50/49: Difference between revisions
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Wikispaces>genewardsmith **Imported revision 244887783 - Original comment: ** |
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{{Interwiki | |||
| en = 50/49 | |||
| de = 50/49 | |||
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{{Infobox Interval | |||
| Name = small septimal diesis, small septimal sixth-tone, septimal tritonic diesis, jubilisma | |||
| Color name = rryy-2, biruyo negative 2nd,<br>Biruyo comma | |||
| Comma = yes | |||
}} | |||
{{Wikipedia|Septimal third tone #Septimal sixth tone}} | |||
'''50/49''', the '''small septimal diesis''' (a.k.a. '''small septimal sixth-tone''' or '''septimal tritonic diesis'''), is a [[7-limit]] [[medium comma]]. It is the only [[superparticular]] [[comma]] in the 7-limit aside from [[126/125]] and [[4375/4374]] which has a numerator which is neither square nor [[triangular number|triangular]], meaning it is not the difference between septimal superparticular rations with numerators differing by either one or two; instead, {{nowrap| 50/49 = ([[10/7]])/([[7/5]]) }}. | |||
== Temperaments == | |||
[[Tempering out]] this comma equates the two septimal tritones (i.e. [[7/5]] and [[10/7]]) with each other, leading to temperaments where [[sqrt(2/1)]] approximates both. In the [[2.5.7 subgroup]], this is known as the jubilic temperament, and the comma is thus known as the '''jubilisma'''. In the full 7-limit, this comma further equates [[15/14]] and [[21/20]] and enables all the [[jubilismic chords]]. | |||
''It cannot be tempered out if all of the consonances of the 7-odd-limit are distinct'', but it ''can'' be equated with other commas; for example: | |||
* ([[36/35]])/(50/49) = [[126/125]] | |||
* ([[45/44]])/(50/49) = [[441/440]] | |||
* ([[49/48]])/(50/49) = [[2401/2400]] | |||
* (50/49)/([[55/54]]) = [[540/539]] | |||
* (50/49)/([[56/55]]) = [[1375/1372]] | |||
* (50/49)/([[64/63]]) = [[225/224]] | |||
* (50/49)/([[65/64]]) = [[640/637]] | |||
* (50/49)/([[66/65]]) = [[1625/1617]] | |||
* (50/49)/([[78/77]]) = [[275/273]] | |||
* (50/49)/([[81/80]]) = [[4000/3969]] | |||
See [[Jubilismic family]] for the rank-3 family where it is tempered out, and [[Jubilismic clan]] for the rank-2 clan where it is tempered out. | |||
Equal temperaments tempering out 50/49 include [[12edo]], [[22edo]], [[26edo]], [[38edo]], [[48edo]], and [[54edo]]. | |||
== Approximations == | |||
{{Interval edo approximation|min_edo=12}} | |||
== Etymology == | |||
The name ''jubilisma'' is likely a reference to the 50-year biblical jubilee cycle. | |||
== See also == | |||
* [[List of superparticular intervals]] | |||
* [[49/48]] – the large septimal sixth-tone | |||
[[Category:Jubilismic]] | |||
[[Category:Commas referencing a famous use of a number]] | |||
Latest revision as of 08:24, 4 March 2026
| Interval information |
small septimal sixth-tone,
septimal tritonic diesis,
jubilisma
Biruyo comma
reduced
50/49, the small septimal diesis (a.k.a. small septimal sixth-tone or septimal tritonic diesis), is a 7-limit medium comma. It is the only superparticular comma in the 7-limit aside from 126/125 and 4375/4374 which has a numerator which is neither square nor triangular, meaning it is not the difference between septimal superparticular rations with numerators differing by either one or two; instead, (10/7)/(7/5).
Temperaments
Tempering out this comma equates the two septimal tritones (i.e. 7/5 and 10/7) with each other, leading to temperaments where sqrt(2/1) approximates both. In the 2.5.7 subgroup, this is known as the jubilic temperament, and the comma is thus known as the jubilisma. In the full 7-limit, this comma further equates 15/14 and 21/20 and enables all the jubilismic chords.
It cannot be tempered out if all of the consonances of the 7-odd-limit are distinct, but it can be equated with other commas; for example:
- (36/35)/(50/49) = 126/125
- (45/44)/(50/49) = 441/440
- (49/48)/(50/49) = 2401/2400
- (50/49)/(55/54) = 540/539
- (50/49)/(56/55) = 1375/1372
- (50/49)/(64/63) = 225/224
- (50/49)/(65/64) = 640/637
- (50/49)/(66/65) = 1625/1617
- (50/49)/(78/77) = 275/273
- (50/49)/(81/80) = 4000/3969
See Jubilismic family for the rank-3 family where it is tempered out, and Jubilismic clan for the rank-2 clan where it is tempered out.
Equal temperaments tempering out 50/49 include 12edo, 22edo, 26edo, 38edo, 48edo, and 54edo.
Approximations
| Edo | Step size | Cents (¢) | Absolute error (¢) | Relative error (%) |
|---|---|---|---|---|
| 31 | 1\31 | 38.71 | +3.73 | +9.65 |
| 32 | 1\32 | 37.50 | +2.52 | +6.73 |
| 33 | 1\33 | 36.36 | +1.39 | +3.82 |
| 34 | 1\34 | 35.29 | +0.32 | +0.90 |
| 35 | 1\35 | 34.29 | -0.69 | -2.01 |
| 36 | 1\36 | 33.33 | -1.64 | -4.93 |
| 37 | 1\37 | 32.43 | -2.54 | -7.84 |
| 66 | 2\66 | 36.36 | +1.39 | +7.63 |
| 67 | 2\67 | 35.82 | +0.85 | +4.72 |
| 68 | 2\68 | 35.29 | +0.32 | +1.80 |
| 69 | 2\69 | 34.78 | -0.19 | -1.11 |
| 70 | 2\70 | 34.29 | -0.69 | -4.02 |
| 71 | 2\71 | 33.80 | -1.17 | -6.94 |
| 72 | 2\72 | 33.33 | -1.64 | -9.85 |
Etymology
The name jubilisma is likely a reference to the 50-year biblical jubilee cycle.
See also
- List of superparticular intervals
- 49/48 – the large septimal sixth-tone