Kite's Thoughts on 41edo Note Names and Key Signatures
How to name any note in any chord on any root in any key?
Background: 12-edo names
First, let's consider conventional 12-edo. A larger chain of fifths (like a 1-D lattice) contains more than 12 names:
Gb Db Ab Eb Bb F C G D A E B F# C# G# D# A#
This chain makes a circle of 12 fifths with duplicate names in spots, but it's easier to read when written out on a line.Any major scale consists of the tonic along with 1 note immediately to the left and 5 notes to the right. In other words, the scale is a 7-note section of the chain of fifths, and the tonic is the 2nd note in the chain.
For example, A major uses the chain from D to G#. But the other 5 notes can also occur, so they need names too. For now, let's get them from the left. That means the names that go with A major are the 12 notes running from Eb to G#.
Here's the same chain of 5ths in relative names, i.e. as intervals not letters:
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 interval ruler selects a portion of the absolute ruler that has ugly double sharps.
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
It makes sense to stick to tonics from only the 12 most central notes, say Ab to C#. But others get used in practice 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 keys to consider. Sonically, 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#
This gives us 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. The names we could make expand 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 15 possible roots. #I and bI are debatable, but they might be used in I - #Im - I or Im - bI - Im.
chord roots: bI bV bII bVI bIII bVII IV I V II VI III VII #IV #I
The interval ruler now shows notes of the chord instead of the scale. Thus, it has to be lengthened to include d7 and A5.
chord intervals: 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: bI 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
Lattices
On a 41edo chain of fifths, we get triple sharps and flats with just the plain 41 names. With enharmonic equivalents, there would easily be quadruple sharps and flats. We can keep simpler names by using ups and downs notation. Instead of a 1-D chain of 5ths, we have a 2-D lattice containing the universe of all possible note names.
note names ^^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#
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 5x35 lattice. The 5 rows are the double-up row, the up row, the plain row, the down row, and the double-down row.
To make sense of this lattice, it's good to know certain enharmonic intervals:
- v3m2 = 5 steps 4thwd and 3 steps down (^^B = vC)
- v4A1 = 7 steps 5thwd and 4 steps down (^^F = vvF#)
- ^d2 = 12 steps 4thwd and 1 step up (vvF# = vGb)
There are 41 possible tonics, with 24 alternate names in parentheses, for a total of 65 tonic names. The alternate names are overlapping keys that allow for modulation to nearby keys without radical respellings.
tonics
(^^Ab ^^Eb ^^Bb) ^^F ^^C ^^G (^^D ^^A)
(^Db ^Ab) ^Eb ^Bb ^F ^C ^G ^D ^A ^E ^B ^F# ^C# (^G# ^D# ^A#)
(Gb Db) Ab Eb Bb F C G D A E B F# C# G# (D# A#)
(vGb vDb vAb) vEb vBb vF vC vG vD vA vE vB vF# vC# (vG# vD#)
(vvG vvD) vvA vvE vvB (vvF# vvC# vvG#)
Because D is so central, this is the pitch that is tuned to A-440. The 13 plain tonics from Ab to G# are the 13 notes in 41edo that are closest to 12edo.
The ^d2 enharmonic connects the start of each row to the end of the row below it. The v4A1 enharmonic connects the top and bottom rows. The v3m2 enharmonic connects the end of the top row with the start of the 4th row. Note that ^^Ab = vvA = ^G#.
The lattice of chord roots. There are 51 root names. The top row and bottom row are identical. It appears twice merely to show the relationship of mid roots to nearby roots. So there is an up row, a plain row, a down row and a mid row.
chord roots
~V ~II ~VI ~III ~VII ~IV
^bI ^bV ^II ^VI ^III ^VII ^IV ^I ^V ^II ^VI ^III ^VII ^#IV ^#I
bI bV II VI III VII IV I V II VI III VII #IV #I
vbI 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. Again, there is an up row, a plain row, a down row, and a (duplicated) mid row. There are 41 default mappings, plus 5 alternate mappings, needed for the ^dim7, vdim7, ^aug, vaug and vhalf-aug chords. The half-aug 5th is shown as both ^^5 and vvA5. In actual chords, the various 2nds will often be 9ths, the 3 minorish 6ths will often be b13ths, and the A4 will usually be a #11.
chord intervals
~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 . (vvA5)
There can be other alternate roots. For example, a major flat-5 chord may be tuned 10/(10:8:7), and the 10/7 may be spelled not as A4 but as ^d5.
Visualize these 3 lattices as written not on wooden rulers but on transparent sheets of plastic. Instead of sliding one ruler alongside another, visualize sliding one sheet on top of another. But before we examine the note names this produces, we need some new terminology.
Upwards and Downwards
In 41-EDO, the conventional duality of major and minor keys tends to disappear, because 7-limit harmony uses both majorish and minorish intervals: Cv7 has a vM3 and a vm7. ("-ish" means ±1 edostep, so majorish includes upmajor, downmajor and plain major.) Even a simple Iv7 - IVv7 - Vv7 song's "footprint" in the interval lattice is so wide that both majorish and minorish are included. The new duality is upward vs. downward. Instead of keys like F major and G minor, we might have C-upwards and D-downwards.
Upward keys use intervals from the up row of the interval lattice, i.e. upmajor and upminor. (Not upperfect, it's too wolfy, and not updiminished, that's just major or augmented). Upmajor is 7-under and upminor is 5-under, thus upward keys are subharmonic or utonal. Downward keys use downmajor and downminor, which are 7-over and 5-over, thus downward keys are otonal. Chords can also be classified as upward or downward. There are a few plain chords like sus4 that are neither, and a few hybrid chords like Cv,^7 that are both.
12-edo keys are heptatonic and extend horizontally either 4thwd (minor, phrygian) or 5thwd (major, lydian) from the tonic. 41-edo keys tend to be both. But they usually extend vertically only one row either upwards or downwards. Obviously, a song needn't stay strictly in one key. Analogous to borrowing from or modulating to the parallel major or minor in 12-edo, one can borrow from or modulate to the parallel upward or downward key. For example, I^m7 - IV^9 - I^m7 - IV^9 - Vv7 - I^m7 borrows a downward chord. Analogous to borrowing from or modulating to the relative major or minor, one can borrowing from or modulate to a relative upward and downward key. See Kite's translation of "Manhattan Island Serenade".
In 12-edo, keys could be classified as sharp, flat or natural depending on the tonic: Eb is a flat key, F and G are natural keys, etc. (This has nothing to do with whether there are sharps or flats in the key signature. Both G major and G minor are natural keys.) In 41-edo, keys are up, down, plain or mid depending on the height of the tonic: ^F is an up key, G is a plain key, and both ^^F and vvB are mid keys. Up/down keys are to upward/downward keys like 12-edo sharp/flat keys are to major/minor keys. Just as 12-edo minor keys are easier to notate in natural or sharp keys than flat keys, 41-edo upward keys are easier to notate in plain or down keys than up keys. An up key that is also upwards will require double-up notes, likewise a downward down key needs double-downs. The notation for vD-downwards is a little awkward, but doubled ups/downs are allowed, and sometimes inevitable. Sometimes one can respell a down key as a plain key, e.g. vC#-downwards can become Db-downwards. Often one can't. If not, in practice one can often simply use a nearby plain key one edostep away. The vocalist certainly won't complain! Thus vD-downwards might become D-downwards as a matter of convenience.
All this can be extended to 11-limit harmonies, which use mid intervals. Mid keys and chords use the mid row of the interval lattice. There is a relative mid, but no parallel mid, because two mid intervals add up to a plain interval.
The Universe of Note Names
Back to visualizing sliding one lattice on top of another. If the tonic is down, the root is down, and the chord is downwards, it might seem a triple-down note name is needed. But triple-downs are avoided by respelling via the v4A1 enharmonic. Thus in vD, a downmajor chord on vII is written not as vvE vvvG# vvB but as vvE ^G vvB or ^^Eb ^G ^^Bb. The former
To avoid lots of double-sharps and double-flats, there are six keys that are double-up or double-down. A double-down tonic like vvB works OK for upwards keys, but vvB-downward creates triple downs. Triple downs aren't allowed: vvBv is spelled vvB - ^D - vvF#. The vvB - ^D interval looks minorish but it's majorish. It's a downmajor 3rd spelled as a triple-up minor 3rd. That's a little misleading, so vvB-downward gets renamed either ^^Bb-downward or ^A#-downward. The chord becomes either ^^Bb - ^D - ^^F or ^A# - Cx - ^E#.
THIS IS A WORK IN PROGRESS. TO BE CONTINUED...