11/8

In 11-limit just intonation, 11/8 is an undecimal superfourth of about 551.3¢. This interval, or rather the tempered version found in 24edo, was dubbed the major fourth by Ivan Wyschnegradsky. Furthermore, as stacks of this interval form a core axis of Alpharabian tuning (see also User:Aura/Aura's Ideas on Tonality #11-limit Axis Functionality), it can also be somewhat similarly dubbed the Axirabian paramajor fourth or even the just paramajor fourth.

Interval information

Ratio	11/8
Factorization	2-3 × 11
Monzo	[-3 0 0 0 1⟩
Size in cents	551.3179 ¢
Names	undecimal superfourth, major fourth, Axirabian paramajor fourth, just paramajor fourth
Color name	1o4, ilo 4th
FJS name	[math]\displaystyle{ \text{P4}^{11} }[/math]
Special properties	reduced, reduced harmonic
Tenney norm (log2 nd)	6.45943
Weil norm (log2 max(n, d))	6.91886
Wilson norm (sopfr(nd))	17
[sound info]
Open this interval in xen-calc

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 overtone, 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 much 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.

Approximations by EDOs

Following EDOs (up to 200) contain good approximations 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
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

• 16/11 – its octave complement
• 12/11 – its fifth complement
• Gallery of just intervals