125/81: Difference between revisions
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Correct math notation; move approximation info to the relevant section |
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'''125/81''', the '''triptolemaic augmented fifth''' or '''Trithagorean ultrafifth''', is a [[5-limit]] just interval equal to the Pythagorean augmented fifth [[6561/4096]] lowered by three syntonic commas (([[81/80]])<sup>3</sup>). It is also equal to [[25/16]] lowered by [[81/80]] and to [[128/81]] lowered by [[128/125]]. It also appears as a notable interval in the [[Trithagorean]] scale / [[3.5 subgroup]], being the [[3/1|tritave]]-reduction of ([[5/3]])<sup>3</sup>. | |||
== Approximation == | == Approximation == | ||
This interval can be closely approximated by [[17/11]] and by 5 steps of [[8edo]]. | |||
{{Interval edo approximation|17/11}} | {{Interval edo approximation|17/11}} | ||
Latest revision as of 11:14, 23 October 2026
| Interval information |
Trithagorean ultrafifth
125/81, the triptolemaic augmented fifth or Trithagorean ultrafifth, is a 5-limit just interval equal to the Pythagorean augmented fifth 6561/4096 lowered by three syntonic commas ((81/80)3). It is also equal to 25/16 lowered by 81/80 and to 128/81 lowered by 128/125. It also appears as a notable interval in the Trithagorean scale / 3.5 subgroup, being the tritave-reduction of (5/3)3.
Approximation
This interval can be closely approximated by 17/11 and by 5 steps of 8edo.
| Edo | Step size | Cents (¢) | Absolute error (¢) | Relative error (%) |
|---|---|---|---|---|
| 8 | 5\8 | 750.00 | -1.12 | -0.75 |
| 16 | 10\16 | 750.00 | -1.12 | -1.49 |
| 24 | 15\24 | 750.00 | -1.12 | -2.24 |
| 27 | 17\27 | 755.56 | +4.43 | +9.98 |
| 32 | 20\32 | 750.00 | -1.12 | -2.99 |
| 35 | 22\35 | 754.29 | +3.16 | +9.23 |
| 40 | 25\40 | 750.00 | -1.12 | -3.74 |
| 43 | 27\43 | 753.49 | +2.37 | +8.48 |
| 48 | 30\48 | 750.00 | -1.12 | -4.48 |
| 51 | 32\51 | 752.94 | +1.82 | +7.74 |
| 56 | 35\56 | 750.00 | -1.12 | -5.23 |
| 59 | 37\59 | 752.54 | +1.42 | +6.99 |
| 64 | 40\64 | 750.00 | -1.12 | -5.98 |
| 67 | 42\67 | 752.24 | +1.12 | +6.24 |
| 72 | 45\72 | 750.00 | -1.12 | -6.73 |
| 75 | 47\75 | 752.00 | +0.88 | +5.49 |
| 80 | 50\80 | 750.00 | -1.12 | -7.47 |