Undim family

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 undim family of temperaments tempers out the undim comma, [41 -20 -4⟩, equating the Pythagorean comma with a stack of four schismas. This makes it a member of the schismic–Pythagorean equivalence continuum, with n = 4.

The name undim was given by Petr Pařízek in 2011 for it is some sort of opposite to diminished.

The second comma of the normal comma list defines which 7-limit family member we are looking at. Septimal undim (140 & 152) tempers out 5120/5103 (hemifamity). Unlit (152 & 316) does 4375/4374 (ragisma) instead. Twilight (152 & 176) adds 6144/6125 (porwell) to the comma list and splits the period into two – 1/8 of an octave.

Undim

Undim features a quarter-octave period, which acts as the ptolemaic augmented second (1215/1024). That and five perfect fourths (i.e. a minor second, ~256/243) give the interval class of 5.

Note that all versions of undim (ones that do not already map 19 differently and more accurately) have an obvious extension to prime 19 by observing that sharpening 1215/1024 by 1216/1215 results in 19/16, thus mapping 19/16 to 1\4. This interpretation is arguably much more harmonically plausible, owing to its simplicity and thereby greater tolerance to mistuning.

Subgroup: 2.3.5

Comma list: [41 -20 -4⟩

Mapping: [⟨4 0 41], ⟨0 1 -5]]

mapping generators: ~1215/1024, ~3

Optimal tunings:

• CTE: ~1215/1024 = 300.0000, ~3/2 = 702.6754

error map: ⟨0.0000 +0.7204 +0.3092]

• POTE: ~1215/1024 = 300.0000, ~3/2 = 702.6054

error map: ⟨0.0000 +0.6504 +0.6591]

Optimal ET sequence: 12, …, 104, 116, 128, 140, 152, 620, 772, 924c, 1076bc, 1228bc

Badness (Smith): 0.241703

Septimal undim

Septimal undim tempers out the dimcomp comma, mapping ~25/21 to the 1/4-octave period. It can be described as 12 & 140, and is the unique temperament that equates a syntonic~septimal comma with a stack of three marvel commas. A Pythagorean comma is then found as a stack of four marvel commas. In some 7-limit adaptive-tuning practice, the marvel comma corresponds to a melodic unit called a kleisma, with three kleismas making a comma, so this temperament may be useful for modeling that. 292edo makes for an excellent tuning.

Subgroup: 2.3.5.7

Comma list: 5120/5103, 390625/388962

Mapping: [⟨4 0 41 81], ⟨0 1 -5 -11]]

Optimal tunings:

• CTE: ~25/21 = 300.0000, ~3/2 = 702.7948

error map: ⟨0.0000 +0.8398 -0.2879 +0.4308]

• POTE: ~25/21 = 300.0000, ~3/2 = 702.7362

error map: ⟨0.0000 +0.7812 +0.0051 +1.0754]

Optimal ET sequence: 140, 152, 292

Badness (Smith): 0.062754

11-limit

Subgroup: 2.3.5.7.11

Comma list: 1375/1372, 5120/5103, 5632/5625

Mapping: [⟨4 0 41 81 128], ⟨0 1 -5 -11 -18]]

Optimal tunings:

• CTE: ~25/21 = 300.0000, ~3/2 = 702.7433
• POTE: ~25/21 = 300.0000, ~3/2 = 702.6886

Optimal ET sequence: 140, 152, 292, 444d, 596d

Badness (Smith): 0.034837

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 625/624, 847/845, 1375/1372

Mapping: [⟨4 0 41 81 128 148], ⟨0 1 -5 -11 -18 -21]]

Optimal tunings:

• CTE: ~25/21 = 300.0000, ~3/2 = 702.7792
• POTE: ~25/21 = 300.0000, ~3/2 = 702.7363

Optimal ET sequence: 140, 152f, 292

Badness (Smith): 0.028172

Unlit

Subgroup: 2.3.5.7

Comma list: 4375/4374, 2199023255552/2179240250625

Mapping: [⟨4 0 41 -160], ⟨0 1 -5 27]]

Optimal tunings:

• CTE: ~1215/1024 = 300.0000, ~3/2 = 702.5556

error map: ⟨0.0000 +0.6006 +0.9081 +0.1761]

• POTE: ~1215/1024 = 300.0000, ~3/2 = 702.5764

error map: ⟨0.0000 +0.6214 +0.8043 +0.7369]

Optimal ET sequence: 152, 316, 468, 620, 1088bcd, 1708bccdd

Badness (Smith): 0.268206

11-limit

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4374, 5767168/5740875

Mapping: [⟨4 0 41 -160 -113], ⟨0 1 -5 27 20]]

Optimal tunings:

• CTE: ~1215/1024 = 300.0000, ~3/2 = 702.5582
• POTE: ~1215/1024 = 300.0000, ~3/2 = 702.5826

Optimal ET sequence: 152, 468, 620

Badness (Smith): 0.070215

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 2080/2079, 3025/3024, 1835008/1828125

Mapping: [⟨4 0 41 -160 -113 -334], ⟨0 1 -5 27 20 55]]

Optimal tunings:

• CTE: ~1215/1024 = 300.0000, ~3/2 = 702.5562
• POTE: ~1215/1024 = 300.0000, ~3/2 = 702.5741

Optimal ET sequence: 152f, 316, 468, 620f, 1088bcdf

Badness (Smith): 0.058390

Twilight

Subgroup: 2.3.5.7

Comma list: 6144/6125, 31470387200/31381059609

Mapping: [⟨8 0 82 -79], ⟨0 1 -5 8]]

mapping generators: ~7168/6561, ~3

Optimal tunings:

• CTE: ~7168/6561 = 150.0000, ~3/2 = 702.4765

error map: ⟨0.0000 +0.5215 +1.3036 +0.9865]

• POTE: ~7168/6561 = 150.0000, ~3/2 = 702.5090

error map: ⟨0.0000 +0.5540 +1.1415 +1.2457]

Optimal ET sequence: 152, 328, 480, 1592bccddd

Badness (Smith): 0.213094

11-limit

Subgroup: 2.3.5.7.11

Comma list: 6144/6125, 9801/9800, 19712/19683

Mapping: [⟨8 0 82 -79 15], ⟨0 1 -5 8 1]]

Optimal tunings:

• CTE: ~12/11 = 150.0000, ~3/2 = 702.4692
• POTE: ~12/11 = 150.0000, ~3/2 = 702.5090

Optimal ET sequence: 152, 328, 480, 1112bccddee, 1592bccdddeee

Badness (Smith): 0.048007

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 2080/2079, 3584/3575, 14641/14625

Mapping: [⟨8 0 82 -79 15 -186], ⟨0 1 -5 8 1 17]]

Optimal tunings:

• CTE: ~12/11 = 150.0000, ~3/2 = 702.4168
• POTE: ~12/11 = 150.0000, ~3/2 = 702.4773

Optimal ET sequence: 152f, 328, 480f, 808cdeff

Badness (Smith): 0.041365