges:
@@ Line 233: / Line 224: @@
 * WE: ~7/5 = 599.9732{{c}}, ~3/2 = 708.8873{{c}}
 * CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 708.9227{{c}}
-<!-- * POTE: ~7/5 = 600.000{{c}}, ~3/2 = 708.919{{c}} -->
 {{Optimal ET sequence|legend=0| 10e, 12, 22 }}
@@ Line 249: / Line 239: @@
 * WE: ~7/5 = 599.8871{{c}}, ~3/2 = 708.6725{{c}}
 * CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 708.8176{{c}}
-<!-- * POTE: ~7/5 = 600.000{{c}}, ~3/2 = 708.806{{c}} -->
 {{Optimal ET sequence|legend=0| 10e, 12, 22 }}
@@ Line 265: / Line 254: @@
 * WE: ~7/5 = 598.7732{{c}}, ~3/2 = 704.6889{{c}}
 * CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 706.3950{{c}}
-<!-- * POTE: ~7/5 = 600.000{{c}}, ~3/2 = 706.133{{c}} -->
 {{Optimal ET sequence|legend=0| 12f, 22, 34d }}
@@ Line 281: / Line 269: @@
 * WE: ~7/5 = 599.0204{{c}}, ~3/2 = 705.2572{{c}}
 * CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 706.5660{{c}}
-<!-- * POTE: ~7/5 = 600.000{{c}}, ~3/2 = 706.410{{c}} -->
 {{Optimal ET sequence|legend=0| 12f, 22, 34d }}
@@ Line 297: / Line 284: @@
 * WE: ~7/5 = 598.3048{{c}}, ~3/2 = 705.4512{{c}}
 * CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 707.9238{{c}}
-<!-- * POTE: ~7/5 = 600.000{{c}}, ~3/2 = 707.450{{c}} -->
 {{Optimal ET sequence|legend=0| 10e, 12f, 22f, 34dff }}
@@ Line 313: / Line 299: @@
 * WE: ~7/5 = 598.6103{{c}}, ~3/2 = 706.3076{{c}}
 * CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 708.2256{{c}}
-<!-- * POTE: ~7/5 = 600.000{{c}}, ~3/2 = 707.947{{c}} -->
 {{Optimal ET sequence|legend=0| 10e, 12f, 22f }}
@@ Line 329: / Line 314: @@
 * WE: ~7/5 = 599.4055{{c}}, ~3/2 = 708.8747{{c}}
 * CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 709.5508{{c}}
-<!-- * POTE: ~7/5 = 600.000{{c}}, ~3/2 = 709.578{{c}} -->
 Tuning ranges:
@@ Line 349: / Line 333: @@
 * WE: ~7/5 = 599.9064{{c}}, ~3/2 = 710.1289{{c}}
 * CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 710.2325{{c}}
-<!-- * POTE: ~7/5 = 600.000{{c}}, ~3/2 = 710.240{{c}} -->
 {{Optimal ET sequence|legend=0| 10, 22 }}
@@ Line 365: / Line 348: @@
 * WE: ~7/5 = 599.8239{{c}}, ~3/2 = 710.0128{{c}}
 * CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 710.2067{{c}}
-<!-- * POTE: ~7/5 = 600.000{{c}}, ~3/2 = 710.221{{c}} -->
 {{Optimal ET sequence|legend=0| 10, 22, 54f, 76bdff }}
@@ Line 381: / Line 363: @@
 * WE: ~7/5 = 598.8257{{c}}, ~3/2 = 709.4266{{c}}
 * CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 710.8414{{c}}
-<!-- * POTE: ~7/5 = 600.000{{c}}, ~3/2 = 710.818{{c}} -->
 {{Optimal ET sequence|legend=0| 10, 22f, 32f }}
@@ Line 397: / Line 378: @@
 * WE: ~7/5 = 598.8865{{c}}, ~3/2 = 709.5472{{c}}
 * CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 710.8704{{c}}
-<!-- * POTE: ~7/5 = 600.000{{c}}, ~3/2 = 710.866{{c}} -->
 {{Optimal ET sequence|legend=0| 10, 22f, 32f }}
@@ Line 413: / Line 393: @@
 * WE: ~7/5 = 597.4807{{c}}, ~3/2 = 702.5616{{c}}
 * CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 706.0542{{c}}
-<!-- * POTE: ~7/5 = 600.000{{c}}, ~3/2 = 705.524{{c}} -->
 {{Optimal ET sequence|legend=0| 10, 12, 22e }}
@@ Line 429: / Line 408: @@
 * WE: ~7/5 = 597.1952{{c}}, ~3/2 = 704.1350{{c}}
 * CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 708.1989{{c}}
-<!-- * POTE: ~7/5 = 600.000{{c}}, ~3/2 = 707.442{{c}} -->
 {{Optimal ET sequence|legend=0| 10, 12f, 22ef }}
@@ Line 445: / Line 423: @@
 * WE: ~7/5 = 597.6509{{c}}, ~3/2 = 705.7702{{c}}
 * CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 708.9719{{c}}
-<!-- * POTE: ~7/5 = 600.000{{c}}, ~3/2 = 708.544{{c}} -->
 {{Optimal ET sequence|legend=0| 10, 12f, 22ef }}
@@ Line 463: / Line 440: @@
 * WE: ~7/5 = 597.6509{{c}}, ~16/11 = 652.7788{{c}}
 * CWE: ~7/5 = 600.0000{{c}}, ~16/11 = 653.7119{{c}}
-<!-- * POTE: ~7/5 = 600.000{{c}}, ~16/11 = 653.617{{c}} -->
 {{Optimal ET sequence|legend=0| 2, 20, 22 }}
@@ Line 481: / Line 457: @@
 * WE: ~7/5 = 597.6509{{c}}, ~55/32 = 950.8475{{c}}
 * CWE: ~7/5 = 600.0000{{c}}, ~55/32 = 953.1172{{c}}
-<!-- * POTE: ~7/5 = 600.000{{c}}, ~55/32 = 953.093{{c}} -->
 {{Optimal ET sequence|legend=0| 10, 24d, 34d }}
@@ Line 497: / Line 472: @@
 * WE: ~7/5 = 598.6748{{c}}, ~26/15 = 950.9691{{c}}
 * CWE: ~7/5 = 600.0000{{c}}, ~26/15 = 953.1052{{c}}
-<!-- * POTE: ~7/5 = 600.000{{c}}, ~26/15 = 953.074{{c}} -->
 {{Optimal ET sequence|legend=0| 10, 24d, 34d }}
@@ Line 513: / Line 487: @@
 * WE: ~7/5 = 598.8411{{c}}, ~26/15 = 951.3687{{c}}
 * CWE: ~7/5 = 600.0000{{c}}, ~26/15 = 953.2169{{c}}
-<!-- * POTE: ~7/5 = 600.000{{c}}, ~26/15 = 953.210{{c}} -->
 {{Optimal ET sequence|legend=0| 10, 24d, 34d }}
@@ Line 522: / Line 495: @@
 {{See also| Srutal vs diaschismic }}
-Srutal can be described as the 34d & 46 temperament, where 7/4 is located at 15 generator steps, or the double-augmented fifth (C–Gx). 80edo and 126edo are among the possible tunings. Srutal, shrutar and bidia have similar 19-limit properties, tempering out 190/189, related to rank-3 [[julius]].
+Srutal can be described as the {{nowrap| 34d & 46 }} temperament, where 7/4 is located at 15 generator steps, or the double-augmented fifth (C–Gx). As such, it weakly extends [[leapfrog]]. 80edo and [[126edo]] are among the possible tunings. Srutal, shrutar and bidia have similar 19-limit properties, tempering out 190/189, related to rank-3 [[julius]].
 [[Subgroup]]: 2.3.5.7
@@ Line 535: / Line 508: @@
 * [[CWE]]: ~45/32 = 600.0000{{c}}, ~3/2 = 704.7646{{c}}
 : error map: {{val| 0.000 +2.810 +4.157 +2.643 }}
-<!-- * [[POTE]]: ~45/32 = 600.000{{c}}, ~3/2 = 704.814{{c}} -->
 [[Tuning ranges]]:
@@ Line 555: / Line 527: @@
 * WE: ~45/32 = 599.4413{{c}}, ~3/2 = 704.1999{{c}}
 * CWE: ~45/32 = 600.0000{{c}}, ~3/2 = 704.8017{{c}}
-<!-- * POTE: ~45/32 = 600.000{{c}}, ~3/2 = 704.856{{c}} -->
 Tuning ranges:
@@ Line 575: / Line 546: @@
 * WE: ~45/32 = 599.5490{{c}}, ~3/2 = 704.3516{{c}}
 * CWE: ~45/32 = 600.0000{{c}}, ~3/2 = 704.8347{{c}}
-<!-- * POTE: ~45/32 = 600.000{{c}}, ~3/2 = 704.881{{c}} -->
 Tuning ranges:
@@ Line 596: / Line 566: @@
 * WE: ~17/12 = 599.6459{{c}}, ~3/2 = 704.4237{{c}}
 * CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 704.8083{{c}}
-<!-- * POTE: ~17/12 = 600.000{{c}}, ~3/2 = 704.840{{c}} -->
 Tuning ranges:
@@ Line 616: / Line 585: @@
 * WE: ~17/12 = 599.6371{{c}}, ~3/2 = 704.4790{{c}}
 * CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 704.8745{{c}}
-<!-- * POTE: ~17/12 = 600.000{{c}}, ~3/2 = 704.905{{c}} -->
 {{Optimal ET sequence|legend=0| 34dh, 46, 80 }}
@@ Line 634: / Line 602: @@
 * WE: ~17/12 = 599.6690{{c}}, ~3/2 = 704.5098{{c}}
 * CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 704.8713{{c}}
-<!-- * POTE: ~17/12 = 600.000{{c}}, ~3/2 = 704.899{{c}} -->
 {{Optimal ET sequence|legend=0| 34dh, 46, 80 }}
@@ Line 650: / Line 617: @@
 * WE: ~17/12 = 599.6664{{c}}, ~3/2 = 704.5138{{c}}
 * CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 704.8807{{c}}
-<!-- * POTE: ~17/12 = 600.000{{c}}, ~3/2 = 704.906{{c}} -->
 {{Optimal ET sequence|legend=0| 34dhj, 46, 80 }}
@@ Line 666: / Line 632: @@
 * WE: ~17/12 = 599.8115{{c}}, ~3/2 = 704.5958{{c}}
 * CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 704.8086{{c}}
-<!-- * POTE: ~17/12 = 600.000{{c}}, ~3/2 = 704.817{{c}} -->
 {{Optimal ET sequence|legend=0| 46, 80, 126 }}
@@ Line 673: / Line 638: @@
 == Keen ==
-Keen adds 875/864 as well as 2240/2187 to the set of commas. It may also be described as the {{nowrap| 22 & 56 }} temperament. [[78edo]] is a good tuning choice, and remains a good one in the 11-limit, where the temperament is really more interesting, adding 100/99 and 385/384 to the list of commas.
+Keen adds 875/864 as well as 2240/2187 to the set of commas. It may also be described as the {{nowrap| 22 & 34 }} temperament. [[78edo]] is a good tuning choice, and remains a good one in the 11-limit, where the temperament is really more interesting, adding 100/99 and 385/384 to the list of commas.
 [[Subgroup]]: 2.3.5.7
@@ Line 686: / Line 651: @@
 * [[CWE]]: ~45/32 = 600.0000{{c}}, ~3/2 = 707.5294{{c}}
 : error map: {{val| 0.000 +5.574 -1.373 -1.061 }}
-<!-- * [[POTE]]: ~45/32 = 600.000{{c}}, ~3/2 = 707.571{{c}} -->
 {{Optimal ET sequence|legend=1| 22, 56, 78, 134b }}
@@ Line 702: / Line 666: @@
 * WE: ~45/32 = 599.6286{{c}}, ~3/2 = 707.1712{{c}}
 * CWE: ~45/32 = 600.0000{{c}}, ~3/2 = 707.5984{{c}}
-<!-- * POTE: ~45/32 = 600.000{{c}}, ~3/2 = 707.609{{c}} -->
 {{Optimal ET sequence|legend=0| 22, 56, 78 }}
@@ Line 718: / Line 681: @@
 * WE: ~45/32 = 599.3498{{c}}, ~3/2 = 706.4009{{c}}
 * CWE: ~45/32 = 600.0000{{c}}, ~3/2 = 707.1309{{c}}
-<!-- * POTE: ~45/32 = 600.000{{c}}, ~3/2 = 707.167{{c}} -->
 {{Optimal ET sequence|legend=0| 22f, 34, 56f }}
@@ Line 734: / Line 696: @@
 * WE: ~17/12 = 599.4053{{c}}, ~3/2 = 706.4544{{c}}
 * CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 707.1243{{c}}
-<!-- * POTE: ~17/12 = 600.000{{c}}, ~3/2 = 707.155{{c}} -->
 {{Optimal ET sequence|legend=0| 22f, 34, 56f }}
@@ Line 750: / Line 711: @@
 * WE: ~45/32 = 599.8547{{c}}, ~3/2 = 707.0858{{c}}
 * CWE: ~45/32 = 600.0000{{c}}, ~3/2 = 707.2596{{c}}
-<!-- * POTE: ~45/32 = 600.000{{c}}, ~3/2 = 707.257{{c}} -->
 {{Optimal ET sequence|legend=0| 22, 34, 56 }}
@@ Line 766: / Line 726: @@
 * WE: ~17/12 = 599.8338{{c}}, ~3/2 = 707.0558{{c}}
 * CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 707.2537{{c}}
-<!-- * POTE: ~17/12 = 600.000{{c}}, ~3/2 = 707.252{{c}} -->
 {{Optimal ET sequence|legend=0| 22, 34, 56 }}
@@ Line 773: / Line 732: @@
 == Bidia ==
-Bidia adds [[3136/3125]] to the commas, splitting the period into 1/4 octave. It may be called the {{nowrap| 12 & 56 }} temperament.
+Bidia adds [[3136/3125]] to the commas, splitting the period into 1/4 octave. It may be called the {{nowrap| 12 & 68 }} temperament; its ploidacot is tetraploid monocot.
 [[Subgroup]]: 2.3.5.7
@@ Line 788: / Line 747: @@
 * [[CWE]]: ~25/21 = 300.0000{{c}}, ~3/2 = 705.5070{{c}}
 : error map: {{val| 0.000 +3.552 +2.672 +3.639 }}
-<!-- * [[POTE]]: ~25/21 = 300.000{{c}}, ~3/2 = 705.364{{c}} -->
 {{Optimal ET sequence|legend=1| 12, …, 56, 68, 80, 148d }}
@@ Line 804: / Line 762: @@
 * WE: ~25/21 = 299.6809{{c}}, ~3/2 = 704.3367{{c}}
 * CWE: ~25/21 = 600.0000{{c}}, ~3/2 = 705.2170{{c}}
-<!-- * POTE: ~25/21 = 300.000{{c}}, ~3/2 = 705.087{{c}} -->
 {{Optimal ET sequence|legend=0| 12, 56e, 68, 80 }}
@@ Line 820: / Line 777: @@
 * WE: ~25/21 = 299.7538{{c}}, ~3/2 = 704.7222{{c}}
 * CWE: ~25/21 = 600.0000{{c}}, ~3/2 = 705.3241{{c}}
-<!-- * POTE: ~25/21 = 300.000{{c}}, ~3/2 = 705.301{{c}} -->
 {{Optimal ET sequence|legend=0| 12, 68, 80, 148d, 228bcd, 376bbcddf }}
@@ Line 836: / Line 792: @@
 * WE: ~25/21 = 299.7883{{c}}, ~3/2 = 704.8365{{c}}
 * CWE: ~25/21 = 600.0000{{c}}, ~3/2 = 705.3496{{c}}
-<!-- * POTE: ~25/21 = 300.000{{c}}, ~3/2 = 705.334{{c}} -->
 {{Optimal ET sequence|legend=0| 12, 68, 80, 148d }}
@@ Line 852: / Line 807: @@
 * WE: ~19/16 = 299.7967{{c}}, ~3/2 = 704.8609{{c}}
 * CWE: ~19/16 = 600.0000{{c}}, ~3/2 = 705.3519{{c}}
-<!-- * POTE: ~19/16 = 300.000{{c}}, ~3/2 = 705.339{{c}} -->
 {{Optimal ET sequence|legend=0| 12, 68, 80, 148d }}
@@ Line 868: / Line 822: @@
 * WE: ~19/16 = 299.7961{{c}}, ~3/2 = 704.8577{{c}}
 * CWE: ~19/16 = 300.0000{{c}}, ~3/2 = 705.3413{{c}}
-<!-- * POTE: ~19/16 = 300.000{{c}}, ~3/2 = 705.337{{c}} -->
 {{Optimal ET sequence|legend=0| 12, 68, 80, 148di }}
 Badness (Sintel): 1.24
-== Echidna ==
-Echidna adds 1728/1715 to the commas and takes 9/7 as a generator. It may be called the {{nowrap| 22 & 58 }} temperament. [[58edo]] or [[80edo]] make for good tunings, or their vals can be added to {{val| 138 219 321 388 }} (138cde). In most of the tunings it has a significantly sharp 7/4 which some prefer.
-Echidna becomes more interesting when extended to be an 11-limit temperament by adding 176/175, 540/539 or 896/891 to the commas, where the same tunings can be used as before. It then is able to represent the entire 11-odd-limit diamond to within about six cents of error, within a compass of 24 notes. The 22-note 2mos gives scope for this, and the 36-note mos much more. Better yet, it is related to three important 11-limit edos: 22edo, a trivial tuning, is the smallest consistent in the 11-odd-limit, corresponding to the merge of this temperament with [[hedgehog]]; [[58edo]] is the smallest tuning that is distinctly consistent in the 11-odd-limit and [[80edo]] is the third smallest distinctly consistent in the 11-odd-limit.
-The generator can be interpreted as 11/10, the period complement of 9/7, as a stack of 11/10 and 9/7 makes [[99/70]] which is extremely close to 600{{cent}} and is equal to it if we temper out [[9801/9800|S99]]. Three 11/10's then make a 4/3 (tempering out [[4000/3993|S10/S11]] thus making 10/9 and 12/11 equidistant from 11/10), implying a flat tuning of 4/3.
-Like most srutal extensions, the 13- and 17-limit interpretations are possible by observing that since we have tempered out [[176/175]], tempering out [[351/350]] and [[352/351]] which sum to 176/175 is very elegant. In the 17-limit we can equate the half-octave with 17/12 and 24/17 and we can take advantage of the sharp fifth by combining echidna with [[srutal archagall]], leading to a particularly beautiful temperament (one that prefers a very slightly less sharp fifth than srutal archagall). This mapping of 13 and 17 is supported by the patent vals of the three main echidna edos of 22, 58 and 80, of which all except 22 are consistent in the [[17-odd-limit]].
-[[Subgroup]]: 2.3.5.7
-[[Comma list]]: 1728/1715, 2048/2025
-{{Mapping|legend=1| 2 1 9 2 | 0 3 -6 5 }}
-: mapping generators: ~45/32, ~9/7
-[[Optimal tuning]]s:
-* [[WE]]: ~45/32 = 599.3056{{c}}, ~9/7 = 434.3524{{c}}
-: [[error map]]: {{val| -1.389 +0.408 +1.322 +1.547 }}
-* [[CWE]]: ~45/32 = 600.0000{{c}}, ~9/7 = 434.8327{{c}}
-: error map: {{val| 0.000 +2.543 +4.690 +5.338 }}
-<!-- * [[POTE]]: ~45/32 = 600.000{{c}}, ~9/7 = 434.856{{c}} -->
-{{Optimal ET sequence|legend=1| 22, 58, 80, 138cd, 218cd }}
-[[Badness]] (Sintel): 1.47
-=== 11-limit ===
-Subgroup: 2.3.5.7.11
-Comma list: 176/175, 540/539, 896/891
-Mapping: {{mapping| 2 1 9 2 12 | 0 3 -6 5 -7 }}
-Optimal tunings:
-* WE: ~45/32 = 599.3085{{c}}, ~9/7 = 434.3511{{c}}
-* CWE: ~45/32 = 600.0000{{c}}, ~9/7 = 434.8647{{c}}
-<!-- * POTE: ~45/32 = 600.000{{c}}, ~9/7 = 434.852{{c}} -->
-Minimax tuning:
-* 11-odd-limit: ~9/7 = {{monzo| 5/12 0 0 1/12 -1/12 }}
-: [{{monzo| 1 0 0 0 0 }}, {{monzo| 7/4 0 0 1/4 -1/4 }}, {{monzo| 2 0 0 -1/2 1/2 }}, {{monzo| 37/12 0 0 5/12 -5/12 }}, {{monzo| 37/12 0 0 -7/12 7/12 }}]
-: unchanged-interval (eigenmonzo) basis: 2.11/7
-{{Optimal ET sequence|legend=0| 22, 58, 80, 138cde, 218cde }}
-Badness (Sintel): 0.859
-=== 13-limit ===
-Subgroup: 2.3.5.7.11.13
-Comma list: 176/175, 351/350, 364/363, 540/539
-Mapping: {{mapping| 2 1 9 2 12 19 | 0 3 -6 5 -7 -16 }}
-Optimal tunings:
-* WE: ~45/32 = 599.3397{{c}}, ~9/7 = 434.2772{{c}}
-* CWE: ~45/32 = 600.0000{{c}}, ~9/7 = 434.7864{{c}}
-<!-- * POTE: ~45/32 = 600.000{{c}}, ~9/7 = 434.756{{c}} -->
-{{Optimal ET sequence|legend=0| 22, 36f, 58, 80, 138cde }}
-Badness (Sintel): 0.978
-=== 17-limit ===
-Subgroup: 2.3.5.7.11.13.17
-Comma list: 136/135, 176/175, 221/220, 256/255, 540/539
-Mapping: {{mapping| 2 1 9 2 12 19 6 | 0 3 -6 5 -7 -16 3 }}
-Optimal tunings:
-* WE: ~45/32 = 599.4645{{c}}, ~9/7 = 434.4282{{c}}
-* CWE: ~45/32 = 600.0000{{c}}, ~9/7 = 434.8340{{c}}
-<!-- * POTE: ~17/12 = 600.000{{c}}, ~9/7 = 434.816{{c}} -->
-{{Optimal ET sequence|legend=0| 22, 36f, 58, 80, 138cde }}
-Badness (Sintel): 1.03
-== Echidnic ==
-[[Subgroup]]: 2.3.5.7
-[[Comma list]]: 686/675, 1029/1024
-{{Mapping|legend=1| 2 2 7 6 | 0 3 -6 -1 }}
-: mapping generators: ~45/32, ~8/7
-[[Optimal tuning]]s:
-* [[WE]]: ~45/32 = 599.7208{{c}}, ~8/7 = 234.8330{{c}}
-: [[error map]]: {{val| -0.558 +1.986 +2.733 -5.334 }}
-* [[CWE]]: ~45/32 = 600.0000{{c}}, ~8/7 = 234.9539{{c}}
-: error map: {{val| 0.000 +2.907 +3.963 -3.780 }}
-<!-- * [[POTE]]: ~45/32 = 600.000{{c}}, ~8/7 = 234.492{{c}} -->
-{{Optimal ET sequence|legend=1| 10, 26c, 36, 46 }}
-[[Badness]] (Sintel): 1.83
-=== 11-limit ===
-Subgroup: 2.3.5.7.11
-Comma list: 385/384, 441/440, 686/675
-Mapping: {{mapping| 2 2 7 6 3 | 0 3 -6 -1 10 }}
-Optimal tunings:
-* WE: ~45/32 = 599.8022{{c}}, ~8/7 = 235.0185{{c}}
-* CWE: ~45/32 = 600.0000{{c}}, ~8/7 = 235.0893{{c}}
-<!-- * POTE: ~45/32 = 600.000{{c}}, ~8/7 = 235.096{{c}} -->
-{{Optimal ET sequence|legend=0| 10, 36e, 46, 102, 148 }}
-Badness (Sintel): 1.49
-=== 13-limit ===
-Subgroup: 2.3.5.7.11.13
-Comma list: 91/90, 169/168, 385/384, 441/440
-Mapping: {{mapping| 2 2 7 6 3 7 | 0 3 -6 -1 10 1 }}
-Optimal tunings:
-* WE: ~45/32 = 599.9570{{c}}, ~8/7 = 235.0708{{c}}
-* CWE: ~45/32 = 600.0000{{c}}, ~8/7 = 235.0862{{c}}
-<!-- * POTE: ~45/32 = 600.000{{c}}, ~8/7 = 235.088{{c}} -->
-{{Optimal ET sequence|legend=0| 10, 36e, 46, 102, 148f }}
-Badness (Sintel): 1.19
-=== 17-limit ===
-Subgroup: 2.3.5.7.11.13.17
-Comma list: 91/90, 136/135, 154/153, 169/168, 256/255
-Mapping: {{mapping| 2 2 7 6 3 7 7 | 0 3 -6 -1 10 1 3 }}
-Optimal tunings:
-* WE: ~17/12 = 599.9571{{c}}, ~8/7 = 235.0709{{c}}
-* CWE: ~17/12 = 600.0000{{c}}, ~8/7 = 235.0860{{c}}
-<!-- * POTE: ~17/12 = 600.000{{c}}, ~8/7 = 235.088{{c}} -->
-{{Optimal ET sequence|legend=0| 10, 36e, 46, 102, 148f }}
-Badness (Sintel): 0.983
-; Music
-* [https://untwelve.org/competition/2011 ''A Stiff Shot of Turpentine''] [https://untwelve.org/static/audio/competition/2011/Kosmorsky-A_Stiff_Shot_of_Turpentine.mp3 play] by [[Peter Kosmorsky]]
-* [https://www.youtube.com/watch?v=VsBXIvBZY6A ''56edo Track (Echidnic16 Scale)''] by [[Budjarn Lambeth]] (2025)
 == Shrutar ==
-Shrutar adds 245/243 to the commas, and also tempers out 6144/6125. It can also be described as {{nowrap| 22 & 46 }}. Its generator can be taken as either 36/35 or 35/24; the latter is interesting since along with 15/14 and 21/20, it connects opposite sides of a hexany. [[68edo]] makes for a good tuning, but another excellent choice is a generator of 14<sup>(1/7)</sup>, making 7's just.
+Shrutar adds 245/243 to the commas, and also tempers out [[6144/6125]]. It can also be described as {{nowrap| 22 & 46 }}. Its generator can be taken as either ~36/35 or ~35/24; the latter is interesting since along with 15/14 and 21/20, it connects opposite sides of a hexany. Its ploidacot is diploid alpha-dicot. [[68edo]] makes for a good tuning, but another excellent choice is a generator of 14<sup>(1/7)</sup>, making 7's just.
 By adding 121/120 or 176/175 to the commas, shrutar can be extended to the 11-limit, which loses a bit of accuracy, but picks up low-complexity 11-limit harmony, making shrutar quite an interesting 11-limit system. 68, 114 or a 14<sup>(1/7)</sup> generator can again be used as tunings.
@@ Line 1,045: / Line 845: @@
 * [[CWE]]: ~45/32 = 600.0000{{c}}, ~35/24 = 652.7736{{c}}
 : error map: {{val| 0.000 +3.592 +2.592 +0.589 }}
-<!-- * [[POTE]]: ~45/32 = 600.000{{c}}, ~35/24 = 652.811{{c}} -->
 {{Optimal ET sequence|legend=1| 22, 46, 68, 182b, 250bc }}
@@ Line 1,061: / Line 860: @@
 * WE: ~45/32 = 599.7721{{c}}, ~16/11 = 652.4321{{c}}
 * CWE: ~45/32 = 600.0000{{c}}, ~16/11 = 652.6672{{c}}
-<!-- * POTE: ~45/32 = 600.000{{c}}, ~16/11 = 652.680{{c}} -->
 {{Optimal ET sequence|legend=0| 22, 46, 68, 114 }}
@@ Line 1,077: / Line 875: @@
 * WE: ~45/32 = 599.7699{{c}}, ~16/11 = 652.4035{{c}}
 * CWE: ~45/32 = 600.0000{{c}}, ~16/11 = 652.6374{{c}}
-<!-- * POTE: ~45/32 = 600.000{{c}}, ~16/11 = 652.654{{c}} -->
 {{Optimal ET sequence|legend=0| 22f, 46, 68, 114 }}
@@ Line 1,093: / Line 890: @@
 * WE: ~17/12 = 599.7995{{c}}, ~16/11 = 652.4287{{c}}
 * CWE: ~17/12 = 600.0000{{c}}, ~16/11 = 652.6334{{c}}
-<!-- * POTE: ~17/12 = 600.000{{c}}, ~16/11 = 652.647{{c}} -->
 {{Optimal ET sequence|legend=0| 22f, 46, 68, 114 }}
@@ Line 1,109: / Line 905: @@
 * WE: ~17/12 = 599.8060{{c}}, ~16/11 = 652.5190{{c}}
 * CWE: ~17/12 = 600.0000{{c}}, ~16/11 = 652.7164{{c}}
-<!-- * POTE: ~17/12 = 600.000{{c}}, ~16/11 = 652.730{{c}} -->
 {{Optimal ET sequence|legend=0| 22fh, 46, 68, 114, 182bef }}
@@ Line 1,125: / Line 920: @@
 * WE: ~17/12 = 599.7879{{c}}, ~16/11 = 652.4776{{c}}
 * CWE: ~17/12 = 600.0000{{c}}, ~16/11 = 652.6926{{c}}
-<!-- * POTE: ~17/12 = 600.000{{c}}, ~16/11 = 652.708{{c}} -->
 {{Optimal ET sequence|legend=0| 22fh, 46, 68, 114 }}
 Badness (Sintel): 1.03
+== Shru ==
+Shru tempers out 392/375 and slices the compound semitone into two generators of ~10/7. Its ploidacot is diploid alpha-dicot, the same as shrutar.
+[[Subgroup]]: 2.3.5.7
+[[Comma list]]: 392/375, 1323/1280
+{{Mapping|legend=1| 2 1 9 11 | 0 2 -4 -5 }}
+: mapping generators: ~45/32, ~10/7
+[[Optimal tuning]]s:
+* [[WE]]: ~45/32 = 600.2519{{c}}, ~10/7 = 650.4083{{c}}
+: [[error map]]: {{val| +0.504 -0.887 +14.321 -18.096 }}
+* [[CWE]]: ~45/32 = 600.0000{{c}}, ~10/7 = 650.1017{{c}}
+: error map: {{val| 0.000 -1.752 +13.279 -19.334 }}
+{{Optimal ET sequence|legend=1| 2, 22d, 24 }}
+[[Badness]] (Sintel): 3.99
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
+Comma list: 56/55, 77/75, 1323/1280
+Mapping: {{mapping| 2 1 9 11 8 | 0 2 -4 -5 -1 }}
+Optimal tunings:
+* WE: ~17/12 = 600.2356{{c}}, ~10/7 = 650.3856{{c}}
+* CWE: ~17/12 = 600.0000{{c}}, ~10/7 = 650.1008{{c}}
+{{Optimal ET sequence|legend=0| 2, 22d, 24 }}
+Badness (Sintel): 2.10
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
+Comma list: 56/55, 77/75, 105/104, 507/500
+Mapping: {{mapping| 2 1 9 11 8 15 | 0 2 -4 -5 -1 -7 }}
+Optimal tunings:
+* WE: ~45/32 = 599.9067{{c}}, ~10/7 = 649.4907{{c}}
+* CWE: ~45/32 = 600.0000{{c}}, ~10/7 = 649.5950{{c}}
+{{Optimal ET sequence|legend=0| 2, 24 }}
+Badness (Sintel): 2.12
 == Sruti ==
+Sruti tempers out 19683/19600, setting itself up as a [[hemipyth]] temperament. It has the same semi-octave period as diaschismic, but the generator can be taken as a neutral third or a hemitwelfth. The temperament can be described as {{nowrap| 24 & 34d }}; its ploidacot is diploid dicot. [[58edo]] may be recommended as a tuning.
 [[Subgroup]]: 2.3.5.7
@@ Line 1,145: / Line 992: @@
 * [[CWE]]: ~45/32 = 600.0000{{c}}, ~140/81 = 951.8227{{c}}
 : error map: {{val| 0.000 +1.690 +6.395 +4.869 }}
-<!-- * [[POTE]]: ~45/32 = 600.000{{c}}, ~140/81 = 951.876{{c}} -->
 {{Optimal ET sequence|legend=1| 24, 34d, 58, 150cd, 208ccdd, 266ccdd }}
@@ Line 1,161: / Line 1,007: @@
 * WE: ~45/32 = 599.1951{{c}}, ~121/70 = 950.5864{{c}}
 * CWE: ~45/32 = 600.0000{{c}}, ~121/70 = 951.7972{{c}}
-<!-- * POTE: ~45/32 = 600.000{{c}}, ~121/70 = 951.863{{c}} -->
 {{Optimal ET sequence|legend=0| 24, 34d, 58, 150cdee, 208ccddee, 266ccddeee }}
@@ Line 1,177: / Line 1,022: @@
 * WE: ~45/32 = 599.1479{{c}}, ~26/15 = 950.5337{{c}}
 * CWE: ~45/32 = 600.0000{{c}}, ~26/15 = 951.8314{{c}}
-<!-- * POTE: ~45/32 = 600.000{{c}}, ~26/15 = 951.886{{c}} -->
 {{Optimal ET sequence|legend=0| 24, 34d, 58, 150cdeef, 208ccddeeff, 266ccddeeefff }}
@@ Line 1,193: / Line 1,037: @@
 * WE: ~17/12 = 599.3003{{c}}, ~26/15 = 950.7465{{c}}
 * CWE: ~17/12 = 600.0000{{c}}, ~26/15 = 951.8142{{c}}
-<!-- * POTE: ~17/12 = 600.000{{c}}, ~26/15 = 951.857{{c}} -->
 {{Optimal ET sequence|legend=0| 24, 34d, 58 }}
@@ Line 1,200: / Line 1,043: @@
 == Anguirus ==
+As another hemipyth temperament, anguirus tempers out 49/48. It can be described as the {{nowrap| 10 & 24 }} temperament; its ploidacot is diploid dicot, the same as sruti.
 [[Subgroup]]: 2.3.5.7
@@ Line 1,213: / Line 1,058: @@
 * [[CWE]]: ~45/32 = 600.0000{{c}}, ~7/4 = 953.0188{{c}}
 : error map: {{val| 0.000 +4.083 +1.611 -15.807 }}
-<!-- * [[POTE]]: ~45/32 = 600.000{{c}}, ~7/4 = 953.021{{c}} -->
 {{Optimal ET sequence|legend=1| 10, 24, 34 }}
@@ Line 1,229: / Line 1,073: @@
 * WE: ~45/32 = 599.9250{{c}}, ~7/4 = 952.0646{{c}}
 * CWE: ~45/32 = 600.0000{{c}}, ~7/4 = 952.1784{{c}}
-<!-- * POTE: ~45/32 = 600.000{{c}}, ~7/4 = 952.184{{c}} -->
 {{Optimal ET sequence|legend=0| 10, 24, 34 }}
@@ Line 1,245: / Line 1,088: @@
 * WE: ~45/32 = 599.7575{{c}}, ~7/4 = 951.9241{{c}}
 * CWE: ~45/32 = 600.0000{{c}}, ~7/4 = 952.2980{{c}}
-<!-- * POTE: ~45/32 = 600.000{{c}}, ~7/4 = 952.309{{c}} -->
 {{Optimal ET sequence|legend=0| 10, 24, 34, 58d, 92ddef }}
@@ Line 1,261: / Line 1,103: @@
 * WE: ~17/12 = 599.7925{{c}}, ~7/4 = 952.0004{{c}}
 * CWE: ~17/12 = 600.0000{{c}}, ~7/4 = 952.3178{{c}}
-<!-- * POTE: ~17/12 = 600.000{{c}}, ~7/4 = 952.330{{c}} -->
 {{Optimal ET sequence|legend=0| 10, 24, 34 }}
@@ Line 1,267: / Line 1,108: @@
 Badness (Sintel): 1.10
-== Shru ==
+== Echidna ==
+Echidna adds 1728/1715 to the commas and takes 9/7 as a generator. It may be called the {{nowrap| 22 & 58 }} temperament; its ploidacot is diploid alpha-tricot. [[58edo]] or [[80edo]] make for good tunings, or their vals can be added to {{val| 138 219 321 388 }} (138cde). In most of the tunings it has a significantly sharp 7/4 which some prefer.
+Echidna becomes more interesting when extended to be an 11-limit temperament by adding 176/175, 540/539 or 896/891 to the commas, where the same tunings can be used as before. It then is able to represent the entire 11-odd-limit diamond to within about six cents of error, within a compass of 24 notes. The 22-note 2mos gives scope for this, and the 36-note mos much more. Better yet, it is related to three important 11-limit edos: 22edo, a trivial tuning, is the smallest consistent in the 11-odd-limit, corresponding to the merge of this temperament with [[hedgehog]]; [[58edo]] is the smallest tuning that is distinctly consistent in the 11-odd-limit and [[80edo]] is the third smallest distinctly consistent in the 11-odd-limit.
+The generator can be interpreted as 11/10, the period complement of 9/7, as a stack of 11/10 and 9/7 makes [[99/70]] which is extremely close to 600{{cent}} and is equal to it if we temper out [[9801/9800|S99]]. Three 11/10's then make a 4/3 (tempering out [[4000/3993|S10/S11]] thus making 10/9 and 12/11 equidistant from 11/10), implying a flat tuning of 4/3.
+Like most srutal extensions, the 13- and 17-limit interpretations are possible by observing that since we have tempered out [[176/175]], tempering out [[351/350]] and [[352/351]] which sum to 176/175 is very elegant. In the 17-limit we can equate the half-octave with 17/12 and 24/17 and we can take advantage of the sharp fifth by combining echidna with [[srutal archagall]], leading to a particularly beautiful temperament (one that prefers a very slightly less sharp fifth than srutal archagall). This mapping of 13 and 17 is supported by the patent vals of the three main echidna edos of 22, 58 and 80, of which all except 22 are consistent in the [[17-odd-limit]].
+[[Subgroup]]: 2.3.5.7
+[[Comma list]]: 1728/1715, 2048/2025
+{{Mapping|legend=1| 2 1 9 2 | 0 3 -6 5 }}
+: mapping generators: ~45/32, ~9/7
+[[Optimal tuning]]s:
+* [[WE]]: ~45/32 = 599.3056{{c}}, ~9/7 = 434.3524{{c}}
+: [[error map]]: {{val| -1.389 +0.408 +1.322 +1.547 }}
+* [[CWE]]: ~45/32 = 600.0000{{c}}, ~9/7 = 434.8327{{c}}
+: error map: {{val| 0.000 +2.543 +4.690 +5.338 }}
+{{Optimal ET sequence|legend=1| 22, 58, 80, 138cd, 218cd }}
+[[Badness]] (Sintel): 1.47
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
+Comma list: 176/175, 540/539, 896/891
+Mapping: {{mapping| 2 1 9 2 12 | 0 3 -6 5 -7 }}
+Optimal tunings:
+* WE: ~45/32 = 599.3085{{c}}, ~9/7 = 434.3511{{c}}
+* CWE: ~45/32 = 600.0000{{c}}, ~9/7 = 434.8647{{c}}
+Minimax tuning:
+* 11-odd-limit: ~9/7 = {{monzo| 5/12 0 0 1/12 -1/12 }}
+: [{{monzo| 1 0 0 0 0 }}, {{monzo| 7/4 0 0 1/4 -1/4 }}, {{monzo| 2 0 0 -1/2 1/2 }}, {{monzo| 37/12 0 0 5/12 -5/12 }}, {{monzo| 37/12 0 0 -7/12 7/12 }}]
+: unchanged-interval (eigenmonzo) basis: 2.11/7
+{{Optimal ET sequence|legend=0| 22, 58, 80, 138cde, 218cde }}
+Badness (Sintel): 0.859
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
+Comma list: 176/175, 351/350, 364/363, 540/539
+Mapping: {{mapping| 2 1 9 2 12 19 | 0 3 -6 5 -7 -16 }}
+Optimal tunings:
+* WE: ~45/32 = 599.3397{{c}}, ~9/7 = 434.2772{{c}}
+* CWE: ~45/32 = 600.0000{{c}}, ~9/7 = 434.7864{{c}}
+{{Optimal ET sequence|legend=0| 22, 36f, 58, 80, 138cde }}
+Badness (Sintel): 0.978
+=== 17-limit ===
+Subgroup: 2.3.5.7.11.13.17
+Comma list: 136/135, 176/175, 221/220, 256/255, 540/539
+Mapping: {{mapping| 2 1 9 2 12 19 6 | 0 3 -6 5 -7 -16 3 }}
+Optimal tunings:
+* WE: ~45/32 = 599.4645{{c}}, ~9/7 = 434.4282{{c}}
+* CWE: ~45/32 = 600.0000{{c}}, ~9/7 = 434.8340{{c}}
+{{Optimal ET sequence|legend=0| 22, 36f, 58, 80, 138cde }}
+Badness (Sintel): 1.03
+== Echidnic ==
+Echidnic tempers out 686/675 and [[1029/1024]]. It has the same semi-octave period as diaschismic, but slices the generator of a fifth into three ~8/7's. It can be described as the {{nowrap| 10 & 46 }} temperament; its ploidacot is diploid tricot.
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: 392/375, 1323/1280
+[[Comma list]]: 686/675, 1029/1024
-{{Mapping|legend=1| 2 1 9 11 | 0 2 -4 -5 }}
+{{Mapping|legend=1| 2 2 7 6 | 0 3 -6 -1 }}
-: mapping generators: ~45/32, ~10/7
+: mapping generators: ~45/32, ~8/7
 [[Optimal tuning]]s:
-* [[WE]]: ~45/32 = 600.2519{{c}}, ~10/7 = 650.4083{{c}}
+* [[WE]]: ~45/32 = 599.7208{{c}}, ~8/7 = 234.8330{{c}}
-: [[error map]]: {{val| +0.504 -0.887 +14.321 -18.096 }}
+: [[error map]]: {{val| -0.558 +1.986 +2.733 -5.334 }}
-* [[CWE]]: ~45/32 = 600.0000{{c}}, ~10/7 = 650.1017{{c}}
+* [[CWE]]: ~45/32 = 600.0000{{c}}, ~8/7 = 234.9539{{c}}
-: error map: {{val| 0.000 -1.752 +13.279 -19.334 }}
+: error map: {{val| 0.000 +2.907 +3.963 -3.780 }}
-<!-- * [[POTE]]: ~45/32 = 600.000{{c}}, ~10/7 = 650.135{{c}} -->
-{{Optimal ET sequence|legend=1| 2, 22d, 24 }}
+{{Optimal ET sequence|legend=1| 10, 26c, 36, 46 }}
-[[Badness]] (Sintel): 3.99
+[[Badness]] (Sintel): 1.83
 === 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 56/55, 77/75, 1323/1280
+Comma list: 385/384, 441/440, 686/675
-Mapping: {{mapping| 2 1 9 11 8 | 0 2 -4 -5 -1 }}
+Mapping: {{mapping| 2 2 7 6 3 | 0 3 -6 -1 10 }}
 Optimal tunings:
-* WE: ~17/12 = 600.2356{{c}}, ~10/7 = 650.3856{{c}}
+* WE: ~45/32 = 599.8022{{c}}, ~8/7 = 235.0185{{c}}
-* CWE: ~17/12 = 600.0000{{c}}, ~10/7 = 650.1008{{c}}
+* CWE: ~45/32 = 600.0000{{c}}, ~8/7 = 235.0893{{c}}
-<!-- * POTE: ~45/32 = 600.000{{c}}, ~10/7 = 650.130{{c}} -->
-{{Optimal ET sequence|legend=0| 2, 22d, 24 }}
+{{Optimal ET sequence|legend=0| 10, 36e, 46, 102, 148 }}
-Badness (Sintel): 2.10
+Badness (Sintel): 1.49
 === 13-limit ===
 Subgroup: 2.3.5.7.11.13
-Comma list: 56/55, 77/75, 105/104, 507/500
+Comma list: 91/90, 169/168, 385/384, 441/440
+Mapping: {{mapping| 2 2 7 6 3 7 | 0 3 -6 -1 10 1 }}
+Optimal tunings:
+* WE: ~45/32 = 599.9570{{c}}, ~8/7 = 235.0708{{c}}
+* CWE: ~45/32 = 600.0000{{c}}, ~8/7 = 235.0862{{c}}
+{{Optimal ET sequence|legend=0| 10, 36e, 46, 102, 148f }}
+Badness (Sintel): 1.19
+=== 17-limit ===
+Subgroup: 2.3.5.7.11.13.17
+Comma list: 91/90, 136/135, 154/153, 169/168, 256/255
-Mapping: {{mapping| 2 1 9 11 8 15 | 0 2 -4 -5 -1 -7 }}
+Mapping: {{mapping| 2 2 7 6 3 7 7 | 0 3 -6 -1 10 1 3 }}
 Optimal tunings:
-* WE: ~45/32 = 599.9067{{c}}, ~10/7 = 649.4907{{c}}
+* WE: ~17/12 = 599.9571{{c}}, ~8/7 = 235.0709{{c}}
-* CWE: ~45/32 = 600.0000{{c}}, ~10/7 = 649.5950{{c}}
+* CWE: ~17/12 = 600.0000{{c}}, ~8/7 = 235.0860{{c}}
-<!-- * POTE: ~45/32 = 600.000{{c}}, ~10/7 = 650.535{{c}} -->
+{{Optimal ET sequence|legend=0| 10, 36e, 46, 102, 148f }}
-{{Optimal ET sequence|legend=0| 2, 24 }}
+Badness (Sintel): 0.983
-Badness (Sintel): 2.12
+; Music
+* [https://untwelve.org/competition/2011 ''A Stiff Shot of Turpentine''] [https://untwelve.org/static/audio/competition/2011/Kosmorsky-A_Stiff_Shot_of_Turpentine.mp3 play] by [[Peter Kosmorsky]]
+* [https://www.youtube.com/watch?v=VsBXIvBZY6A ''56edo Track (Echidnic16 Scale)''] by [[Budjarn Lambeth]] (2025)
 == Quadrasruta ==
+Named by [[Xenllium]] in 2022, quadrasruta tempers out 2401/2400, the breedsma, and extends [[buzzard]]. It may be described as {{nowrap| 58 & 68 }}; its ploidacot is diploid alpha-tetracot. 126edo may be recommended as a tuning.
 [[Subgroup]]: 2.3.5.7
@@ Line 1,333: / Line 1,271: @@
 * [[CWE]]: ~45/32 = 600.0000{{c}}, ~21/16 = 476.2394{{c}}
 : error map: {{val| 0.000 +3.003 +3.771 +2.456 }}
-<!-- * [[POTE]]: ~45/32 = 600.000{{c}}, ~21/16 = 476.216{{c}} -->
 {{Optimal ET sequence|legend=1| 10, …, 58, 68, 126, 446bbccd }}
@@ Line 1,349: / Line 1,286: @@
 * WE: ~45/32 = 599.4648{{c}}, ~21/16 = 475.6929{{c}}
 * CWE: ~45/32 = 600.0000{{c}}, ~21/16 = 476.1507{{c}}
-<!-- * POTE: ~45/32 = 600.000{{c}}, ~21/16 = 476.118{{c}} -->
 {{Optimal ET sequence|legend=0| 10e, …, 58, 126, 184c, 310bccde }}
@@ Line 1,365: / Line 1,301: @@
 * WE: ~45/32 = 599.3787{{c}}, ~21/16 = 475.6065{{c}}
 * CWE: ~45/32 = 600.0000{{c}}, ~21/16 = 476.1345{{c}}
-<!-- * POTE: ~45/32 = 600.000{{c}}, ~21/16 = 476.099{{c}} -->
 {{Optimal ET sequence|legend=0| 10e, …, 58, 126f, 184cff }}
@@ Line 1,381: / Line 1,316: @@
 * WE: ~17/12 = 599.5077{{c}}, ~21/16 = 475.7713{{c}}
 * CWE: ~17/12 = 600.0000{{c}}, ~21/16 = 476.1814{{c}}
-<!-- * POTE: ~17/12 = 600.000{{c}}, ~21/16 = 476.162{{c}} -->
 {{Optimal ET sequence|legend=0| 10e, 58, 126f }}
@@ Line 1,397: / Line 1,331: @@
 * WE: ~45/32 = 599.2593{{c}}, ~21/16 = 475.4292{{c}}
 * CWE: ~45/32 = 600.0000{{c}}, ~21/16 = 476.0088{{c}}
-<!-- * POTE: ~45/32 = 600.000{{c}}, ~21/16 = 476.017{{c}} -->
 {{Optimal ET sequence|legend=0| 10, 48c, 58, 184cee, 242ccdeee }}
@@ Line 1,413: / Line 1,346: @@
 * WE: ~45/32 = 599.2147{{c}}, ~21/16 = 475.4052{{c}}
 * CWE: ~45/32 = 600.0000{{c}}, ~21/16 = 476.0253{{c}}
-<!-- * POTE: ~45/32 = 600.000{{c}}, ~21/16 = 476.028{{c}} -->
 {{Optimal ET sequence|legend=0| 10, 48c, 58, 126eef, 184ceeff, 242ccdeeeff }}
@@ Line 1,429: / Line 1,361: @@
 * WE: ~17/12 = 599.3353{{c}}, ~21/16 = 475.5495{{c}}
 * CWE: ~17/12 = 600.0000{{c}}, ~21/16 = 476.0691{{c}}
-<!-- * POTE: ~17/12 = 600.000{{c}}, ~21/16 = 476.077{{c}} -->
 {{Optimal ET sequence|legend=0| 10, 48c, 58 }}