How to read 41-equal scores
A crash course for non-guitarists, using the score conventions of the Kite guitar community.
Note names
The octave is divided into 41 equal steps, a tuning called 41-equal or 41edo or 41-ET or 41-TET. Whereas 12-equal has 100¢ steps, 41-equal has steps of 29.27¢. We can round this off to 30¢ for convenience, since a cent or two doesn't matter much in practice. This 30¢ interval is called an arrow, because the little arrows by the noteheads raise or lower the pitch by 30¢. Notes are called up-E, down-F-sharp, etc., written ^E and vF#. A note that has no ups or downs is called plain. The 7 plain natural notes are close to 12-equal, but they do deviate slightly. The pattern is easy to see when the notes are arranged in chain-of-5ths order:
| Ab | Eb | Bb | F | C | G | D | A | E | B | F# | C# | G# |
| -15¢ | -12.5¢ | -10¢ | -7.5¢ | -5¢ | -2.5¢ | 0¢ | +2.5¢ | +5¢ | +7.5¢ | +10¢ | +12.5¢ | +15¢ |
All the notes with sharps are extra-sharp. All the notes with flats are extra-flat. Note that G# is sharper than Ab by one arrow. Thus G# is also ^Ab, and Ab is also vG#. (Likewise C# = ^Db, vD# = Eb, etc.)
Let's use that 30¢ figure to find some pitches.
- D is 0¢, so ^D is 30¢ sharp and vD is 30¢ flat
- Bb is -10¢, so ^Bb is +20¢
- C is -5¢, so vC is -35¢
In the table above, D is the anchor note that agrees with standard tuning exactly. D is an ideal anchor because it makes the table symmetrical. But C, G, A and E have also been used.
Sometimes double arrows are needed. ^^C is called dup-C and vvC# is called dud-C-sharp (rhymes with cup and cud). Note that ^^C = vvC#. Thus 4 arrows = a sharp. One more equivalence: ^^C = vDb. Thus 3 arrows = a minor 2nd. These equivalences are useful, because if you've already found vDb on your instrument, and you see ^^C or vvC# on the score, you know what to play.
A major 2nd is 7 arrows. In this table, the plain notes are bolded.
| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|---|---|---|---|---|---|---|---|
| C | ^C | ^^C = vvC# | vC# | C# | ^C# | ^^C# | |
| vvDb | vDb | Db | ^Db | ^^Db = vvD | vD | D |
The interval from C# to Db is a diminished 2nd. Since this interval is 0¢ in 12-equal, musicians don't think about it much. But in 41-equal, a diminished 2nd is actually -30¢, and a descending dim 2nd = 1 arrow.
| arrows | frets | interval | equivalence |
|---|---|---|---|
| 1 | half a fret | a quarter-sharp or a descending dim 2nd | ^C = B# |
| 2 | 1 fret | half a sharp | ^^C = vvC# |
| 3 | 1.5 frets | a minor 2nd | ^^^C = Db |
| 4 | 2 frets | 1 sharp, i.e. an augmented unison | ^^^^C = C# |
In 12-equal, there are 3 "versions" of a note, for example Db, D and D# (not counting the occasional Dbb or D##). In 41-equal, there are 15 versions, running from vvDb to ^^D# (counting ^^Db and vvD separately even though they are the same note, likewise with ^^D and vvD#).
| ^^Db | ^^D | ^^D# |
| ^Db | ^D | ^D# |
| Db | D | D# |
| vDb | vD | vD# |
| vvDb | vvD | vvD# |
So how exactly would a vocalist or violinist or trombonist tune a precise number of cents sharp or flat? It helps to borrow a Kite guitar and familiarize oneself with the sound of the various intervals. It also helps to understand just intonation. See the "What it is -- long explanation" page for an overview.
In practice, use the ups and downs as a rough guide, then listen to the other parts and try to blend. For example, the downmajor 3rd from D to vF# is 380¢, very close to the just intonation 5/4 of 386¢. So to play vF# over a D, one simply flattens the 12-equal F# until the interference beats go away. But wait, should the 3rd be 380¢ or 386¢? In general, deviating slightly from 41-equal is fine, if it makes the chord smoother. The just harmonic 7th is only 3¢ sharp of the 41-equal downminor 7th, so such deviation is even less of an issue.
Beware, not all composers use simple just intonation ratios in their harmonies!
| C anchor | G anchor | D anchor | A anchor | E anchor | ||
|---|---|---|---|---|---|---|
| C | C +0¢ | C -2¢ | C -5¢ | C -7¢ | C -10¢ | C |
| ^C | C +29¢ | C +27¢ | C +24¢ | C +22¢ | C +20¢ | ^C |
| ^^C / vvC# / vDb | C# -41¢ | C# -44¢ | C# -46¢ | C# -49¢ | C +49¢ | ^^C / vvC# / vDb |
| vC# / Db | C# -12¢ | C# -15¢ | C# -17¢ | C# -20¢ | C# -22¢ | vC# / Db |
| #C / ^Db | C# +17¢ | C# +15¢ | C# +12¢ | C# +10¢ | C# +7¢ | #C / ^Db |
| ^C# / ^^Db / vvD | C# +46¢ | C# +44¢ | C# +41¢ | C# +39¢ | C# +37¢ | ^C# / ^^Db / vvD |
| vD | D -24¢ | D -27¢ | D -29¢ | D -32¢ | D -34¢ | vD |
| D | D +5¢ | D +2¢ | D +0¢ | D -2¢ | D -5¢ | D |
| ^D | D +34¢ | D +32¢ | D +29¢ | D +27¢ | D +24¢ | ^D |
| ^^D / vvD# / vEb | Eb -37¢ | Eb -39¢ | Eb -41¢ | Eb -44¢ | Eb -46¢ | ^^D / vvD# / vEb |
| vD# / Eb | Eb -7¢ | Eb -10¢ | Eb -12¢ | Eb -15¢ | Eb -17¢ | vD# / Eb |
| D# / ^Eb | Eb +22¢ | Eb +20¢ | Eb +17¢ | Eb +15¢ | Eb +12¢ | D# / ^Eb |
| ^D# / ^^Eb / vvE | E -49¢ | Eb +49¢ | Eb +46¢ | Eb +44¢ | Eb +41¢ | ^D# / ^^Eb / vvE |
| vE | E -20¢ | E -22¢ | E -24¢ | E -27¢ | E -29¢ | vE |
| E | E +10¢ | E +7¢ | E +5¢ | E +2¢ | E +0¢ | E |
| ^E / vvF | E +39¢ | E +37¢ | E +34¢ | E +32¢ | E +29¢ | ^E / vvF |
| ^^E / vF | F -32¢ | F -34¢ | F -37¢ | F -39¢ | F -41¢ | ^^E / vF |
| F | F -2¢ | F -5¢ | F -7¢ | F -10¢ | F -12¢ | F |
| ^F | F +27¢ | F +24¢ | F +22¢ | F +20¢ | F +17¢ | ^F |
| ^^F / vvF# / vGb | F# -44¢ | F# -46¢ | F# -49¢ | F +49¢ | F +46¢ | ^^F / vvF# / vGb |
| vF# / Gb | F# -15¢ | F# -17¢ | F# -20¢ | F# -22¢ | F# -24¢ | vF# / Gb |
| #F / ^Gb | F# +15¢ | F# +12¢ | F# +10¢ | F# +7¢ | F# +5¢ | #F / ^Gb |
| ^F# / ^^Gb / vvG | F# +44¢ | F# +41¢ | F# +39¢ | F# +37¢ | F# +34¢ | ^F# / ^^Gb / vvG |
| vG | G -27¢ | G -29¢ | G -32¢ | G -34¢ | G -37¢ | vG |
| G | G +2¢ | G +0¢ | G -2¢ | G -5¢ | G -7¢ | G |
| ^G | G +32¢ | G +29¢ | G +27¢ | G +24¢ | G +22¢ | ^G |
| ^^G / vvG# / vAb | G# -39¢ | G# -41¢ | G# -44¢ | G# -46¢ | G# -49¢ | ^^G / vvG# / vAb |
| vG# / Ab | G# -10¢ | G# -12¢ | G# -15¢ | G# -17¢ | G# -20¢ | vG# / Ab |
| #G / ^Ab | G# +20¢ | G# +17¢ | G# +15¢ | G# +12¢ | G# +10¢ | #G / ^Ab |
| ^G# / ^^Ab / vvA | G# +49¢ | G# +46¢ | G# +44¢ | G# +41¢ | G# +39¢ | ^G# / ^^Ab / vvA |
| vA | A -22¢ | A -24¢ | A -27¢ | A -29¢ | A -32¢ | vA |
| A | A +7¢ | A +5¢ | A +2¢ | A +0¢ | A -2¢ | A |
| ^A | A +37¢ | A +34¢ | A +32¢ | A +29¢ | A +27¢ | ^A |
| ^^A / vvA# / vBb | Bb -34¢ | Bb -37¢ | Bb -39¢ | Bb -41¢ | Bb -44¢ | ^^A / vvA# / vBb |
| vA# / Bb | Bb -5¢ | Bb -7¢ | Bb -10¢ | Bb -12¢ | Bb -15¢ | vA# / Bb |
| A# / ^Bb | Bb +24¢ | Bb +22¢ | Bb +20¢ | Bb +17¢ | Bb +15¢ | A# / ^Bb |
| ^A# / ^^Bb / vvB | B -46¢ | B -49¢ | Bb +49¢ | Bb +46¢ | Bb +44¢ | ^A# / ^^Bb / vvB |
| vB | B -17¢ | B -20¢ | B -22¢ | B -24¢ | B -27¢ | vB |
| B | B +12¢ | B +10¢ | B +7¢ | B +5¢ | B +2¢ | B |
| ^B / vvC | B +41¢ | B +39¢ | B +37¢ | B +34¢ | B +32¢ | ^B / vvC |
| ^^B / vC | C -29¢ | C -32¢ | C -34¢ | C -37¢ | C -39¢ | ^^B / vC |
| C | C +0¢ | C -2¢ | C -5¢ | C -7¢ | C -10¢ | C |
| C anchor | G anchor | D anchor | A anchor | E anchor |
See also: Ups and downs notation
Scales and key signatures
Any scale without arrows is just as one would expect. D minor is still D E F G A Bb C D. Adding up or down to the scale name alters the 3rd, 6th and 7th. Thus D upminor is D E ^F G A ^Bb ^C D. If the tonic has an arrow, it's added to every note: vD minor = vD vE vF vG vA vBb vC vD. Sometimes the scale's arrows cancel out the tonic's arrow; see ^D downmajor and vD upminor below.
The key signature is divided into two regions. The sharp/flat region is as usual, except it can also have double-sharps or double-flats (e.g. Db minor has Bbb). The arrow region consists of up to two stacks, a quadruple stack for the tonic, 2nd 4th and 5th, and/or a triple stack for the 3rd, 6th and 7th.
| tonic | |||
|---|---|---|---|
| D | up-D | down-D | |
| major
scale |
D E F# G A B C# D |
^D ^E ^F# ^G ^A ^B ^C# ^D |
vD vE vF# vG vA vB vC# vD |
| downmajor
scale |
D E vF# G A vB vC# D |
^D ^E F# ^G ^A B C# ^D |
vD vE vvF# vG vA vvB vvC# vD |
| minor
scale |
D E F G A Bb C D |
^D ^E ^F ^G ^A ^Bb ^C ^D |
vD vE vF vG vA vBb vC vD |
| upminor
scale |
D E ^F G A ^Bb ^C D |
^D ^E ^^F ^G ^A ^^Bb ^^C ^D |
vD vE F vG vA Bb C vD |
Upmajor and downminor work similarly.
The quadruple stack always has the exact same shape: two above and two below. The upper two arrows are always a 4th above the lower two. This standardized shape enables one to parse the stack as a whole. The only variations are whether the arrows point up or down, and whether they are single or double. The triple stack (two above, one below) can likewise be parsed as a whole.
The lowest arrow in the quadruple stack always indicates the tonic. The triple stack's lowest arrow indicates the 3rd, and its highest arrow indicates the 7th. Thus it's easy to deduce the tonic from either stack. Note that C downmajor and vA upminor have slightly different key signatures. The former has plain D and the latter has down D.
Modal key signatures are possible, e.g. A downmixolydian is A B vC# D E vF# vG A. Its key signature has 2 sharps and one triple stack of downs. It's like D downmajor's key signature, except that the triple stack is a 5th higher.
See also: Kite's Thoughts on 41edo Note Names and Key Signatures
Cancelling rules
Arrows behave just like accidentals. If a middle-C on the score has an up, any middle-Cs following it in the same measure inherit that up. Any C in another octave does not. If an up-C is followed by a down-C, the down-arrow cancels the up-arrow.
But what happens when accidentals are mixed with arrows? What if the key signature makes that upped C be sharp? Or what if there is a C with a sharp just before the upped C? Does the up-arrow override or "cancel" the sharp? And what if an upped C is followed by a sharpened C?
There are several possible ways to handle this issue. The default is the simplest way, to explicitly specify both arrows and accidentals every time. Thus any accidental or arrow cancels any previous ones. An arrow by itself implies a natural sign.
| start with
this |
turn it into this | |||||
|---|---|---|---|---|---|---|
| C | ^C | ^^C | C# | ^C# | ^^C# | |
| C | ^ | ^^ | # | ^# | ^^# | |
| ^C | ♮ | ^^ | # | ^# | ^^# | |
| ^^C | ♮ | ^ | # | ^# | ^^# | |
| C# | ♮ | ^ | ^^ | ^# | ^^# | |
| ^C# | ♮ | ^ | ^^ | # | ^^# | |
| ^^C# | ♮ | ^ | ^^ | # | ^# | |
Another approach has arrows and accidentals behave independently of each other, and not cancel each other. This approach is always indicated at the top of the score. Remember the 15 versions of D?
| ^^Db | ^^D | ^^D# |
| ^Db | ^D | ^D# |
| Db | D | D# |
| vDb | vD | vD# |
| vvDb | vvD | vvD# |
Adding an accidental moves you sideways. Adding an arrow moves you up or down. Thus a sharpened D followed by a downed D is vD#. This approach requires the plain sign ◇ (mnemonic: it looks like an up and a down fused together), which is analogous to a natural sign.
| start with
this |
turn it into this | |||||
|---|---|---|---|---|---|---|
| C | ^C | ^^C | C# | ^C# | ^^C# | |
| C | ^ | ^^ | # | ^# | ^^# | |
| ^C | ◇ | ^^ | ◇# | # | ^^# | |
| ^^C | ◇ | ^ | ◇# | ^# | # | |
| C# | ♮ | ^♮ | ^^♮ | ^ | ^^ | |
| ^C# | ◇♮ | ♮ | ^^♮ | ◇ | ^^ | |
| ^^C# | ◇♮ | ^♮ | ♮ | ◇ | ^ | |
This approach often minimizes clutter greatly. See also: Kite's Thoughts on 41edo Note Names and Key Signatures
Interval names
Intervals also use arrows. Since D is 0¢ and E is +5¢, a major 2nd is 205¢. Since an arrow is about 30¢, an upmajor 2nd is about 235¢.
This table lists all the intervals. There are 7 versions of every scale degree: 3 major versions, 3 minor versions, and a mid version midway between major and minor (e.g. ~2). The 7 fourths overlap with the 7 fifths, making 4 tritones. There are only 3 versions of the octave. Just like the notes, most intervals have alternate names. A few are shown here, e.g. aug unisons and aug 5ths.
11 basic frequency ratios are shown. There are 8 familiar ratios that should "lock" right away, and 3 bolded ratios (7/6, 7/5 and 7/4) that will lock with a little practice. There are many more ratios, enough to fill the whole column!
| cents | ratio | degree | name | alternate name | |||
|---|---|---|---|---|---|---|---|
| 0 | 0¢ | 1/1 | 1sns | P1 | perfect 1sn | ^d2 | updim 2nd |
| 1 | 29 | ^1 | aug 1sn | vvm2 | dudminor 2nd | ||
| 2 | 59 | 2nds | vm2 | downminor 2nd | ^^1 / vvA1 | dup 1sn | |
| 3 | 88 | m2 | minor 2nd | vA1 | downaug 1sn | ||
| 4 | 117 | ^m2 | upminor 2nd | A1 | aug 1sn | ||
| 5 | 146 | ~2 | mid 2nd | ^A1 | upaug 1sn | ||
| 6 | 176 | vM2 | downmajor 2nd | ||||
| 7 | 205 | M2 | major 2nd | ||||
| 8 | 234 | ^M2 | upmajor 2nd | ||||
| 9 | 263 | 7/6 | 3rds | vm3 | downminor 3rd | ||
| 10 | 293 | m3 | minor 3rd | ||||
| 11 | 322 | 6/5 | ^m3 | upminor 3rd | |||
| 12 | 351 | ~3 | mid 3rd | ||||
| 13 | 380 | 5/4 | vM3 | downmajor 3rd | |||
| 14 | 410 | M3 | major 3rd | ||||
| 15 | 439 | ^M3 | upmajor 3rd | ||||
| 16 | 468 | 4ths | v4 | down 4th | |||
| 17 | 498 | 4/3 | P4 | perfect 4th | |||
| 18 | 527 | ^4 | up 4th | ||||
| 19 | 556 | tritones | ~4 | mid 4th | vd5 | downdim 5th | |
| 20 | 585 | 7/5 | vA4 | downaug 4th | d5 | dim 5th | |
| 21 | 615 | A4 | aug 4th | ^d5 | updim 5th | ||
| 22 | 644 | ^A4 | upaug 4th | ~5 | mid 5th | ||
| 23 | 673 | 5ths | v5 | down 5th | |||
| 24 | 702 | 3/2 | P5 | perfect 5th | |||
| 25 | 732 | ^5 | up 5th | ||||
| 26 | 761 | 6ths | vm6 | downminor 6th | ^^5 / vvA5 | dup 5th | |
| 27 | 790 | m6 | minor 6th | vA5 | downaug 5th | ||
| 28 | 820 | 8/5 | ^m6 | upminor 6th | A5 | aug 5th | |
| 29 | 849 | ~6 | mid 6th | ^A5 | upaug 5th | ||
| 30 | 878 | 5/3 | vM6 | downmajor 6th | |||
| 31 | 907 | M6 | major 6th | ||||
| 32 | 937 | ^M6 | upmajor 6th | ||||
| 33 | 966 | 7/4 | 7ths | vm7 | downminor 7th | ||
| 34 | 995 | m7 | minor 7th | ||||
| 35 | 1024 | ^m7 | upminor 7th | ||||
| 36 | 1054 | ~7 | mid 7th | ||||
| 37 | 1083 | vM7 | downmajor 7th | ||||
| 38 | 1112 | M7 | major 7th | ||||
| 39 | 1141 | ^M7 | upmajor 7th | ||||
| 40 | 1171 | 8ves | v8 | down 8ve | |||
| 41 | 1200 | 2/1 | P8 | 8ve | |||
Chord names
Any chord name without arrows is as expected. Cm7 is still C Eb G Bb. But in practice most chords have arrows in them. An arrow between the chord root and the chord type (e.g. C^m7) raises or lowers the 3rd, and also the 6th, 7th or 11th, if present. Thus C down-seven is the usual C7 chord with the 3rd and 7th downed: Cv7 = C vE G vBb. Mnemonic: every other note of a stacked-3rds chord with a 6th below the root is affected: 6th - root - 3rd - 5th - 7th - 9th - 11th - 13th. Note that the 6th is affected, but the 13th is not.
In a dom7 chord, if the 3rd is downed but the 7th is not, the chord is named C-down add7, written Cv,7. The comma before the 7 means "add".
Analogous to M and m, "a" means augmented and "d" means diminished. Thus C^a = C ^E G# and Bvd7 = B vD F vAb. Half-diminished chords are named as dim add7 chords. For example, C ^Eb Gb ^Bb is C updim up-7, written C^d^7.
Alterations are enclosed in parentheses, additions never are. For example Cv(vv#5) is C vE vvG#. However sus chords are written C4 or C2 as usual.
Chord roots often have arrows. The arrows add up and cancel out as expected. Thus vAvm = vA vvC vE, and vA^m = vA C vE.
See also:
The circle of fifths
These charts show the notes and intervals of 12-equal as a circle of 5ths.
But in 41-equal, G# ≠ Ab, and the circle of fifths opens up into a spiral. Because this spiral is really a circle of 41 fifths, the innermost and outermost few notes are duplicates. In the 2nd chart, "-ish" means ±1 arrow.
