11/8: Difference between revisions
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| Monzo = -3 0 0 0 1 | | Monzo = -3 0 0 0 1 | ||
| Cents = 551.31794 | | Cents = 551.31794 | ||
| Name = undecimal superfourth, <br>major fourth, <br> | | Name = undecimal superfourth, <br>major fourth, <br>Axirabian paramajor fourth, <br>just paramajor fourth | ||
| Color name = 1o4, ilo 4th | | Color name = 1o4, ilo 4th | ||
| FJS name = P4<sup>11</sup> | | FJS name = P4<sup>11</sup> | ||
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In [[11-limit]] [[just intonation]], '''11/8''' is an '''undecimal [[superfourth]]''' of about 551.3[[cent|¢]]. This interval has [https://en.wikipedia.org/wiki/Major_fourth_and_minor_fifth also been referred to] as the '''major fourth'''. 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 ''' | In [[11-limit]] [[just intonation]], '''11/8''' is an '''undecimal [[superfourth]]''' of about 551.3[[cent|¢]]. This interval has [https://en.wikipedia.org/wiki/Major_fourth_and_minor_fifth also been referred to] as the '''major fourth'''. 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'''. | ||
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). | 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). | ||
Revision as of 01:26, 26 December 2021
| Interval information |
major fourth,
Axirabian paramajor fourth,
just paramajor fourth
reduced harmonic
[sound info]
In 11-limit just intonation, 11/8 is an undecimal superfourth of about 551.3¢. This interval has also been referred to as the major fourth. 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.
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[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 | ↑ |