2023/2000

2023/2000, the pentagoth comma, is the interval between a stack of two 20/17 minor thirds and one 7/5 tritone.

Interval information

Ratio	2023/2000
Factorization	2-4 × 5-3 × 7 × 172
Monzo	[-4 0 -3 1 0 0 2⟩
Size in cents	19.79558¢
Name	pentagoth comma
Color name	17oozg33 sosozotrigu 3rd Sosozotrigu comma
FJS name	[math]\displaystyle{ \text{ddd3}^{7,17,17}_{5,5,5} }[/math]
Special properties	reduced
Tenney norm (log2 nd)	21.9481
Weil norm (log2 max(n, d))	21.9646
Wilson norm (sopfr(nd))	64
Comma size	small
Open this interval in xen-calc

Tempering out this comma alone in the 2.5.7.17 subgroup leads to the pentagoth temperament.

Temperaments

Note: groundfault uses extensions that identify the ~20/17~13/11 with ~19/16 and tempering out 209/208; however, this leads to considerable damage to prime 19.

Pentagoth

Subgroup: 2.5.7.17

Comma list: 2023/2000

Mapping: [⟨1 2 2 4], ⟨0 1 1 1], ⟨0 0 2 -1]]

mapping generators: ~2, ~5/4, ~20/17

Optimal tunings:

• WE: ~2 = 1200.566 ¢, ~5/4 = 387.475 ¢, ~20/17 = 289.545 ¢

error map: ⟨+0.566 2.294 -1.127 -4.760]

• CWE: ~2 = 1200.0000 ¢, ~5/4 = 388.049 ¢, ~5/4 = 289.369 ¢

error map: ⟨0.000 1.735 -2.038 -6.275]

Badness (Sintel): 0.509

2.5.7.13/11.17.23

Subgroup: 2.5.7.13/11.17.23

Comma list: 221/220, 161/160, 1309/1300

Subgroup-val mapping: [⟨1 2 2 0 4 5], ⟨0 1 1 0 1 0], ⟨0 0 2 1 -1 -2]]

Optimal tuning (CWE): ~2 = 1\1, ~5/4 = 387.513, ~13/11 = 288.786

Badness (Sintel): 0.191

No-3 no-19 23-limit

Subgroup: 2.5.7.11.13.17.23

Comma list: 176/175, 221/220, 161/160, 1309/1300

Subgroup-val mapping: [⟨1 2 2 2 2 4 5], ⟨0 1 1 3 3 1 0], ⟨0 0 2 2 3 -1 -2]]

Optimal tuning (CWE): ~2 = 1\1, ~5/4 = 390.453, ~13/11 = 288.844

Badness (Sintel): 0.497

Pentagoth clan

Rank-2 temperaments supporting pentagoth

Vengeance

Main article: Vengeance

A lower-error replica of mavila, with the fifth being ~25/17 instead of ~3/2.

Subgroup: 2.5.7.17

Comma list: 2023/2000, 78608/78125

Subgroup-val mapping: [⟨1 1 5 1], ⟨0 3 -5 7]]

Optimal tuning (CWE): ~2 = 1\1, ~34/25 = 527.718

Optimal ET sequence: 7g, 9, 25, 34, 93, 127, 288, 415

Badness (Sintel): 0.720

Sidewalk

Sidewalk, 2.5.7.17[21 & 25], tempers out 823543/800000 in the 2.5.7 subgroup.

Subgroup: 2.5.7.17

Comma list: 343/340, 2023/2000

Subgroup-val mapping: [⟨1 4 4 6], ⟨0 -7 -5 -8]]

Optimal tuning (CWE): ~2 = 1\1, ~20/17 = 287.377

Badness (Sintel): 0.447

2.5.7.13/11.17.23

Subgroup: 2.5.7.13/11.17.23

Comma list: 221/220, 161/160, 1309/1300, 343/340

Subgroup-val mapping: [⟨1 4 4 0 6 5], ⟨0 -7 -5 1 -8 -2]]

Optimal tuning (CWE): ~2 = 1\1, ~13/11 = 287.412

Badness (Sintel): 0.256

No-3 no-19 23-limit

Subgroup: 2.5.7.11.13.17.23

Comma list: 176/175, 221/220, 161/160, 1309/1300, 343/340

Subgroup-val mapping: [⟨1 4 4 8 8 6 5], ⟨0 -7 -5 -19 -18 -8 -2]]

Optimal tuning (CWE): ~2 = 1\1, ~13/11 = 286.821

Badness (Sintel): 0.671

Etymology

As of June 6, 2026, pentagoth is the name given to the rank-3 temperament 2.5.7.17[2023/2000] by groundfault. It originally referred to a rank-2 temperament now called vengeance, which as a 2.5.7.17 temperament tempers out this comma.