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== How to read the scores ==
== How to read the scores ==
This is a crash course for non-guitarists.
A crash course for non-guitarists.


===Notes and intervals ===
===Notes and intervals ===
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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.
But in 41-equal, G# ≠ Ab, 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.
[[File:41-edo spiral with notes and cents.png|left|thumb|400x400px]]
[[File:41-edo spiral with notes and cents.png|left|thumb|400x400px]]
[[File:41-edo spiral.png|center|thumb|400x400px]]
[[File:41-edo spiral.png|center|thumb|400x400px]]
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In the tables and charts 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 by microtonalists historically.  
In the tables and charts 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.  


In certain situations, 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#. 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.
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#. 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.


*1 arrow = half a fret = a quarter-sharp or quarter-flat
*1 arrow = half a fret = a quarter-sharp or quarter-flat
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Of course, not all composers use simple JI ratios in their harmonies!   
Of course, not all composers use simple JI ratios in their harmonies!   


This table lists all the notes and intervals. Mid intervals, written with "~", are midway between major and minor. A few of the more important frequency ratios are shown.
This table lists all the notes and 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 [[Ratio|frequency ratios]] are shown. There are 8 familiar ones that should "lock" right away, and 3 others (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!
{| class="wikitable right-3 center-all"
{| class="wikitable right-3 center-all"
!41-equal note
!41-equal note
! colspan="2" |tuning
! colspan="2" |tuning
! colspan="2" |interval from D
! colspan="4" |interval from D
|-
|-
|D
|D
|D
|D
| +0¢
| +0¢
| rowspan="2" |1sns
|P1
|P1
|^d2
|1/1
|1/1
|-
|-
|^D
|^D
|D
|D
| +29¢
| +29¢
|^1 / vvm2 / ^^d2
|^1
|vvm2
|
|
|-
|-
|^^D / vvD# / vEb
|^^D / vvD# / vEb
|Eb
|Eb
| -41¢
| -41¢
|^^1 / vvA1 / vm2
| rowspan="7" |2nds
|vm2
|^^1 / vvA1
|
|
|-
|-
|vD# / Eb
|vD# / Eb
|Eb
|Eb
| -12¢
| -12¢
|vA1 / m2
|m2
|vA1
|
|
|-
|-
| D# / ^Eb
| D# / ^Eb
|Eb
|Eb
| +17¢
| +17¢
|A1 / ^m2
|^m2
|A1
|
|
|-
|-
|^D# / ^^Eb / vvE
|^D# / ^^Eb / vvE
|Eb
|Eb
| +46¢
| +46¢
|^A1 / ~2
|~2
|^A1
|
|
|-
|-
|vE
|vE
|E
|E
| -24¢
| -24¢
| vM2
| vM2
|
|
|
|-
|-
|E
|E
|E
|E
| +5¢
| +5¢
| M2
| M2
|
|
|
|-
|-
|^E / vvF
|^E / vvF
|E
|E
| +34¢
| +34¢
|^M2
|^M2
|
|
|
|-
|-
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|F
|F
|  -37¢
|  -37¢
| rowspan="7" |3rds
|vm3
|vm3
|
|7/6
|7/6
|-
|-
|F
|F
| F
| F
| -7¢
| -7¢
|m3
|m3
|
|
|
|-
|-
|^F
|^F
|F
|F
| +22¢
| +22¢
|^m3
|^m3
|
|6/5
|6/5
|-
|-
|^^F / vvF# / vGb
|^^F / vvF# / vGb
|F#
|F#
| -49¢
| -49¢
|~3
|~3
|
|
|
|-
|-
|vF# / Gb
|vF# / Gb
| F#
| F#
| -20¢
| -20¢
|vM3
|vM3
|
|5/4
|5/4
|-
|-
|#F / ^Gb
|#F / ^Gb
|F#
|F#
| +10¢
| +10¢
|M3
|M3
|
|
|
|-
|-
|^F# / ^^Gb / vvG
|^F# / ^^Gb / vvG
|F#
|F#
| +39¢
| +39¢
|^M3
|^M3
|
|
|
|-
|-
|vG
|vG
|G
|G
| -32¢
| -32¢
| rowspan="3" |4ths
|v4
|v4
|
|
|
|-
|-
| G
| G
|G
|G
| -2¢
| -2¢
|P4
|P4
|
|4/3
|4/3
|-
|-
|^G
|^G
| G
| G
| +27¢
| +27¢
|^4
|^4
|
|
|
|-
|-
|^^G / vvG# / vAb
|^^G / vvG# / vAb
| G#
| G#
| -44¢
| -44¢
| rowspan="4" |tri-tones
|~4 / vd5
|~4 / vd5
|
|
|
|-
|-
|vG# / Ab
|vG# / Ab
|G#
|G#
| -15¢
| -15¢
|vA4 / d5
|vA4 / d5
|
|7/5
|7/5
|-
|-
|#G / ^Ab
|#G / ^Ab
|G#
|G#
| +15¢
| +15¢
|A4 / ^d5
|A4 / ^d5
|
|
|
|-
|-
|^G# / ^^Ab / vvA
|^G# / ^^Ab / vvA
|G#
|G#
| +44¢
| +44¢
|^A4 / ~5
|^A4 / ~5
|
|
|
|-
|-
|vA
|vA
|A
|A
| -27¢
| -27¢
| rowspan="3" |5ths
|v5
|v5
|
|
|
|-
|-
|A
|A
|A
|A
| +2¢
| +2¢
|P5
|P5
|
|3/2
|3/2
|-
|-
|^A
|^A
|A
|A
| +32¢
| +32¢
|^5
|^5
|
|
|
|-
|-
|^^A / vvA# / vBb
|^^A / vvA# / vBb
|Bb
|Bb
| -39¢
| -39¢
|^^5 / vvA5 / vm6
| rowspan="7" |6ths
|vm6
|^^5 / vvA5
|
|
|-
|-
|vA# / Bb
|vA# / Bb
|Bb
|Bb
| -10¢
| -10¢
|vA5 / m6
|m6
|vA5
|
|
|-
|-
|A# / ^Bb
|A# / ^Bb
|Bb
|Bb
| +20¢
| +20¢
|A5 / ^m6
|^m6
|A5
|8/5
|8/5
|-
|-
|^A# / ^^Bb / vvB
|^A# / ^^Bb / vvB
|Bb
|Bb
| +49¢
| +49¢
|^A5 / ~6
|~6
|^A5
|
|
|-
|-
|vB
|vB
|B
|B
| -22¢
| -22¢
|vM6
|vM6
|
|5/3
|5/3
|-
|-
| B
| B
|B
|B
| +7¢
| +7¢
|M6
|M6
|
|
|
|-
|-
|^B / vvC
|^B / vvC
|B
|B
| +37¢
| +37¢
| ^M6
| ^M6
|
|
|
|-
|-
|^^B / vC
|^^B / vC
|C
|C
| -34¢
| -34¢
| rowspan="7" |7ths
|vm7
|vm7
|
|7/4
|7/4
|-
|-
|C
|C
|C
|C
| -5¢
| -5¢
|m7
|m7
|
|
|
|-
|-
|^C
|^C
|C
|C
| +24¢
| +24¢
|^m7
|^m7
|
|
|
|-
|-
|^^C / vvC# / vDb
|^^C / vvC# / vDb
|C#
|C#
| -46¢
| -46¢
| ~7
| ~7
|
|
|
|-
|-
|vC# / Db
|vC# / Db
|C#
|C#
| -17¢
| -17¢
|vM7
|vM7
|
|
|
|-
|-
|#C / ^Db
|#C / ^Db
|C#
|C#
| +12¢
| +12¢
|M7
|M7
|
|
|
|-
|-
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|  +41¢
|  +41¢
|^M7
|^M7
|
|
|
|-
|-
|vD
|vD
|D
|D
| -29¢
| -29¢
| rowspan="2" |8ves
|v8
|v8
|
|
|
|-
|-
|D
|D
|D
|D
| +0¢
| +0¢
|P8
|P8
|
| 2/1
| 2/1
|}
|}
See also: [[Ups and downs notation]]  
See also: [[Ups and downs notation]]  
===Key signatures===
===Scales and key signatures===
The key signature is divided into two regions. The sharp/flat region is as usual, except it can also have ## and bb. The arrow region consists of up to two stacks, a '''quadruple stack''' for the tonic, 2nd 4th and 5th, and a '''triple stack''' for the 3rd, 6th and 7th. Both stacks always have the exact same shape. 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 quite easy to deduce the tonic. Note that C downmajor and vA upminor have different 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 <u>3rd, 6th and 7th</u>. Thus D upminor is D E ^F G A ^Bb ^C D. If the tonic has an up or down, it's added to every note: vD upminor = vD vE F vG vA Bb C vD. The key signature is divided into two regions. The sharp/flat region is as usual, except it can also have ## and bb. The arrow region consists of up to two stacks, a '''quadruple stack''' for the tonic, 2nd 4th and 5th, and a '''triple stack''' for the 3rd, 6th and 7th. Both stacks always have the exact same shape. 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 quite easy to deduce the tonic. Note that C downmajor and vA upminor have different key signatures.


TO DO: complete the table
TO DO: complete the table
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|+
|+
!
!
!C
!D
!up-C
!up-D
!down-C
!down-D
!A
!up-A
!down-A
|-
|-
!plain major
!plain major
|C D E F G A B C
|D E F# G A B C# D
|^C ^D ^E ^F ^G ^A ^B ^C
|^D ^E ^F# ^G ^A ^B ^C# ^D
|
|vD vE vF# vG vA vB vC# vD
|
|-
|
!downmajor
|
|D E vF# G A vB vC# D
|^D ^E F# ^G ^A B C# ^D
|vD vE vvF# vG vA vvB vvC# vD
|-
|-
!plain minor
!plain minor
|C D Eb F G Ab Bb C
|D E F G A Bb C D
|^C ^D ^Eb ^F ^G ^Ab ^Bb ^C
|^D ^E ^F ^G ^A ^Bb ^C ^D
|
|vD vE vF vG vA vBb vC vD
|
|
|
|-
|-
!upmajor
!upminor
|C D ^E F G ^A ^B C
|D E ^F G A ^Bb ^C D
|^C ^D ^^E ^F ^G ^^A ^^B ^C
|^D ^E ^^F ^G ^A ^^Bb ^^C ^D
|
|vD vE F vG vA Bb C vD
|
|}
|
{| class="wikitable"
|
|+
!
!plain major
!downmajor
!plain minor
!upminor
|-
|-
!upminor
!D
|C D E F G A B C
|C D vE F G vA vB C
|C D Eb F G Ab Bb C
|C D ^Eb F G ^Ab ^Bb C
|C D ^Eb F G ^Ab ^Bb C
|-
!up-D
|^C ^D ^E ^F ^G ^A ^B ^C
|^C ^D E ^F ^G A B ^C
|^C ^D ^Eb ^F ^G ^Ab ^Bb ^C
|^C ^D ^^Eb ^F ^G ^^Ab ^^Bb ^C
|^C ^D ^^Eb ^F ^G ^^Ab ^^Bb ^C
|
|
|
|
|-
|-
!downmajor
!down-D
|C D vE F G vA vB C
|vA vB vC# vD vE vF# vG# vA
|^C ^D E ^F ^G A B ^C
|
|
|
|[[File:VA vMajor keysig.png]]
|[[File:VA vMajor keysig.png]]


vA vB vvC# vD vE vvF# vvG#
vA vB vvC# vD vE vvF# vvG# vA
|-
|vA vB vC vD vE vF vG vA
!downminor
|vA vB C vD vE F G vA
|C D vEb F G vAb vBb C
|^C ^D Eb ^F ^G Ab Bb ^C
|
|
|
|
|}
|}
Occasionally key signatures will have double-sharps or double-flats, e.g. Db upminor.
Upmajor and downminor work similarly.
 
Occasionally key signatures will have double-sharps or double-flats, e.g. vDb upminor = vDb vEb Fb vGb vAb Bbb Cb vDb.


See also: [[Kite's Thoughts on 41edo Note Names and Key Signatures]]
See also: [[Kite's Thoughts on 41edo Note Names and Key Signatures]]
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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.
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 cancel the sharp? And what if an upped C is followed by a sharpened C?
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.  
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.  
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|
|
|}
|}
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. Consider these 15 versions of D:
A 3rd 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. Consider these 15 versions of D:
{| class="wikitable"
{| class="wikitable"
|+
|+
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|
|
|}
|}
This approach often minimizes clutter greatly because of how 41edo works. The key of D downmajor has vC#. Of all the various C-naturals, the most likely one is vC.
This approach often minimizes clutter greatly because of how 41edo works. The key of D downmajor has vC#. Of all the various C-naturals, often the most likely one is vC.


See also: [[Kite's Thoughts on 41edo Note Names and Key Signatures]]
See also: [[Kite's Thoughts on 41edo Note Names and Key Signatures]]
===Chord names===
===Chord names===
Any chord name without ups or downs is exactly what one would expect it to be. 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.  
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.  


If the 3rd is downed but the minor 7th is not, the chord is named C-down add7, written Cv,7. The comma before the 7 means "add".  
If the 3rd is downed but the minor 7th is not, the chord is named C-down add7, written Cv,7. The comma before the 7 means "add".  
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Alterations are enclosed in parentheses, additions never are. For example Cv(vv#5) is C vE vvG#.
Alterations are enclosed in parentheses, additions never are. For example Cv(vv#5) is C vE vvG#.


Chord roots often have arrows. The arrows add up and cancel out as expected. Thus vA^m7 = vA C vE G, and vAvm = vA vvC vE vvG.
Chord roots often have arrows. The arrows add up and cancel out as expected. Thus vA^m = vA C vE, and vAvm = vA vvC vE.


See also:
See also:
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