So, what's the point of all this? Well, if you want to write 5-limit music, but you're bored of the same old scales, the scale you want will probably be in this list. It should be quicker to look down the page to find some likely looking scales than to spend ages working out the scale you want from first principles.
Step sizes are given in 12- and 19-equal. From these, the scale can be reconstructed in any meantone. All the types are distinct in 19-equal, and therefore in any meantone other then 12-equal.
No, I didn't think them all up myself. The first one should be familiar for a start. Maybe I should credit John Chalmers, because I discussed the list as I drew it up, and I'm sure he had a similar one at his end.
This first group of scales have at least 4 consonant triads with distinct roots.
They can all be defined on this subset of the 5-limit lattice:
D-----A-----E-----B \ / \ / \ / \ \ / \ / \ / \ F-----C-----G-----D-----A / \ / \ / \ / \ / / \ / \ / \ / \ / Db----Ab----Eb----Bb----F
Type 1: C D E F G A B C D E F G A B C D E F G A B C D E 2 2 1 2 2 2 1 G A B C D E F G A B C D E F G A 3 3 2 3 3 3 2 Type 2: C D Eb F G A B C D Eb F G A B C D 2 1 2 2 2 2 1 G A B C D Eb F G 3 2 3 3 3 3 2 Type 3: C D Eb F G Ab B C G Ab B C D Eb F G 2 2 1 2 1 3 1 C D E F G Ab B C 3 3 2 3 2 4 2 G Ab B C D E F GTypes 1 and 2 are the only ones on this page that lose chords if converted from meantone (or 19-equal) to JI. Type 2 is the only one that wouldn't fulfil the criteria in JI. The rest of the scales in this group are a subset of the following structure:
D-----A-----E-----B \ / \ / \ / \ / \ / \ / F-----C-----G / \ / \ / \ / \ / \ / \ Db----Ab----Eb----Bb
Type 4: C Db E F G Ab B C 1 3 1 2 1 3 1 2 4 2 3 2 4 2 Type 5: C Eb E F G Ab A C 3 1 1 2 1 1 3 5 1 2 3 2 1 5 Type 6: C Eb E F G A Bb C C D Eb F G Ab A C 3 1 1 2 2 1 2 5 1 2 3 3 2 3 Type 7: E G A Bb B C Eb E E G Ab A Bb C Eb E 2 2 1 1 1 3 1 3 3 2 1 2 5 1 Type 8: C Db E F G Ab A C C Eb E F G Ab B C 1 3 1 2 1 1 3 2 4 2 3 2 1 5 Type 9: Eb E G Ab Bb B C Eb E G Ab A B C Eb E 1 3 1 2 1 1 3 1 5 2 3 1 2 5 Type 10: C Db Eb E F G Ab C Ab C Db Eb E F G Ab 1 2 1 1 2 1 4 C E F G Ab A B C 2 3 1 3 3 2 6 E F G Ab A B C E Type 11: C Db Eb F G Ab A C Ab A C Db Eb F G Ab 1 2 2 2 1 1 3 C Eb E F G A B C 2 3 3 3 2 1 5 E F G A B C Eb E Type 12: C D Eb E F G A C A C D Eb E F G A 2 1 1 1 2 2 3 C Eb F G Ab A Bb C 3 2 1 2 3 3 5 Eb F G Ab A Bb C Eb Type 13: C Eb E F G Ab Bb C Eb E F G Ab Bb C Eb 3 1 1 2 1 2 2 C D E F G Ab A C 5 1 2 3 2 3 3 A C D E F G Ab AThe next group of scales have more than 4 consonant triads, but less than 4 distinct roots. This makes it more dificult to construct convincing chord sequences. Well, I say that to make it sound like this has something to do with music rather than being an abstract mathematical exercise with some pretty lattice diagrams. You decide for youself how the chord sequences work.
All the scales from now on are a subset of this structure:
C#----G#
/ \ / \
/ \ / \
A-----E-----B
/ \ / \ /
/ \ / \ /
F-----C-----G
\ / \ / \
\ / \ / \
Ab----Eb----Bb
\ / \ /
\ / \ /
Cb----Gb
Type 14: C C# E F G Ab A C C Eb E F G Ab Cb C 1 3 1 2 1 1 3 same as 8&9 1 5 2 3 2 1 5 Type 15: C C# Eb E F G A C C Eb F G Ab A Cb C 1 2 1 1 2 2 3 1 4 1 2 3 3 5All the scales above have a consonant third, fourth, fifth and sixth from the tonic. For the rest, this is not the case. This is important in characterising the scales.
Type 16: C C# Eb E G A Bb C C Eb E G Ab A B C 1 2 1 3 2 1 2 1 4 1 5 3 2 3 Type 17: C Eb E G G# Ab B C C Eb E Gb G A Bb C 3 1 3 1 0 3 1 5 1 5 1 1 4 2 Type 18: C C# Eb E G G# A C C C# Eb E G Ab A C 1 2 1 3 1 1 3 same as 4 1 4 1 5 1 2 5 Type 19: C Eb E G G# Bb B C C Eb E G Ab A Cb C 3 1 3 1 2 1 1 same as 8,9&14 5 1 5 1 3 2 2All the scales so far have have more then four consonant chords, except for types 2 and 3. The rest have only 4 triads with less than four roots.
Type 20: C C# Eb E G# Ab A C C Eb E G# Ab A Cb C 1 2 1 4 0 1 3 1 4 1 6 1 1 5 Type 21: C C# Eb E G G# Bb C C Eb E Gb G A Cb C 1 2 1 3 1 2 2 1 4 1 5 1 3 3 Type 22: C C# Eb E Ab A Cb C 1 2 1 4 1 2 1 1 4 1 5 1 4 1That's it for 5-limit consonance unless you only want 3 triads in which case you can work the scales out yourself. If you can find any scale types with 4 triads that I missed, well, e-mail me and I'll correct the omission. I've revised the commentary on types 2 and 3 because I'm sure it was wrong before, but it seems nobody noticed. There are also a lot of 7-limit scales that demand attention.