11/8
Ratio | 11/8 |
Factorization | 2^{-3} × 11 |
Monzo | [-3 0 0 0 1⟩ |
Size in cents | 551.31794¢ |
Names | undecimal superfourth, undecimal semiaugmented fourth, harmonic semiaugmented fourth, Axirabian paramajor fourth, just paramajor fourth, undecimal major fourth, harmonic fourth |
Color name | 1o4, ilo 4th |
FJS name | [math]\text{P4}^{11}[/math] |
Special properties | reduced, reduced harmonic |
Tenney height (log_{2} nd) | 6.45943 |
Weil height (log_{2} max(n, d)) | 6.91886 |
Wilson height (sopfr (nd)) | 17 |
Harmonic entropy (Shannon, [math]\sqrt{nd}[/math]) |
~4.24125 bits |
[sound info] | |
open this interval in xen-calc |
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).
It 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 root motion 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.
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. 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 – see the history of Aura's Ideas on Functional Harmony for explanation of the modified names. 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. 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.
Approximations by EDOs
Following EDOs (up to 200) contain good approximations^{[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 ^{[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 | ↑ |