Syntonic–kleismic equivalence continuum

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The syntonic–kleismic equivalence continuum (or syntonic–enneadecal equivalence continuum) is a continuum of 5-limit temperaments which equate a number of syntonic commas (81/80) with the 19-comma ([-30 19⟩).

All temperaments in the continuum satisfy (81/80)ⁿ ~ [-30 19⟩. Varying n results in different temperaments listed in the table below. It converges to meantone as n approaches infinity. If we allow non-integer and infinite n, the continuum describes the set of all 5-limit temperaments supported by 19edo (due to it being the unique equal temperament that tempers out both commas and thus tempers out all combinations of them). The just value of n is approximately 6.376…, and temperaments having n near this value tend to be the most accurate ones.

This continuum can also be expressed as the relationship between 81/80 and the enneadeca ([-14 -19 19⟩). That is, (81/80)^k ~ [-14 -19 19⟩. In this case, k = 3n − 19.

Temperaments with integer n

n	Temperament	Comma
		Ratio	Monzo
0	19 & 19c	1162261467/1073741824	[-30 19⟩
1	7c & 12c	71744535/67108864	[-26 15 1⟩
2	Hogzilla	4428675/4194304	[-22 11 2⟩
3	Stump	273375/262144	[-18 7 3⟩
4	Negri	16875/16384	[-14 3 4⟩
5	Magic	3125/3072	[-10 -1 5⟩
6	Hanson	15625/15552	[-6 -5 6⟩
7	Sensipent	78732/78125	[2 9 -7⟩
8	Unicorn	1594323/1562500	[-2 13 -8⟩
9	19 & 51c	129140163/125000000	[-6 17 -9⟩
…	…	…	…
∞	Meantone	81/80	[-4 4 -1⟩

Examples of temperaments with fractional values of k:

Notable temperaments of fractional n

Temperament	n	Comma
Unsmate	9/2 = 4.5	[-24 2 9⟩
Sycamore	11/2 = 5.5	[-16 -6 11⟩
Counterhanson	25/4 = 6.25	[-20 -24 25⟩
Enneadecal	19/3 = 6.3	[-14 -19 19⟩
Egads	51/8 = 6.375	[-36 -52 51⟩
Acrokleismic	32/5 = 6.4	[22 33 -32⟩
Parakleismic	13/2 = 6.5	[8 14 -13⟩
Countermeantone	20/3 = 6.6	[10 23 -20⟩
Mowgli	15/2 = 7.5	[0 22 -15⟩

Negri

For extensions, see Semaphoresmic clan #Negri.

The 5-limit version of negri tempers out the negri comma, spliting a perfect fourth into four ~16/15 generators. It corresponds to n = 4. The only 7-limit extension that make any sense to use is to map the hemifourth to 7/6~8/7.

Subgroup: 2.3.5

Comma list: 16875/16384

Mapping: [⟨1 2 2], ⟨0 -4 3]]

mapping generators: ~2, ~16/15

Optimal tunings:

• WE: ~2 = 1202.3403 ¢, ~16/15 = 126.0002 ¢

error map: ⟨+2.340 -1.275 -3.633]

• CWE: ~2 = 1200.0000 ¢, ~16/15 = 125.6610 ¢

error map: ⟨0.000 -4.599 -9.331]

Optimal ET sequence: 9, 10, 19, 67c, 86c, 105c

Badness (Sintel): 2.04

Lalasepyo (8c & 11)

Subgroup: 2.3.5

Comma list: 4613203125/4294967296

Mapping: [⟨1 -1 6], ⟨0 7 -10]]

mapping generators: ~2, ~675/512

Optimal tunings:

• WE: ~2 = 1202.5641 ¢, ~675/512 = 443.2124 ¢

error map: ⟨+2.564 -2.033 -3.053]

• CWE: ~2 = 1200.0000 ¢, ~675/512 = 442.2692 ¢

error map: ⟨0.000 -6.071 -9.006]

Optimal ET sequence: 8c, 11, 19

Badness (Sintel): 24.9

The temperament finder - 5-limit 19 & 8c

Parakleismic (5-limit)

Main article: Parakleismic

For extensions, see Ragismic microtemperaments #Parakleismic.

The 5-limit version of parakleismic tempers out the parakleisma. It corresponds to n = 13/2, and 13 generator steps give the interval class of 3.

Subgroup: 2.3.5

Comma list: 1224440064/1220703125

Mapping: [⟨1 -8 -8], ⟨0 13 14]]

mapping generators: ~2, ~5/3

Optimal tunings:

• WE: ~2 = 1199.971 ¢, ~5/3 = 884.7383 ¢

error map: ⟨-0.029 -0.127 +0.253]

• CWE: ~2 = 1200.000 ¢, ~5/3 = 884.8576 ¢

error map: ⟨0.000 -0.106 +0.293]

Optimal ET sequence: 19, 61, 80, 99, 118, 453, 571, 689, 1496

Badness (Sintel): 1.02

Mowgli

For extensions, see Hemimean clan #Mowglic.

TE, CTE and POTE coincide at 126.7237 ¢ with pure octaves since prime 2 is not involved in the comma to begin with.

Subgroup: 2.3.5

Comma list: 31381059609/30517578125

Mapping: [⟨1 0 0], ⟨0 15 22]]

mapping generators: ~2, ~27/25

Optimal tunings:

• WE: ~2 = 1199.9478 ¢, ~27/25 = 126.7236 ¢

error map: ⟨-0.001 -1.100 +1.606]

• CWE: ~2 = 1200.0000 ¢, ~27/25 = 126.7237 ¢

error map: ⟨0.000 -1.100 +1.607]

Optimal ET sequence: 19, 85c, 104c, 123, 142, 161, 303

Badness (Sintel): 15.3

Countermeantone

Subgroup: 2.3.5

Comma list: [10 23 -20⟩

Mapping: [⟨1 -10 -11], ⟨0 20 23]]

mapping generators: ~2, ~78125/52488

Optimal tunings:

• WE: ~2 = 1199.9478 ¢, ~78125/52488 = 695.0566 ¢

error map: ⟨-0.052 -0.301 +0.562]

• CWE: ~2 = 1200.0000 ¢, ~78125/52488 = 695.0846 ¢

error map: ⟨0.000 -0.264 +0.631]

Optimal ET sequence: 19, …, 126, 145, 164, 183, 713, 896c, 1079c, 1262c, 1445c

Badness (Sintel): 8.76

Counterhanson

For extensions, see Ragismic microtemperaments #Counterkleismic.

Subgroup: 2.3.5

Comma list: [-20 -24 25⟩

Mapping: [⟨1 -5 -4], ⟨0 25 24]]

mapping generators: ~2, ~6/5

Optimal tunings:

• WE: ~2 = 1200.0419 ¢, ~6/5 = 316.0916 ¢

error map: ⟨+0.042 +0.126 -0.282]

• CWE: ~2 = 1200.0000 ¢, ~6/5 = 316.0021 ¢

error map: ⟨0.000 +0.097 -0.344]

Optimal ET sequence: 19, …, 148, 167, 186, 205, 224, 429, 653, 1082, 1735c

Badness (Sintel): 7.45

Oviminor

See also: Ragismic microtemperaments § Oviminor

Oviminor is named after the facts that it takes 184 minor thirds of 6/5 to reach 4/3, the Roman consul was Eggius in the year 184 AD, and the Latin word for egg is ovum, and with prefix ovi-. It sets a new record of complexity for a chain of nineteen 6/5's past egads, though it is less accurate.

Subgroup: 2.3.5

Comma list: [-134 -185 184⟩

Mapping: [⟨1 -134 -134], ⟨0 184 185]]

mapping generators: ~2, ~5/3

Optimal tunings:

• WE: ~2 = 1200.0094 ¢, ~5/3 = 884.2568 ¢

error map: ⟨+0.009 +0.033 -0.069]

• CWE: ~2 = 1200.0000 ¢, ~5/3 = 884.2499 ¢

error map: ⟨0.000 +0.026 -0.083]

Optimal ET sequence: 19, …, 1600, 3219, 4819

Badness (Sintel): 751