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 15 possible roots. #I and bI are debatable, they might be used in I - #Im - I or Im - bI - Im.

bI  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:                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

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 containing 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#

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. 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. 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

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.

         (^^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.

     ~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 chords, the various 2nds will often be 9ths, the 3 minorish 6ths will often be b13ths, and the A4 will usually be a #11.

               ~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)

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.

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. Instead of major and minor keys, keys tend to be upward or downward. Instead of F major and G minor, we might have C-upwards and D-downwards.

Upward keys use notes 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.

Keys are up, down or plain depending on the height of the tonic: ^F is an up key, G is a plain key, etc. In fact we could say ^F is an up *tonic*. Up/down tonics are to upward/downward keys like sharp/flat tonics are to major/minor keys.

Just as minor keys tend to be sharp not flat (C# minor not Db minor), upward keys tend to be down not up. An up key that is also upwards will require double-up notes, likewise a downward down key makes double-downs. So vD-downwards is a little awkward, and either D-downwards or ^D-downwards is better.

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...