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]
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Major fourth and minor fifth

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 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