11/8
| Interval information |
harmonic fourth,
undecimal tritone,
undecimal major fourth,
undecimal semiaugmented fourth,
harmonic semiaugmented fourth
reduced harmonic
(Shannon, [math]\displaystyle{ \sqrt{nd} }[/math])
[sound info]
In 11-limit just intonation, 11/8 is an undecimal superfourth of about 551.3 ¢. This interval is close (~3 ¢) to exactly between a perfect fourth and augmented fourth, the latter of which is the augmented version of the Pythagorean diatonic generator, therefore may be called the harmonic/undecimal semiaugmented fourth.
This interval is the simplest superfourth in JI, and as it falls about halfway between 12edo's perfect fourth and tritone, it is very xenharmonic. As an octave-reduced harmonic, it is a basis of consonance in 11-limit JI, alongside the lower odd numbers 9, 7, 5 and 3. It can be found in harmonic series chords such as 4:5:6:7:8:9:10:11:12, sitting somewhere between the stronger and more familiar consonances of 10 (prime 5) and 12 (prime 3).
Terminology
The naming pattern from undecimal neutral third and undecimal neutral second and their octave complements can be rigorously generalized and results in the somewhat unconventional harmonic/undecimal neutral fourth. This interval has also been termed the undecimal major fourth since the tempered version found in 24edo was dubbed the "major fourth" by Ivan Wyschnegradsky, although this may be confusing in diatonic contexts. Because it is right between the diatonic fourth and tritone, it may also be called the (lesser) undecimal tritone.[1] More recently, Zhea Erose has suggested calling it something more simple: the harmonic fourth – under the idea that it is the simplest harmonic that is in the general (very) rough range of "fourths" when octave-reduced. Furthermore, as stacks of this interval form a core axis of Alpharabian tuning, it has also been dubbed the Axirabian paramajor fourth or more simply the just paramajor fourth.
Potential usage
This interval is very well-represented in 24edo, making that system especially good for approximations of JI chords involving primes 3 and 11 such as 8:9:11:12. Not only that, but composers who have experience with 24edo may find it very useful not only as a fantastic addition to major chords, but also as an interesting chord root both for chord progressions within a key, and for modulations to key signatures that are not in the same chain of fifths. Furthermore, these same useful functions can carry over to higher EDOs with good 11-limit representation such as 159edo.
In more tonal music, 11/8 relative to the tonic ends up being used as the chord root for what amounts to a voicing variation of a 1/1-9/8-225/176-3/2 chord, which, is preceded by a variation on a 1/1-5/4-3/2-225/128 chord built on 16/15 relative to the tonic (basically, a type of Neapolitan chord), and, followed up by a variation on the 1/1-5/4-3/2-16/9 dominant seventh chord (or potentially even a 1/1-5/4-3/2-16/9-16/15 dominant ninth chord) built on 3/2 relative to the tonic for a special type of half cadence. This is a dramatic musical gesture that Aura has named the "simul half cadence".
Approximations by EDOs
Following EDOs (up to 200) contain good approximations[note 1] of the interval 11/8. Errors are given by magnitude, the arrows in the table show if the EDO representation is sharp (↑) or flat (↓).
| EDO | deg\edo | Absolute error (¢) |
Relative error (r¢) |
↕ | Equally acceptable multiples [note 2] |
|---|---|---|---|---|---|
| 11 | 5\11 | 5.8634 | 5.3748 | ↓ | |
| 13 | 6\13 | 2.5282 | 2.7389 | ↑ | 12\26 |
| 24 | 11\24 | 1.3179 | 2.6359 | ↓ | 22\48 |
| 37 | 17\37 | 0.0334 | 0.1030 | ↑ | 34\74, 51\111, 68\148, 85\185 |
| 50 | 23\50 | 0.6821 | 2.8419 | ↑ | 46\100 |
| 61 | 28\61 | 0.4983 | 2.5329 | ↓ | 56\122 |
| 63 | 29\63 | 1.0630 | 5.5808 | ↑ | |
| 85 | 39\85 | 0.7297 | 5.1688 | ↓ | |
| 87 | 40\87 | 0.4062 | 2.9449 | ↑ | 80\174 |
| 98 | 45\98 | 0.2975 | 2.4299 | ↓ | 90\196 |
| 124 | 57\124 | 0.2950 | 3.0479 | ↑ | |
| 135 | 62\135 | 0.2068 | 2.3269 | ↓ | |
| 137 | 63\137 | 0.5069 | 5.7868 | ↑ | |
| 159 | 73\159 | 0.3745 | 4.9627 | ↓ | |
| 161 | 74\161 | 0.2349 | 3.1509 | ↑ | |
| 172 | 79\172 | 0.1552 | 2.2238 | ↓ | |
| 198 | 91\198 | 0.1972 | 3.2540 | ↑ |
See also
References
- ↑ Kyle Gann (1998) Anatomy of an Octave