49/48: Difference between revisions
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{{Infobox Interval | {{Infobox Interval | ||
| Name = large septimal diesis, large septimal sixth-tone, slendro diesis, semaphore comma, semaphoresma | |||
| Color name = zz2, zozo 2nd,<br>zzM, zozoma | |||
| Name = large septimal diesis, | |||
| Color name = zz2, zozo | |||
| Sound = Ji-49-48-csound-foscil-220hz.mp3 | | Sound = Ji-49-48-csound-foscil-220hz.mp3 | ||
| Comma = yes | |||
}} | }} | ||
{{Wikipedia|Septimal diesis}} | |||
'''49/48''', the '''large septimal diesis''' (a.k.a. '''large septimal sixth-tone''' or '''slendro diesis'''), is a [[7-limit]] [[superparticular]] ratio spanning the small distance between a subminor third ([[7/6]]) and a supermajor second ([[8/7]]) or between the supermajor sixth ([[12/7]]) and the harmonic seventh ([[7/4]]). Measuring about 35.7{{cent}}, it is a [[medium comma]]; however, in classical Western music, this interval is not known as a [[comma]] as it is not tempered out in [[12edo|12tet]]. | |||
This interval has a function similar to [[25/24]] in that it separates the [[7/6]] and [[8/7]] intervals in a [[6:7:8]] triad, similarly to how [[25/24]] separates [[5/4]] and [[6/5]] in a [[4:5:6]] triad. The 6:7:8 triad consists of odd [[harmonic]]s [[1/1|1]], [[3/1|3]], and [[7/1|7]] [[octave reduced]] to span the [[4/3|perfect fourth]], while the 4:5:6 triad consists of odd harmonics 1, 3, and 5 octave reduced to span the [[3/2|perfect fifth]]. In that regard, tempering out 49/48 can be considered a form of [[exotemperament|exotempering]] that neutralizes the 6:7:8 chord and equates it with its inverse [[21:24:28|1/(8:7:6)]], just like how [[dicot]], which tempers out 25/24, neutralizes the 4:5:6 chord and equates it with its inverse [[10:12:15|1/(6:5:4)]]. | |||
In | == Temperaments == | ||
49/48 is [[tempered out]] in [[15edo]] and [[19edo]], where 7/6 and 8/7 are equated, and the fourth is split in a perfect half. 3/1 is also split into two [[7/4]]~[[12/7]]'s. In the 2.3.7 [[subgroup]], this is known as the [[semaphore]] temperament, and the comma is thus known as the '''semaphore comma''' or '''semaphoresma'''. | |||
''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: | |||
* (49/48)/([[81/80]]) = [[245/243]] | |||
* (49/48)/([[64/63]]) = [[1029/1024]] | |||
* (49/48)/([[3125/3072]]) = [[3136/3125]] | |||
* (49/48)/([[50/49]]) = [[2401/2400]] | |||
* ([[128/125]])/(49/48) = [[6144/6125]] | |||
* ([[36/35]])/(49/48) = [[1728/1715]] | |||
See [[Semaphoresmic family]] for the rank-3 family where it is tempered out. See [[Semaphoresmic clan]] for the rank-2 clan where it is tempered out. | |||
== Approximations == | |||
{{interval edo approximation|min_edo=5}} | |||
== See also == | == See also == | ||
* [[Medium comma]] | * [[Medium comma]] | ||
* [[List of superparticular intervals]] | * [[List of superparticular intervals]] | ||
* [[Gallery of just intervals]] | * [[Gallery of just intervals]] | ||
[[Category: | [[Category:Semaphore]] | ||
[[Category: | [[Category:Semaphoresmic]] | ||
[[Category: | [[Category:Commas named for how they divide the fourth]] | ||
[[Category: | [[Category:Commas named after musical traditions]] | ||
Latest revision as of 09:36, 15 May 2026
| Interval information |
large septimal sixth-tone,
slendro diesis,
semaphore comma,
semaphoresma
zzM, zozoma
reduced
[sound info]
49/48, the large septimal diesis (a.k.a. large septimal sixth-tone or slendro diesis), is a 7-limit superparticular ratio spanning the small distance between a subminor third (7/6) and a supermajor second (8/7) or between the supermajor sixth (12/7) and the harmonic seventh (7/4). Measuring about 35.7 ¢, it is a medium comma; however, in classical Western music, this interval is not known as a comma as it is not tempered out in 12tet.
This interval has a function similar to 25/24 in that it separates the 7/6 and 8/7 intervals in a 6:7:8 triad, similarly to how 25/24 separates 5/4 and 6/5 in a 4:5:6 triad. The 6:7:8 triad consists of odd harmonics 1, 3, and 7 octave reduced to span the perfect fourth, while the 4:5:6 triad consists of odd harmonics 1, 3, and 5 octave reduced to span the perfect fifth. In that regard, tempering out 49/48 can be considered a form of exotempering that neutralizes the 6:7:8 chord and equates it with its inverse 1/(8:7:6), just like how dicot, which tempers out 25/24, neutralizes the 4:5:6 chord and equates it with its inverse 1/(6:5:4).
Temperaments
49/48 is tempered out in 15edo and 19edo, where 7/6 and 8/7 are equated, and the fourth is split in a perfect half. 3/1 is also split into two 7/4~12/7's. In the 2.3.7 subgroup, this is known as the semaphore temperament, and the comma is thus known as the semaphore comma or semaphoresma.
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:
- (49/48)/(81/80) = 245/243
- (49/48)/(64/63) = 1029/1024
- (49/48)/(3125/3072) = 3136/3125
- (49/48)/(50/49) = 2401/2400
- (128/125)/(49/48) = 6144/6125
- (36/35)/(49/48) = 1728/1715
See Semaphoresmic family for the rank-3 family where it is tempered out. See Semaphoresmic clan for the rank-2 clan where it is tempered out.
Approximations
| Edo | Step size | Cents (¢) | Absolute error (¢) | Relative error (%) |
|---|---|---|---|---|
| 31 | 1\31 | 38.71 | +3.01 | +7.78 |
| 32 | 1\32 | 37.50 | +1.80 | +4.81 |
| 33 | 1\33 | 36.36 | +0.67 | +1.83 |
| 34 | 1\34 | 35.29 | -0.40 | -1.14 |
| 35 | 1\35 | 34.29 | -1.41 | -4.12 |
| 36 | 1\36 | 33.33 | -2.36 | -7.09 |
| 64 | 2\64 | 37.50 | +1.80 | +9.62 |
| 65 | 2\65 | 36.92 | +1.23 | +6.64 |
| 66 | 2\66 | 36.36 | +0.67 | +3.67 |
| 67 | 2\67 | 35.82 | +0.12 | +0.69 |
| 68 | 2\68 | 35.29 | -0.40 | -2.28 |
| 69 | 2\69 | 34.78 | -0.91 | -5.26 |
| 70 | 2\70 | 34.29 | -1.41 | -8.23 |