Kite's Thoughts on 41edo Note Names and Key Signatures
How to name every note in 41-edo? In certain styles of music, e.g. free jazz or avant-guarde classical, there are not well-defined chords, and the exact spelling of a note doesn't really matter. In these styles, one can get away with only one name for each note. But in other styles, there is a clear sense of key, with definite chords and chord progressions. In those cases, it's important to spell e.g. C minor as C Eb G and not C D# G. This article assumes the latter style of music, and covers how to name a note so that it agrees with both the key and the chord root. It also covers key signatures. Consider it a style guide for tonal 41-edo music. First, let's consider conventional 12-edo, and see what lessons we can draw from it.
Background: 12-edo names
Even though there's only 12 notes, the names are selected from a much larger chain of fifths (like a 1-D lattice):
the chain of fifths ...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 of 12-edo 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 using relative names not absolute names, i.e. intervals not letters:
the relative chain of fifths ...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 (absolute): Gb Db Ab Eb Bb F C G D A E B F# C# G# D# A# interval ruler (relative): 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
One might think we could limit possible tonics to only the 12 most central notes, say Ab to C#. But other tonics are needed 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 their counterparts D# minor and Eb minor as well, for a total of 26 keys to consider. There are only 24 actual keys, 12 of each type, but 2 keys have 2 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#.
all possible tonics: 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 15 key signatures (see below). This also 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 universe of possible note names expands to run from Gbb (dim 5th in Cb major) to Gx (major 7th in A# minor), 29 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 from the tonic" 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 of notes in a chord, we need a third 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 A6
The d4 and d8 never appear in chords, but the m6 appears as a b13. The A4 shows up as a #11. The A1 or A8 almost never appears. The M2 shows up in sus2 chords, but also in 9th chords as a M9. The m2 appears as a flat-9. The aug 9th appears in the Hendrix chord, and the aug 6th in aug 6th chords.
Every chord type (sus4, min7, etc.) can be mapped to this ruler. The root is always P1. The min7 chord falls mostly on the fourthward half of the ruler.
min7: d7 d4 d8 d5 m9 m13 m3 m7 P4 P1 P5 M2 M6 M3 M7 A11 A1 A5 A9 A6
The dim7 chord is even more fourthward.
dim7: d7 d4 d8 d5 m9 m13 m3 m7 P4 P1 P5 M2 M6 M3 M7 A11 A1 A5 A9 A6
The aug chord is very fifthward:
aug: d7 d4 d8 d5 m9 m13 m3 m7 P4 P1 P5 M2 M6 M3 M7 A11 A1 A5 A9 A6
The sus4 chord is dead center:
sus4: d7 d4 d8 d5 m9 m13 m3 m7 P4 P1 P5 M2 M6 M3 M7 A11 A1 A5 A9 A6
Some chord mappings like the sus4 are quite compact, and some like the dim7 are quite farflung. Every chord type has a fifthspan which measures this. The fifthspans of these examples are: min7 = 4, dim7 = 9, aug = 8 and sus4 = 2.
Spelling a Chord
To spell a chord, we use all 3 rulers. Let's do the Vaug example in A. First put the roots ruler below the notes ruler so that the I lines up with the tonic.
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
Then put the interval ruler below that, with P1 lining up with the appropriate roman numeral.
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 and the root is V or E. The aug chord has a major 3rd which is G# and an augmented 5th which is B#.
We're used to defining a chord by the intervals from the root to each note. But there are also intervals among the non-root notes. For example, a tetrad has six internal intervals. Ideally, all the internal intervals are spelled according to their default mapping, i.e. not as augmented or diminished intervals (except the tritone of course). Alternate mappings are inevitable for chords with a fifthspan of 7 or more. In such chords, the interval between the most 4thwd note and the most 5thwd note is forced to be aug or dim. An obvious example is the various augmented and diminished chords. Another example is M7b9. The interval from the 7th up to the 9th is spelled as a d3.
A curious special case is the Hendrix chord. It's usually spelled P1 M3 m7 A9. The interval from the 7th up to the 9th is a P4, but it's spelled as an A3. Spelling it as a P4 would make the A9 become a m10. But that would make the interval from the 3rd to the 9th change from a M7 to a d8. Thus we must choose between an A3 and a d8. Looking at the interval lattice, the A9 is five steps away from the nearest chord note (the M3), but the m10 is only one step away from m7. The m10 would make a smaller fifthspan. But there is a rule that a chord can have only one note of each degree. In other words, each chord note must have its own letter name. In C, the M3 and m10 would both use the letter name E.
Although chords with a large fifthspan may force the use of internal aug and dim intervals, they should never be used otherwise. No Bb C# F chord!
How bad do the note names get? 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! But in practice, one generally avoids triple sharps and flats by using enharmonic equivalents. (Triple sharps and flats do appear in classical music, but very rarely.)
For symmetrical chords like aug and dim7, we can respell the chord as one of its homonyms without changing the chord type. The d7 of the Gbbdim7 chord would be spelled as a M6, as Ebb. This respells Gbbdim7 as its homonym Ebbdim7. Alternatively, every note in the chord might be respelled by respelling the root. The bV root might become #IV, and Gbbdim7 might become Fdim7.
How many note names do we need? Avoiding triple sharps and flats, there are up to 35 different note names for only 12 notes, almost 3 times as many names as notes!
Atonal music that doesn't use a specific scale or have an identifiable tonic needn't use key signatures, and usually doesn't. The rest of this section concerns tonal music.
There are many possible key signatures. Each of the seven degrees can be sharp, flat or natural, thus the possibilities number 3^7 = 2187. However the number is limited to only 15. One reason for this is that many arbitrary combinations of sharps and flats won't be meaningful musically, like E# and Fb, or B# and C. But besides that, every key signature indicates a scale that the musician is expected to practice and memorize. Thus key signatures are limited to major and minor scales. (It's possible but rare for key signatures to indicate other diatonic modes, such as using a two-sharp key signature for E dorian. In the Baroque era this was more common.) There are 15 major and minor key signatures for a total of 30 scales.
|key signature||major key||major scale||minor key||minor scale|
|♭♭♭♭♭♭♭||C♭ major||C♭ D♭ E♭ F♭ G♭ A♭ B♭ C♭||A♭ minor||A♭ B♭ C♭ D♭ E♭ F♭ G♭ A♭|
|♭♭♭♭♭♭||G♭ major||G♭ A♭ B♭ C♭ D♭ E♭ F G♭||E♭ minor||E♭ F G♭ A♭ B♭ C♭ D♭ E♭|
|♭♭♭♭♭||D♭ major||D♭ E♭ F G♭ A♭ B♭ C D♭||B♭ minor||B♭ C D♭ E♭ F G♭ A♭ B♭|
|♭♭♭♭||A♭ major||A♭ B♭ C D♭ E♭ F G A♭||F minor||F G A♭ B♭ C D♭ E♭ F|
|♭♭♭||E♭ major||E♭ F G A♭ B♭ C D E♭||C minor||C D E♭ F G A♭ B♭ C|
|♭♭||B♭ major||B♭ C D E♭ F G A B♭||G minor||G A B♭ C D E♭ F G|
|♭||F major||F G A B♭ C D E F||D minor||D E F G A B♭ C D|
|no sharps / flats||C major||C D E F G A B C||A minor||A B C D E F G A|
|♯||G major||G A B C D E F♯ G||E minor||E F♯ G A B C D E|
|♯ ♯||D major||D E F♯ G A B C♯ D||B minor||B C♯ D E F♯ G A B|
|♯ ♯ ♯||A major||A B C♯ D E F♯ G♯ A||F♯ minor||F♯ G♯ A B C♯ D E F♯|
|♯ ♯ ♯ ♯||E major||E F♯ G♯ A B C♯ D♯ E||C♯ minor||C♯ D♯ E F♯ G♯ A B C♯|
|♯ ♯ ♯ ♯ ♯||B major||B C♯ D♯ E F♯ G♯ A♯ B||G♯ minor||G♯ A♯ B C♯ D♯ E♯ F G♯|
|♯ ♯ ♯ ♯ ♯ ♯||F♯ major||F♯ G♯ A♯ B C♯ D♯ E♯ F♯||D♯ minor||D♯ E♯ F♯ G♯ A♯ B C♯ D♯|
|♯ ♯ ♯ ♯ ♯ ♯ ♯||C♯ major||C♯ D♯ E♯ F♯ G♯ A♯ B♯ C♯||A♯ minor||A♯ B♯ C♯ D♯ E♯ F♯ G♯ A♯|
On a 41edo chain of fifths, we get triple sharps and flats with just 41 notes. With enharmonic equivalents, there would easily be quadruple sharps and flats. We can avoid this 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.
note names ^^Fbb ^^Cbb ^^Gbb ^^Dbb ^^Abb ^^Ebb ^^Bbb ^^Fb ^^Cb ^^Gb ^^Db ^^Ab ^^Eb ^^Bb ^^F ^^C ^^G ^^D ^^A ^^E ^^B ^^F# ^^C# ^^G# ^^D# ^^A# ^^E# ^^B# ^^Fx ^^Cx ^^Gx ^^Dx ^^Ax ^^Ex ^^Bx ^Fbb ^Cbb ^Gbb ^Dbb ^Abb ^Ebb ^Bbb ^Fb ^Cb ^Gb ^Db ^Ab ^Eb ^Bb ^F ^C ^G ^D ^A ^E ^B ^F# ^C# ^G# ^D# ^A# ^E# ^B# ^Fx ^Cx ^Gx ^Dx ^Ax ^Ex ^Bx Fbb Cbb Gbb Dbb Abb Ebb Bbb Fb Cb Gb Db Ab Eb Bb F C G D A E B F# C# G# D# A# E# B# Fx Cx Gx Dx Ax Ex Bx vFbb vCbb vGbb vDbb vAbb vEbb vBbb vFb vCb vGb vDb vAb vEb vBb vF vC vG vD vA vE vB vF# vC# vG# vD# vA# vE# vB# vFx vCx vGx vDx vAx vEx vBx vvFbb vvCbb vvGbb vvDbb vvAbb vvEbb vvBbb vvFb vvCb vvGb vvDb vvAb vvEb vvBb vvF vvC vvG vvD vvA vvE vvB vvF# vvC# vvG# vvD# vvA# vvE# vvB# vvFx vvCx vvGx vvDx vvAx vvEx vvBx
Unless your screen is very wide, that last lattice is probably too wide to fit, and wraps around in an ugly way. So here's a screenshot:
The 5 rows are the double-up row, the up row, the plain row, the down row, and the double-down row. The lattice has five 5x7 blocks: double-flat, flat, natural, sharp and double-sharp. Tonics come from the middle three blocks only. The outer two blocks are only partially used, and Fbb, Cbb, Ex and Bx would probably never be needed.
To make sense of this lattice, it's good to know certain enharmonic equivalences:
- 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)
Each letter A through G appears in the lattice 25 times. This table shows the 25 different "versions" of each letter:
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 the A-440 standard. 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#.
Here's 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 ^bII ^bVI ^bIII ^bVII ^IV ^I ^V ^II ^VI ^III ^VII ^#IV ^#I bI bV bII bVI bIII bVII IV I V II VI III VII #IV #I vbI vbV vbII vbVI vbIII vbVII vIV vI vV vII vVI vIII vVII v#IV v#I ~V ~II ~VI ~III ~VII ~IV
Here's the lattice of intervals. Again, there is an up row, a plain row, a down row, and a (duplicated) mid row. Every possible chord type maps to this lattice. There are 41 default mappings, plus 5 alternate mappings, needed for the ^dim7, vdim7, ^aug, vaug, ^half-aug and vhalf-aug chords. The half-aug 5th is shown as both ^^5 and vvA5. The spelling matches the 3rd. 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 v8 v5 vM2 vM6 vM3 vM7 ~5 ~2 ~6 ~3 ~7 ~4 . (vvA5)
There can be other alternate mappings. For example, a major flat-5 chord may be tuned 10/(10:8:7), and the 10/7 may need to be spelled not as A4 but as ^d5.
In 12-edo, to avoid triple-sharps/flats, one can rename symmetrical chords like aug or dim7 as a homonym, i.e. Gbbdim7 becomes Ebbdim7. But 41 is a prime number, and there are no symmetrical chords in 41-edo.
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 via the ^d2 enharmonic). 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, depending on how they map to the interval lattice. The first syllable of the chord name tells you how to classify it: upminor, downdim, etc. There are a few plain chords like sus4 that are neither, and a few hybrid chords like v,^7 that are both. Just as a major key song can have minor chords, so can a downward key song have upward chords.
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. For example, a simple Iv7 - IVv7 - Vv7 song is in a downward key.
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", which starts in E-upward and ends in G-downward.
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 would be 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 about one edostep! 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 mid 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. For a plain natural tonic, this works fairly well. But what if the tonic is down, the root is down, and the chord is downwards? Can we avoid a triple-down note name? Yes, by respelling the entire chord via the v4A1 enharmonic. Thus in vD, a downmajor chord on vII is written not as vvE vvvG# vvB but as ^^Eb ^G ^^Bb. This is a little misleading, because the root is not vII but ^^^bII. Thus what should be a majorish II appears as a minorish II. To clarify things, one might add a note on the staff just above the chord "^^Eb = vvE" or perhaps "^^Ebv = vvEv".
It would be possible to write the vIIv chord as vvE ^G vvB. But this is like writing 12-edo C minor as C D# G. The intervals within a chord must always be spelled properly!
In 12-edo, three root movements of a major 3rd lead back to the tonic. This would be written somewhat misleadingly as CM7 - EM7 - AbM7 - CM7. But the musician is expected to recognize the relationship between EM7 and AbM7, and to be aware of the common tones they have. Here too, one might clarify with "Ab = G#".
The universe of note names, whether in 12-edo or 41-edo, is only so large. When a song goes past the edge of the universe, the notation has to adjust. Another example is a 41-edo Latrizo aka Slendric pump: Cv7 - ^Dv7 - vFv7 - Gv7 - Cv7. Just as the 12-edo musician is expected to know that CM7 - EM7 - AbM7 is the exact same root movement twice, an accomplished 41-edo musician should see Cv7 - ^Dv7 - vFv7 similarly. The score might have "vF = ^^E".
So we can use the v4A1 enharmonic to remove any triple-ups or downs. But there are times when we need to also use v3m2 or ^d2. For example, in the key of vEb, a vbIIvm chord would be vvFb vvvAbb vvCb. To remove the triple up, subtract a v4A1. That gives us ^^Fbb ^Abbb ^^Cbb. To remove the triple-flat, subtract an ^d2 to get ^Eb Gb ^Bb. The staff might have "vvFb = ^Eb". Rather than subtracting twice, we could have simply subtracted v3m2 from vvFbvm once. Note that the vbII root is spelled as ^^I.
Every chord type maps to the interval lattice. In 12edo, the width of this mapping gives us the chord's fifthspan. In 41edo, besides the fifthspan, there is a rowspan that is the height of the mapping. You can tell how many rows a chord maps to directly from its name. Most chords map to two rows. All-plain chords like sus2 or sus4 map to one row. The v^7 chord (P1 vM3 P5 ^m7) ia a hybrid chord that is both upwards and downwards, and is a three-row chord. There are other three-row chords, for example the two half-aug chords ^(^^5) and v(vv#5). Note that the half-aug 5th of 26\41 is spelled differently in these two chords to avoid internal triple-up/down intervals.
Three-row chords are tricky to notate because one of the four rows of the root lattice won't work. For example, in the key of ^^C, a Vv^7 is ^^G ^B ^^D ^^^F. But in vvC#, it's vvG# vvvB# vvD# vF#. Either way, triples are needed. Because when it goes over the edge, moving it four rows vertically simply causes it to go over the other edge. The solution is to use the ^d2 to turn it into a vAb chord, spelled vAb vvC vEb Gb. But it won't do to have the V chord of ^^C or vvC# be a vAb chord! So the entire key must be respelled as vDb. But a Vv(vv#5) chord in this key would be vAb vvC vvvE, with triple-downs. (In ^^C it would be a much neater ^^G ^B D#.) So I^m - Vv^7 - I^m - Vv(vv#5) - I in this key breaks the notation, even though it's a reasonable chord progression. In practice, the halfaug chord might be written as vAb vvC ^Eb.
A four-row chord like ^mv7(~5) is possible but rare. One of the internal intervals must have a triple up/down.
When using ups and downs on the staff, arrows (ups, downs and plains) can be either dependent or independent. If dependent, they can never appear alone. They must appear with a conventional accidental (a sharp, flat or natural). A conventional accidental cancels an arrow. So if the key signature or an earlier arrow sets F to downsharp, a natural sign sets it to plain F natural.
If independent, arrows can appear alone, and change the height (up, down, plain, etc.) without affecting the sharpness. A new symbol is required, a plain sign, which is a small square rotated 45 degrees, i.e. a diamond shape. (Mnemonic: it looks like an up fused to a down.) If F defaults to vF#, F with an up sign is ^F#. Taking this further, a conventional accidental doesn't affect the height, so F with a natural sign is vF. To get plain F, one puts both a natural and a plain in front of the F notehead. Thus arrows and conventional accidental function as two completely separate layers.
There is a long history of using dependent arrows, in fact 81/80 arrows are often literally fused to a conventional accidental. But independent arrows result in a less cluttered score, at least for 41-edo.
Before we can discuss key signatures, we need to decide which 7-note scales they should indicate. Two obvious choices are the plain major and plain minor scales, which are exactly the same as their 12-equal counterparts. Another obvious one is the 5-limit Ptolomeic scale. Ups and downs notation has a convention that adding "up-" or "down-" to a chord name or a scale name affects only the 3rd, 6th and 7th. So the Ptolomeic scale is the downmajor scale. It's utonal counterpart is the upminor scale. The obvious 7-limit no-fives scales are the upmajor and downminor scales. These 6 scales imply a 7th one, an 11-limit no-fives no-sevens scale, the mid scale, which can be written out as either vvMajor or ^^Minor.
How to choose between the two mid key sigs? Triple arrows are avoided in the key sig, so if the tonic is an upped note, avoid ^^Minor. If it's downed, avoid vvMajor. If it's plain, in general choose the key sig with the fewest sharps or flats. For example, D mid is usually better as D ^^Minor (one flat) than as D vvMajor (two sharps).
The lattice image of these 7 scales is shown here. The 4 bolded notes are common to all scales.
dbl-upminor --> ^^m6 ^^m3 ^^m7 upminor --> ^m6 ^m3 ^m7 ^M6 ^M3 ^M7 <-- upmajor minor --> m6 m3 m7 P4 P1 P5 M2 M6 M3 M7 <-- major downminor --> vm6 vm3 vm7 vM6 vM3 vM7 <-- downmajor vvM6 vvM3 vvM7 <-- dbl-downmajor
For yaza music, using independent intervals lets one borrow from the parallel major/minor mode of the same height with a minimum of clutter. For example, in downmajor, one can borrow from downminor without using any downs. In C downmajor, E, A and B are downed, and Eb, Ab and Bb are automatically downed as well. Cv7 appears on the staff as C E G Bb, without any downs.
As with 12-edo, there are many possible key signatures, but in practice they are limited to only major and minor scales. As with 12-edo, sharp keys have sharps running fifthward from F#, and flat keys have flats running fourthward from Bb. Arrows are added in.
I initially tried (as other have done) integrating the arrows into the sharps and flats. For example, A downmajor's key sig has three down-sharps in the F, C and G spots. D downmajor added a down-natural before the F spot on the B line: vNatB v#F v#C. vE vMinor has vF# vvC vvG vvD vA vE vB. The main drawback to this approach is that deducing the tonic and the scale from the key sig takes a lot of effort.
Then Aaron Wolf had a great idea: separate the arrows from the sharps and flats. The key sig is divided into two regions. The #/b region is as usual, except it can also have ## and bb. The arrow region consists of up to two arrow stacks, a quadruple one for the tonic, 2nd 4th and 5th, and a triple one for the 3rd, 6th and 7th. Both stacks always have the same shape. The quadruple stack has 2 columns of arrows with the lowest arrow indicating the tonic. The triple stack has the lowest arrow on the 3rd. vA vMajor looks like this:
The scale is vA vB vvC# vD vE vvF# vvG# vA. Down-A ^Major would omit the triple stack. Plain-A vvMajor would omit the quadruple stack. Plain-A plain major would omit both. Down-A plain major would have a triple stack with single downs instead of double downs.
The two arrow stacks always have the exact same shape, and each should be parsed as a single object. The only variation is the height: ^^ vs. ^ vs. v vs. vv. For other tonics besides A, the arrow stacks are placed vertically so that ledger lines are avoided. The placement of each stack's lowest arrow ranges from the low D up to the central C. For tonics D, E, F, G and A (including sharpened/flattened/upped/downed versions of these 5 letters), the triple stack is higher than the quadruple stack. For all versions of tonics B and C, it's lower.
The tonic's letter is easily found from the quadruple stack: it's the lowest arrow. if there is no quadruple stack, the tonic is one step above the triple stack's highest arrow, or two steps below its lowest arrow. The tonic's height is the same as the height of the quadruple stack. If there is no quadruple stack, the tonic is plain. Once the tonic letter is found, the sharpness/flatness (e.g. B vs. Bb) is obvious from the #/b region. The mode (major vs. minor) is found as usual from the number of sharps or flats. Finally, the type of major/minor is the difference in height between the two arrow stacks.
For example, in the above picture, the quadruple stack tells us that the tonic is some kind of A, in fact vA because the stack has single downs. The #/b region has 3 sharps, so we have vA major. Finally the triple stack has double-downs, which is one more down than the quadruple stack's single-downs, so it's vA downmajor.
Speaking of modes, note that vA ^Minor (vA vB vD vE) is distinct from C vMajor (vE vA vB). Thus the tonic is immediately apparent, unlike conventional notation in which one has to decide between major and minor.
In general, other modes (dorian, lydian, etc.) should be avoided, for simplicity's sake. But they could possibly be indicated by adding or removing sharps and flats. For example, removing 1 sharp changes the key from vA vMajor to vA vMixolydian = vA vB vvC# vD vE vvF# vvG vA.
How many keys are there? 7 scales x 41 tonics = 287 musically distinct keys. How many key signatures are there? There are 65 possible tonics, and 8 scales x 65 = 520 key sigs. 8 scales not 7 because vvMajor and ^^Minor are different key sigs. But not all of these would be used, because they would result in triple arrows. Excluding those makes 440 key sigs.
A Software Solution
EarTrainer41 doubles as a 41-edo style guide. You can select a tonic and see all 8 key sigs (plain/up/down major, plain/up/down minor, plus two mid ones). All 41 notes are named relative to the tonic. You can toggle the tonic to display enharmonic equivalences, e.g. you can go from G# downmajor to ^Ab downmajor.