FloraC: Move

2026-08-09T13:48:34Z

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 Badness (Sintel): 1.67
 [[Category:Parakleismic family| ]] <!-- main article -->
 [[Category:Temperament families]]
 [[Category:Rank 2]]

FloraC: Birth

2026-08-09T13:27:20Z

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{{Technical data page}}

The '''parakleismic family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[parakleisma]] ({{monzo|legend=1| 8 14 -13 }}, [[ratio]]: 1 224 440 064 / 1 220 703 125).

== Parakleismic ==
{{Main| Parakleismic }}

Parakleismic has a generator a very slightly flat [[6/5|classical minor third]]. Note that in the data that follow, the generator is its [[octave complement]], [[~]][[5/3]], thirteen of which give the interval class of [[3/1|3]], and fourteen give the interval class of [[5/1|5]]. It is a member of the [[syntonic–kleismic equivalence continuum]], with equivalence number {{nowrap| ''n'' {{=}} 13/2 }}. It is an undoubted [[microtemperament]] in the 5-limit, with the [[118edo]] tuning giving errors well under a cent.

[[Subgroup]]: 2.3.5

[[Comma list]]: 1224440064/1220703125

{{Mapping|legend=1| 1 -8 -8 | 0 13 14 }}
: mapping generators: ~2, ~5/3

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.971{{c}}, ~5/3 = 884.7383{{c}}
: [[error map]]: {{val| -0.029 -0.127 +0.253 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~5/3 = 884.8576{{c}}
: error map: {{val| 0.000 -0.106 +0.293 }}

{{Optimal ET sequence|legend=1| 19, 61, 80, 99, 118, 453, 571, 689, 1496 }}

[[Badness]] (Sintel): 1.02

== Septimal parakleismic ==
{{Main| Parakleismic }}

While [[118edo]] no longer has better than a cent of accuracy in the 7-limit, it is a decent temperament there nonetheless, and this allows an extension adding [[3136/3125]] and [[4375/4374]], for which [[99edo]], 118edo, and especially [[217edo]] are accurate tunings.

Parakleismic does not extend easily to the 11- or 13-limit. Possible 11-limit extensions include undecimal parakleismic (99 & 118), paralytic (99e & 118), parkleismic (80 & 99), and paradigmic (80 & 99e).

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 3136/3125, 4375/4374

{{Mapping|legend=1| 1 -8 -8 -23 | 0 13 14 35 }}
: mapping generators: ~2, ~5/3

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.7820{{c}}, ~5/3 = 884.6581{{c}}
: [[error map]]: {{val| -0.218 +0.344 +0.644 -0.779 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/3 = 884.8088{{c}}
: error map: {{val| 0.000 +0.560 +1.010 -0.516 }}

{{Optimal ET sequence|legend=1| 19, 61d, 80, 99, 217, 316, 415 }}

[[Badness]] (Sintel): 0.694

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 385/384, 3136/3125, 4375/4374

Mapping: {{mapping| 1 -8 -8 -23 30 | 0 13 14 35 -36 }}

Optimal tunings:
* WE: ~2 = 1200.3296{{c}}, ~5/3 = 884.9921{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.7519{{c}}

{{Optimal ET sequence|legend=0| 19, 99, 118 }}

Badness (Sintel): 1.64

=== Paralytic ===
Paralytic (99e & 118) tempers out [[441/440]], [[5632/5625]], and [[19712/19683]]. In 13-limit, 118 & 217 tempers out 1001/1000, 1575/1573, and 3584/3575.

Subgroup: 2.3.5.7.11

Comma list: 441/440, 3136/3125, 4375/4374

Mapping: {{mapping| 1 -8 -8 -23 -57 | 0 13 14 35 82 }}

Optimal tunings:
* WE: ~2 = 1199.9940{{c}}, ~5/3 = 884.7757{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.7800{{c}}

{{Optimal ET sequence|legend=0| 19e, …, 99e, 118, 217, 335 }}

Badness (Sintel): 1.19

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 441/440, 1001/1000, 3136/3125, 4375/4374

Mapping: {{mapping| 1 -8 -8 -23 -57 59 | 0 13 14 35 82 -75 }}

Optimal tunings:
* WE: ~2 = 1199.9218{{c}}, ~5/3 = 884.7285{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.7858{{c}}

{{Optimal ET sequence|legend=0| 99e, 118, 217 }}

Badness (Sintel): 1.85

==== Paraklein ====
Paraklein (19e & 118) is another 13-limit extension of paralytic, which equates [[13/11]] with [[32/27]], [[14/13]] with [[15/14]], [[25/24]] with [[26/25]], and [[27/26]] with [[28/27]].

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 352/351, 625/624, 729/728

Mapping: {{mapping| 1 -8 -8 -23 -57 -28 | 0 13 14 35 82 43 }}

Optimal tunings:
* WE: ~2 = 1199.8239{{c}}, ~5/3 = 884.6449{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.7709{{c}}

{{Optimal ET sequence|legend=0| 19e, …, 99ef, 118 }}

Badness (Sintel): 1.55

=== Parkleismic ===
Subgroup: 2.3.5.7.11

Comma list: 176/175, 1375/1372, 2200/2187

Mapping: {{mapping| 1 -8 -8 -23 -43 | 0 13 14 35 63 }}

Optimal tunings:
* WE: ~2 = 1199.1848{{c}}, ~5/3 = 884.3386{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.9158{{c}}

{{Optimal ET sequence|legend=0| 19e, 61de, 80, 179, 259cd }}

Badness (Sintel): 1.85

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 176/175, 325/324, 1375/1372

Mapping: {{mapping| 1 -8 -8 -23 -43 -14 | 0 13 14 35 63 24 }}

Optimal tunings:
* WE: ~2 = 1199.5318{{c}}, ~5/3 = 884.5800{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.9118{{c}}

{{Optimal ET sequence|legend=0| 19e, 61de, 80, 179 }}

Badness (Sintel): 1.51

=== Paradigmic ===
Subgroup: 2.3.5.7.11

Comma list: 540/539, 896/891, 3136/3125

Mapping: {{mapping| 1 -8 -8 -23 16 | 0 13 14 35 -17 }}

Optimal tunings:
* WE: ~2 = 1199.0616{{c}}, ~5/3 = 884.2124{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.8877{{c}}

{{Optimal ET sequence|legend=0| 19, 61d, 80, 99e, 179e, 457bcddeeee }}

Badness (Sintel): 1.38

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 325/324, 540/539, 832/825

Mapping: {{mapping| 1 -8 -8 -23 16 -14 | 0 13 14 35 -17 24 }}

Optimal tunings:
* WE: ~2 = 1199.2683{{c}}, ~5/3 = 884.3805{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.9061{{c}}

{{Optimal ET sequence|legend=0| 19, 61d, 80, 99e }}

Badness (Sintel): 1.48

=== Semiparakleismic ===
Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 3136/3125, 4375/4374

Mapping: {{mapping| 2 -3 -2 -11 -4 | 0 13 14 35 23 }}
: mapping generators: ~99/70, ~33/28

Optimal tunings:
* WE: ~99/70 = 599.9270{{c}}, ~33/28 = 284.7841{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~33/28 = 284.8119{{c}}

{{Optimal ET sequence|legend=0| 80, 118, 198, 316, 514c }}

Badness (Sintel): 1.13

==== Semiparamint ====
This extension was named ''semiparakleismic'' in the earlier materials.

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 1001/1000, 3025/3024, 4375/4374

Mapping: {{mapping| 2 -3 -2 -11 -4 15 | 0 13 14 35 23 -16 }}

Optimal tunings:
* WE: ~99/70 = 599.8253{{c}}, ~33/28 = 284.7608{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~33/28 = 284.8366{{c}}

{{Optimal ET sequence|legend=0| 80, 118, 198 }}

Badness (Sintel): 1.40

==== Semiparawolf ====
This extension was named ''gentsemiparakleismic'' in the earlier materials.

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 325/324, 364/363, 3136/3125

Mapping: {{mapping| 2 -3 -2 -11 -4 -4 | 0 13 14 35 23 24 }}

Optimal tunings:
* WE: ~99/70 = 600.0569{{c}}, ~13/11 = 284.8431{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~13/11 = 284.8216{{c}}

{{Optimal ET sequence|legend=0| 80, 118f, 198f }}

Badness (Sintel): 1.67

[[Category:Parakleismic family| ]] 
[[Category:Temperament families]]
[[Category:Rank 2]]

Parakleismic family - Revision history

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