This is a table for converting between steps in 31-equal and ratio space coordinates. At the moment, it includes most of the intervals you're likely to need when using 7-limit harmony. The others can usually be worked out fairly easily when in context. Extended 11-limit intervals have now been added. As I've been using the 4-D table for years, I can vouch for its accuracy, unless I copied it wrong. There may be 5-D mistakes, though.
| 34 | 0 1 1 -1 | 1 3 -2 0 | 4 -3 2 -1 | -2 -2 0 1 1 | ||
| 33 | -2 -1 2 0 | 3 -3 0 1 | -1 1 -1 1 | 2 -1 0 -1 1 | ||
| 32 | 3 2 -1 -1 | -1 2 1 0 -1 | 4 0 -1 1 -1 | -4 1 0 0 1 | ||
| 31 | 1 0 0 | -4 2 2 -1 | ||||
| 30 | -1 -2 1 1 | 3 -2 -1 0 1 | -2 0 1 -1 1 | |||
| 29 | 0 1 0 1 -1 | 4 1 -2 0 | -1 3 0 -1 | 3 -1 1 -1 | ||
| 28 | -3 1 1 | 1 -3 2 0 | 2 -1 -1 1 | -2 3 -2 1 | 1 1 -1 -1 1 | |
| 27 | -1 -1 0 0 1 | 6 0 -1 -1 | 2 0 1 0 -1 | |||
| 26 | 0 2 -1 | 4 -2 0 | -1 0 2 -1 | -4 2 1 1 -1 | ||
| 25 | -2 0 0 1 | 1 -2 1 -1 1 | 5 1 -1 0 -1 | -3 2 0 -1 1 | ||
| 24 | 2 1 0 -1 | 0 -2 -1 1 1 | -5 0 1 0 1 | 3 -1 0 1 -1 | ||
| 23 | 0 -1 1 | -4 3 0 0 | 1 1 -2 1 | 4 -1 -1 -1 1 | ||
| 22 | -2 -3 2 1 | 1 2 0 0 -1 | -2 1 -1 0 1 | 5 -2 1 0 -1 | ||
| 21 | 3 0 -1 | 2 -2 2 -1 | -2 2 1 -1 | -4 -1 0 1 1 | -1 0 1 1 -1 | |
| 20 | -4 0 2 0 | 1 -2 0 1 | 0 0 0 -1 1 | -3 2 -1 1 | ||
| 19 | 1 3 -1 -1 | -2 -2 1 0 1 | 2 1 -1 1 -1 | |||
| 18 | -1 1 0 | 3 -3 1 0 | 4 -1 -2 1 | |||
| 17 | -3 -1 1 1 | 4 0 0 0 -1 | 1 -1 -1 0 1 | -4 1 1 -1 1 | ||
| 16 | 1 0 1 -1 | 2 2 -2 0 | -2 2 0 1 -1 | 2 -2 1 1 -1 | ||
| 15 | 0 0 -1 1 | -1 -2 2 0 | 3 -2 0 -1 1 | -1 2 -1 -1 1 | ||
| 14 | 4 1 -1 -1 | -3 0 0 0 1 | 0 1 1 0 -1 | 5 -1 -1 1 -1 | ||
| 13 | 2 -1 0 | -2 3 -1 0 | -3 1 2 -1 | -5 -2 1 1 1 | ||
| 12 | -4 1 0 1 | 0 -3 1 1 | 3 2 -1 0 -1 | -1 -1 1 -1 1 | ||
| 11 | 0 2 0 -1 | 1 0 0 1 -1 | 4 -2 1 -1 | -2 -1 -1 1 1 | ||
| 10 | -2 0 1 | -1 2 -2 1 | 3 -2 -1 1 | 5 1 0 -1 -1 | 2 0 -1 -1 1 | |
| 9 | 0 -2 0 0 1 | 3 3 -2 -1 | -4 -2 2 1 | 3 -1 1 0 -1 | ||
| 8 | 1 1 -1 | -4 3 1 -1 | 0 -1 2 -1 | -3 1 1 1 -1 | ||
| 7 | -1 -1 0 1 | 1 2 1 -1 -1 | -2 1 0 -1 1 | |||
| 6 | 3 0 0 -1 | 0 2 -1 1 -1 | -4 -1 1 0 1 | 4 -2 0 1 -1 | ||
| 5 | -3 2 0 | 1 -2 1 | 2 0 -2 1 | 5 -2 -1 -1 1 | ||
| 4 | -1 0 -1 0 1 | 2 1 0 0 -1 | -5 0 1 1 | |||
| 3 | 4 -1 -1 | -1 1 1 -1 | 0 3 -2 0 | 3 -3 2 -1 | 0 -1 1 1 -1 | |
| 2 | -2 1 -1 1 | -3 -1 2 0 | 2 -3 0 1 | 1 -1 0 -1 1 | ||
| 1 | 2 2 -1 -1 | -2 2 1 0 -1 | 3 0 -1 1 -1 | -5 1 0 0 1 | ||
| 0 | 0 0 0 | |||||
| -1 | -2 -2 1 1 | 2 -2 -1 0 1 | -3 0 1 -1 1 | 5 -1 0 0 -1 | ||
| -2 | 2 -1 1 -1 | 3 1 -2 0 | -2 3 0 -1 | -1 1 0 1 -1 | ||
| -3 | -4 1 1 | 1 -1 -1 1 | 0 -3 2 0 | -3 3 -2 1 | 0 1 -1 -1 1 | |
| -4 | -2 -1 0 0 1 | 1 0 1 0 -1 | 5 0 -1 -1 | |||
| -5 | 3 -2 0 | -1 2 -1 | -2 0 2 -1 | -5 2 1 1 -1 | ||
| -6 | -3 0 0 1 | 0 -2 1 -1 1 | 4 1 -1 0 -1 | -4 2 0 -1 1 | ||
| -7 | 1 1 0 -1 | -1 -2 -1 1 1 | 2 -1 0 1 -1 | |||
| -8 | -1 -1 1 | 4 -3 -1 1 | 0 1 -2 1 | 3 -1 -1 -1 1 | ||
| -9 | 0 2 0 0 -1 | -3 -3 2 1 | 4 2 -2 -1 | -3 1 -1 0 1 | ||
| -10 | 2 0 -1 | 1 -2 2 -1 | -3 2 1 -1 | -5 -1 0 1 1 | -2 0 1 1 -1 | |
| -11 | 0 -2 0 1 | -1 0 0 -1 1 | -4 2 -1 1 | 2 1 1 -1 -1 | ||
| -12 | 4 -1 0 -1 | 0 3 -1 -1 | -3 -2 1 0 1 | 1 1 -1 1 -1 | ||
| -13 | -2 1 0 | 2 -3 1 0 | 3 -1 -2 1 | 5 2 -1 -1 -1 | ||
| -14 | 3 0 0 0 -1 | 0 -1 -1 0 1 | -4 -1 1 1 | -5 1 1 -1 1 | ||
| -15 | 0 0 1 -1 | 1 2 -2 0 | -3 2 0 1 -1 | 1 -2 1 1 -1 | ||
| -16 | -1 0 -1 1 | -2 -2 1 0 | 2 -2 0 -1 1 | -2 2 -1 -1 1 | ||
| -17 | 3 1 -1 -1 | -4 0 0 1 | -1 1 1 0 -1 | 4 -1 -1 1 -1 | ||
| -18 | 1 -1 0 | -3 3 -1 0 | -4 1 2 -1 | |||
| -19 | -1 -3 1 1 | 2 2 -1 0 -1 | -2 -1 1 -1 1 | |||
| -20 | 0 0 0 1 -1 | -1 2 0 -1 | 4 0 -2 0 | 3 -2 1 -1 | ||
| -21 | -3 0 1 | -2 2 -2 1 | 2 -2 -2 1 | 4 1 0 -1 -1 | -1 0 1 1 -1 | |
| -22 | -1 -2 0 0 1 | 2 -1 1 0 -1 | 2 3 -2 -1 | -5 2 -1 0 1 | ||
| -23 | 0 1 -1 | 4 -3 0 0 | -1 -1 2 -1 | -4 1 1 1 -1 | ||
| -24 | -2 -1 0 1 | 0 2 1 -1 -1 | 5 0 -1 0 -1 | -3 1 0 -1 1 | ||
| -25 | 2 0 0 -1 | -1 2 -1 1 -1 | -5 -1 1 0 1 | 3 -2 0 1 -1 | ||
| -26 | 0 -2 1 | -4 2 0 | 1 0 -2 1 | 4 -2 -1 -1 1 | ||
| -27 | 1 1 0 0 -1 | -2 0 -1 0 1 | -6 0 1 1 | |||
| -28 | 3 -1 -1 | -1 3 -2 0 | -2 1 1 -1 | 2 -3 2 -1 | -1 -1 1 1 -1 | |
| -29 | 0 -1 0 -1 1 | -4 -1 2 0 | 1 -3 0 1 | -3 1 -1 1 | ||
| -30 | 1 2 -1 -1 | -3 2 1 0 -1 | 2 0 -1 1 -1 | |||
| -31 | -1 0 0 | 4 -2 -2 1 | ||||
| -32 | 1 -2 -1 0 1 | 4 -1 0 0 -1 | -3 -2 1 1 | -4 0 1 -1 1 | ||
| -33 | 2 1 -2 0 | -3 3 0 -1 | 1 -1 1 -1 | -2 1 0 1 -1 | ||
| -34 | 0 -1 -1 1 | -1 -3 2 0 | -4 3 -2 1 | 2 2 0 -1 -1 |
Note that I'm not suggesting that all these intervals are heard as the ratios those coordinates correspond to. The complexity is there to handle modulations and the like.
Okay, now here's a Java applet to convert from a vector to a number of steps from 31=.
Type in the values you want for the octave-equivalent coordinates. The octave coordinate and the number of steps in 31= are then calculated automatically when you press return. You can tab between fields. Note this works up to the 13-limit! You can use it to check the table above if you really want to.
Using this page, you can notate JI music in 31-equal. This may mean writing music in staff notation, and assuming performers can work out what the chords are supposed to be. Or, it might only be a way of giving names to intervals. The next table, then, gives names to intervals in 31-equal.
| 31= | 12= | 7= | Diatonic | Chromatic | Microtonal |
| 31 | 12 | 7 | octave | perfect octave | perfect octave |
| 30 | 12 | 6 | augmented seventh | sub-octave | |
| 29 | 11 | 7 | diminished octave | supermajor seventh | |
| 28 | 11 | 6 | major seventh | major seventh | major seventh |
| 27 | 10 | 7 | neutral seventh | ||
| 26 | 10 | 6 | minor seventh | minor seventh | minor seventh |
| 25 | 10 | 5 | augmented sixth | subminor seventh | |
| 24 | 9 | 6 | diminished seventh | supermajor sixth | |
| 23 | 9 | 5 | major sixth | major sixth | major sixth |
| 22 | 9 | 4 | neutral sixth | ||
| 21 | 8 | 5 | minor sixth | minor sixth | minor sixth |
| 20 | 8 | 4 | augmented fifth | subminor sixth | |
| 19 | 7 | 5 | diminished sixth | super-fifth | |
| 18 | 7 | 4 | fifth | perfect fifth | perfect fifth |
| 17 | 7 | 3 | sub-fifth | ||
| 16 | 6 | 4 | diminished fifth | greater tritone | |
| 15 | 6 | 3 | augmented fourth | lesser tritone | |
| 14 | 5 | 4 | super-fourth | ||
| 13 | 5 | 3 | fourth | perfect fourth | perfect fourth |
| 12 | 5 | 2 | augmented third | sub-fourth | |
| 11 | 4 | 3 | diminished fourth | supermajor third | |
| 10 | 4 | 2 | major third | major third | major third |
| 9 | 3 | 3 | neutral third | ||
| 8 | 3 | 2 | minor third | minor third | minor third |
| 7 | 3 | 1 | augmented second | subminor third | |
| 6 | 2 | 2 | diminished third | supermajor second | |
| 5 | 2 | 1 | major second | major second | tone |
| 4 | 2 | 0 | neutral second | ||
| 3 | 1 | 1 | minor second | minor second | diatonic semitone |
| 2 | 1 | 0 | augmented unison | chromatic semitone | |
| 1 | 0 | 1 | diminished second | quartertone | |
| 0 | 0 | 0 | unison | perfect unison | unison |
David Keenan uses the same kind of names.
The first column obviously shows the number of steps in 31-equal. The next columns show the number of chromatic semitones and "quartertones" that make up the interval. These uniquely define an interval in a general meantone temperament. In each case the simplest mapping is used. The number of chromatic semitones is also the number of steps in 12 note equal temperament. The number of "quartertones" is the number of steps on a diatonic scale. So, an interval of an "nth" is n-1 "quartertones". I would like to call these chromatic and diatonic steps respectively. However, there may be confusion as diatonic semitones are sometimes called diatonic steps.
The diatonic column shows what these intervals are called when they occur in a diatonic scale. These names are frequently and sometimes imprecisely (note, I do not say incorrectly) used. The difference between a major and minor interval is always a chromatic semitone.
In the chromatic column, intervals are named according to shifts of chromatic semitones from diatonic intervals. An augmented interval is raised by a chromatic semitone, and a diminished interval lowered by one. This corresponds to adding a sharp or flat in staff notation, which must be the most popular form of music notation in the world. The fact that it is consistent with meantone temperament is a good argument for using meantones.
The microtonal column shows a more intuitive way of naming the intervals. The idea here is that there is a range of intervals that can be called thirds, fourths and so on. These names make most sense when describing high-prime intervals. For example, 7/6 or (-1 -1 0 1)H is slightly flat of a minor third, why not call it a subminor third rather than an augmented tone? Some alternative names in common use are also given.
The neutral intervals are characteristic of 13-limit intervals. So, 13/8 is a neutral sixth, 11/9 is a neutral third, 11/10 is a neutral second and 11/6 is a neutral seventh. A neutral third could be written as either a doubly augmented tone or a doubly diminished fourth. In practice, when you're using a general meantone, both can be used for an 11/9.
The prefixes sub and super are used to mean raise or lower by the "quartertone" or diminished second. Whether this should be called a quartertone, as it's really a fifth of a tone, is a matter of some debate. So, I usually put it in quotes. I've come up with the alternative demisemitone to mean half of a chromatic semitone, but that is cumbersome.
31-equal nomenclature is about as precise as you need to get for instinctive purposes. You could argue that it's too precise, but you don't need all the names. If you're deliberately composing in JI, it makes sense to specify the ratios. However, if you hear a super-fourth, you should describe it as such rather than forcing the interpretation 11/8 onto it.
As a disclaimer, though, I'll state that 31-equal is not sufficiently precise that you can approximate other scales to it. I tried this a long time ago, and it doesn't work. Although you can't say which notes are wrong, let alone in which direction, the essential character of a scale can be killed by that sort of thing.
Staff notation is merely a system of conventions. So, there is nothing to stop you basing your nomenclature around, say, 17 or 22-equal. This way happens to be the most convenient one for me.
Boundaries between pitch classes will be fuzzy. You may say an interval is somewhere between a perfect and super-fifth. A lot of the time it depends on context. You could even say that fifths are much more important than super-fifths, and so and interval in the middle is a slightly sharpened fifth. That makes more sense to me than calling it a slightly flattened super fifth.
I originally intended to put more stuff here, but that was a while ago. I can't remember what now. So I'll end it here.