Kite's Thoughts on 41edo Note Names and Key Signatures
How to name any note in any chord on any root in any key? First, let's consider 12-edo note names.
Background: 12-edo names
Even though there's only 12 notes, the names are selected from a much larger 1-D lattice of notes, this chain of fifths:
Gb Db Ab Eb Bb F C G D A E B F# C# G# D# A#
This chain is of course actually a circle of 12 fifths, but its easier to read written out this way. Especially later when we lengthen it.
Whatever your tonic is, you select the closest notes in the lattice. For a major scale, you select the tonic, the note immediately to the left, and the 5 notes immediately to the right. Thus A major has notes starting at D and going to G#. But the other 5 notes can also occur, so they need names too. For now, let's get them from the left, so A major is the 12 notes running from Eb to G#. Here's the same chain of 5ths in relative notation, i.e. as intervals not notes:
d5 m2 m6 m3 m7 P4 P1 P5 M2 M6 M3 M7
Imagine that both chains are written out on two long strips of wood, like two rulers. We can line up the P1 with the tonic A to find the 12 note names:
note ruler: Gb Db Ab Eb Bb F C G D A E B F# C# G# D# A# interval ruler: d5 m2 m6 m3 m7 P4 P1 P5 M2 M6 M3 M7
Which notes can be used for the tonic? In theory, any of them, even A#. But the relative ruler selects a portion of the absolute ruler that has double sharps, which are ugly.
notes: Gb Db Ab Eb Bb F C G D A E B F# C# G# D# A# E# B# Fx Cx Gx intervals: d5 m2 m6 m3 m7 P4 P1 P5 M2 M6 M3 M7
Instead we use A#'s 12-edo equivalent, Bb.
notes: Fb Cb Gb Db Ab Eb Bb F C G D A E B F# C# G# D# A# intervals: d5 m2 m6 m3 m7 P4 P1 P5 M2 M6 M3 M7
For tonics, we could use only the 12 most central notes, say Ab to C#. But actually more are used, because Ab minor would have Cb and Fb. We're used to the black keys on the piano having two names, but we expect a white key to have only one name. Cb and Fb are better named B and E, and Ab minor becomes G# minor. Likewise C# major would have E# and B#, so it becomes Db major. Strictly avoiding misnamed white keys gives us 11 major keys Db to B and 11 minor keys Bb to G#. The 12th key inevitably has a misnamed white key: F# major has E# and Gb major has Cb. So we include both, and include Eb minor and D# minor as well, for a total of 26 possible keys. Of course musically there are only 24 keys, 12 of each type, but some keys have two names.
There are 16 possible tonics, running from Gb to D#. But modulation to an adjacent key in the chain of 5ths is very common, and we don't want to modulate from a key with 6 sharps to one with 5 flats. In other words, musical context sometimes forces us to use additional keys. So we need some overlapping keys.Two extra major and minor keys gives us 15 major keys and 15 minor keys. There are 18 possible tonics running from Cb to A#.
major keys: Cb Gb Db Ab Eb Bb F C G D A E B F# C# minor keys: Ab Eb Bb F C G D A E B F# C# G# D# A#
Thus there are 30 keys using 21 notes, running from Fb to B#. But a major key often borrows from the parallel minor, and vice versa. In other words, major keys often use blue notes, and minor keys often use the harmonic or melodic minor scale. So the universe of possible notes expands to run from Abb to Gx, 27 notes. More than twice the number of notes in the edo!
But it's even worse than that, because a note can be spelled differently depending on which chord it's part of. For example, the note 3 semitones above the tonic is usually spelled as a minor 3rd. But over a Vaug chord, it's spelled as an augmented 2nd. In A, over an Eaug chord, C is spelled B#. The dim7 chord provides a common example in the fourthward direction. In A, IVdim7 has D F Ab and Cb.
Thus the default mapping of 3 semitones to the chain of 5ths is m3, and an alternate mapping is A2. Most of the 12 notes have a default mapping that is major, minor or perfect, and an alternate mapping that is augmented or diminished. The exception is the tritone, which has no default mapping and two alternate mappings A4 and d5.
To determine the proper spelling, we need a 3rd ruler, which has all the possible chord roots. There are 12 of them, but some of them have two names, for the same reason that F# major is also Gb major. Thus there are 14 possible roots:
bV bII bVI bIII bVII IV I V II VI III VII #IV #I
The interval ruler now shows not notes of the scale but notes of the chord. Thus it has to be lengthened to include d7 and A5.
d7 d4 d8 d5 m9 m13 m3 m7 P4 P1 P5 M2 M6 M3 M7 A11 A1 A5 A9
The d4 and d8 never appear in chords, but the m6 appears as a b13. The A4 shows up as a #11. The A1 never appears. The M2 shows up in sus2 chords, but also in 9th chords as a M9. The m2 appears as a flat-9. We might as well include the sharp 9th too as A9.
To spell a chord, first put the roots ruler below the notes ruler so that I lines up with the tonic. Then put the interval ruler below that, with P1 lining up with the appropriate roman numeral. The Eaug example looks like this:
notes: Cb Gb Db Ab Eb Bb F C G D A E B F# C# G# D# A# E# B# Fx Cx roots: bV bII bVI bIII bVII IV I V II VI III VII #IV #I intervals: d7 d4 d8 d5 m9 m13 m3 m7 P4 P1 P5 M9 M6 M3 M7 A11 A1 A5
The tonic is A, the root is V or E, and the aug 5th of the V chord is B#.
How many note names do we need? On one extreme, in Cb major, with a bVdim7 chord, the chord root is Gbb, and the 7th of the chord is Fbbb! On the other extreme, in A# minor, with a VIIaug chord, the root is Gx, and the 5th is D#x! From Fbbb to D#x is 46 notes!! But in practice, one would avoid triple sharps and flats by using enharmonic equivalents. The d7 would be misspelled as a M6, as Ebb. Still, there are 35 different note names for only 12 notes, almost 3 times as many names as notes.
41-edo names
In 41edo, even with only 41 names, there are triple sharps and flats. With enharmonic equivalents, there would easily be quadruple sharps and flats. This can be avoided by using ups and downs notation. Instead of a 1-D chain of 5ths, we have a 2-D lattice of the universe of all possible notes.
^^Fb ^^Cb ^^Gb ^^Db ^^Ab ^^Eb ^^Bb ^^F ^^C ^^G ^^D ^^A ^^E ^^B ^^F# ^^C# ^^G# ^^D# ^^A# ^^E# ^^B# ^Fb ^Cb ^Gb ^Db ^Ab ^Eb ^Bb ^F ^C ^G ^D ^A ^E ^B ^F# ^C# ^G# ^D# ^A# ^E# ^B# Fb Cb Gb Db Ab Eb Bb F C G D A E B F# C# G# D# A# E# B# vFb vCb vGb vDb vAb vEb vBb vF vC vG vD vA vE vB vF# vC# vG# vD# vA# vE# vB# vvFb vvCb vvGb vvDb vvAb vvEb vvBb vvF vvC vvG vvD vvA vvE vvB vvF# vvC# vvG# vvD# vvA# vvE# vvB#
To make sense of this lattice, it's good to know certain enharmonic intervals. Going 12 steps 5thwd (desc dim 2nd) and 1 step down (v1) returns you to the same note. Thus an ^d2 can be added or subtracted from any note name. Starting at F, 12 steps right = E# and 1 step down = vE#, thus vE# = F. Other enharmonic equivalences:
- v3m2 = 5 steps 4thwd and 3 steps down
- v4A1 = 7 steps 5thwd and 4 steps down
This lattice extends into double sharps/flats in certain keys. Triple sharps/flats and triple ups/downs are avoided. Thus the full universe of note names fits inside a a 5x35 lattice, about 150 names.
There are 41 possible tonics, 18 of which have alternate names in parentheses, for a total of 59 tonic names. The alternate names are overlapping keys that allow for modulation to nearby keys without radical respellings.
(^^Ab ^^Eb ^^Bb)^^F ^^C ^^G
(^Db ^Ab) ^Eb ^Bb ^F ^C ^G ^D ^A ^E ^B ^F# ^C# (^G# ^D#)
(Gb Db) Ab Eb Bb F C G D A E B F# C# G# (D# A#)
(vDb vAb) vEb vBb vF vC vG vD vA vE vB vF# vC# (vG# vD#)
vvA vvE vvB(vvF# vvC# vvG#)
- The 3 alternate names on the top row correspond to the 3 tonics on the bottom row (^^Ab = vvA)
- The 2 alternate names at the start of the 2nd row correspond to the last 2 tonics of the 3rd row (^Db = C#)
- The 2 alternate names at the end of the 2nd row correspond to the first 2 tonics of the 3rd row (^Db = C#)
The lattice of chord roots. There are 48 root names. The 6 mid roots appear twice, to show their relationship to nearby roots.
~V ~II ~VI ~III ~VII ~IV
^bV ^II ^VI ^III ^VII ^IV ^I ^V ^II ^VI ^III ^VII ^#IV ^#I
bV II VI III VII IV I V II VI III VII #IV #I
vbV vII vVI vIII vVII vIV vI vV vII vVI vIII vVII v#IV v#I
~V ~II ~VI ~III ~VII ~IV
The lattice of intervals. There are 41 default mappings, plus 4 alternate mappings, needed for the ^dim7, vdim7, ^aug, vaug and ^half-aug chords.
~5 ~2 ~6 ~3 ~7 ~4 (^^5)
(^d7) ^m2 ^m6 ^m3 ^m7 ^4 ^1 ^5 ^M2 ^M6 ^M3 ^M7
. . d5 m2 m6 m3 m7 P4 P1 P5 M2 M6 M3 M7 A4 . (A5)
(vd7) vm2 vm6 vm3 vm7 v4 v1 v5 vM2 vM6 vM3 vM7
~5 ~2 ~6 ~3 ~7 ~4
Visualize these 3 lattices as written not on wooden rulers but on transparent sheets of plastic. Instead of sliding one ruler alongside another, we slide one sheet on top of another.
THIS IS A WORK IN PROGRESS. TO BE CONTINUED...