Footnotes

[1] (M2 means Major 2nd which equals 2 semitones, and P4 means Perfect 4th which equals 5 semitones).

[2] When we say all notes we mean all typical notes in our Western 12-tone, equal temperament system or 12TET, which is the default tuning of most midi systems.

[3] Tip: you can hover over the space combobox to show a tooltip explaining the intervals for any chosen space.

[4] A pitch class is an offset within the octave. It is simply a number between 0 and 11. The pitch class of any midi note is its midi note number modulo 12.

[5] maybe you observed already that there are more keys in tonespace (18) than pitch classes (12), meaning that two keys can end up starting at the same pitch class or offset. For instance, the keys of C# and Db both start at pitch class 1. Therefore the colored grid will be exactly the same for both, with the same pattern of colored and disabled cells. Why bother then in distinguishing these two keys at all? Well, it appears that musicians do make a distinction between two such keys, but the distinction only affects note naming, but not which notes are allowed. So it affects not which absolute pitches are included in the key, but only the labels applied to those pitches. Again this has to do with the fact that traditional note names are just a viewpoint of a musician on an underlying world of absolute pitches.

[6] This is because the geometric chord fitting algorithm will try to make ‘compact’ shapes in the space, i.e. will try to make a shape consisting of cells that are close to eachother. For this to sound pleasantly, both the horizontal and vertical semitone distance between nearby cells should best be equal to common semitone distances between chord notes. With a [1:12] space this is not the case, since not many chords have 1 or 12 semitones distances between their notes. With a [4:3] space it is.