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== How to read the scores ==
== How to read the scores ==
This guide is meant for non-guitarists.


===Notes ===
===Notes and intervals ===
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:
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:
{| class="wikitable center-all"
{| class="wikitable center-all"
Line 53: Line 54:
| style="width:40px;" |F#
| style="width:40px;" |F#
| style="width:40px;" |C#
| style="width:40px;" |C#
| style="width:40px;" |G#
| style="width:40px;" | G#
|-
|-
| -15¢
| -15¢
Line 69: Line 70:
| +15¢
| +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.) Since G# ≠ Ab, the familiar circle of 12 fifths opens up into a spiral. Because this spiral is really a circle of 41 fifths, the innermost and outermost few notes are duplicates.
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¢
 
Intervals also use arrows. D is 0¢ and E is +5¢. This tells us that a major 2nd is 205¢, so an upmajor 2nd is 235¢. 
 
These charts show the notes and intervals of 12-equal as a circle of 5ths.
[[File:12-edo circle with notes.png|left|thumb|400x400px]]
[[File:12-edo circle of 5ths.png|center|thumb|400x400px]]
 
 
 
 
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.
[[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]]
Line 78: Line 95:
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 by microtonalists historically.  


Assuming D is the anchor note, let's use that 30¢ figure to find some pitches.
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.
 
* 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 certain situations, double arrows are needed. ^^C is called '''dup'''-C and vvC# is '''dud'''-C-sharp. 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
* 2 arrows = 1 fret = half a sharp/flat
*2 arrows = 1 fret = half a sharp/flat
* 3 arrows = 1.5 frets = a minor 2nd
*3 arrows = 1.5 frets = a minor 2nd
*4 arrows = 2 frets = 1 sharp/flat = an augmented unison
* 4 arrows = 2 frets = 1 sharp/flat = an augmented unison
Since a minor 2nd and an augmented unison add up to a major 2nd, there are 7 arrows from C to D. In this table, the plain notes are bolded.
Since a minor 2nd and an augmented unison add up to a major 2nd, there are 7 arrows from C to D. In this table, the plain notes are bolded.
{| class="wikitable center-all"
{| class="wikitable center-all"
Line 104: Line 115:
| style="width:75px;" |'''C'''
| style="width:75px;" |'''C'''
| style="width:75px;" |^C
| style="width:75px;" |^C
| style="width:75px;" |^^C = vvC#
| style="width:75px;" | ^^C = vvC#
| style="width:75px;" |vC#
| style="width:75px;" |vC#
| style="width:75px;" |'''C#'''
| style="width:75px;" |'''C#'''
| style="width:75px;" | ^C#
| style="width:75px;" |^C#
| style="width:75px;" | ^^C#
| style="width:75px;" |^^C#
| style="width:75px;" |
| style="width:75px;" |
|-
|-
|
|
|vvDb
| vvDb
|vDb
|vDb
|'''Db'''
|'''Db'''
|^Db
|^Db
|^^Db = vvD
|^^Db = vvD
|vD
| vD
|'''D'''
|'''D'''
|}
|}
The three equivalences show up clearly in the 3rd column:


* ^^C = vvC# (aug 1sn = 4 arrows)
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 <u>descending</u> dim 2nd = 1 arrow.
* ^^C = vDb (min 2nd = 3 arrows)
* vvC# = vDb (descending dim 2nd = 1 arrow)


This next 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.
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 [https://kiteguitar.com/what-it-is/ "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 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.
 
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.
{| 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="2" |interval from D
|-
|-
Line 150: Line 164:
|
|
|-
|-
| vD# / Eb
|vD# / Eb
|Eb
|Eb
| -12¢
| -12¢
Line 156: Line 170:
|
|
|-
|-
|D# / ^Eb
| D# / ^Eb
|Eb
|Eb
| +17¢
| +17¢
| A1 / ^m2
|A1 / ^m2
|
|
|-
|-
|^D# / ^^Eb / vvE
|^D# / ^^Eb / vvE
| Eb
|Eb
| +46¢
| +46¢
|^A1 / ~2
|^A1 / ~2
Line 169: Line 183:
|-
|-
|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
|
|
Line 188: Line 202:
|^^E / vF
|^^E / vF
|F
|F
| -37¢
| -37¢
|vm3
|vm3
|7/6
|7/6
|-
|-
|F
|F
|F
| F
| -7¢
| -7¢
|m3
|m3
Line 199: Line 213:
|-
|-
|^F
|^F
| F
|F
| +22¢
| +22¢
|^m3
|^m3
Line 205: Line 219:
|-
|-
|^^F / vvF# / vGb
|^^F / vvF# / vGb
| F#
|F#
| -49¢
| -49¢
|~3
|~3
Line 211: Line 225:
|-
|-
|vF# / Gb
|vF# / Gb
|F#
| F#
| -20¢
| -20¢
|vM3
|vM3
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|vG
|vG
|G
|G
| -32¢
| -32¢
| v4
|v4
|
|
|-
|-
|G
| G
|G
|G
| -2¢
| -2¢
| P4
|P4
|4/3
|4/3
|-
|-
|^G
|^G
|G
| G
| +27¢
| +27¢
|^4
|^4
Line 247: Line 261:
|-
|-
|^^G / vvG# / vAb
|^^G / vvG# / vAb
|G#
| G#
| -44¢
| -44¢
|~4 / vd5
|~4 / vd5
Line 265: Line 279:
|-
|-
|^G# / ^^Ab / vvA
|^G# / ^^Ab / vvA
| G#
|G#
| +44¢
| +44¢
|^A4 / ~5
|^A4 / ~5
Line 271: Line 285:
|-
|-
|vA
|vA
| A
|A
| -27¢
| -27¢
|v5
|v5
Line 277: Line 291:
|-
|-
|A
|A
| A
|A
| +2¢
| +2¢
| P5
|P5
|3/2
|3/2
|-
|-
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|Bb
|Bb
| +49¢
| +49¢
| ^A5 / ~6
|^A5 / ~6
|
|
|-
|-
|vB
|vB
| B
|B
| -22¢
| -22¢
|vM6
|vM6
|5/3
|5/3
|-
|-
|B
| B
|B
|B
| +7¢
| +7¢
Line 327: Line 341:
|B
|B
| +37¢
| +37¢
|^M6
| ^M6
|
|
|-
|-
Line 351: Line 365:
|C#
|C#
| -46¢
| -46¢
|~7
| ~7
|
|
|-
|-
Line 368: Line 382:
|^C# / ^^Db / vvD
|^C# / ^^Db / vvD
|C#
|C#
| +41¢
| +41¢
|^M7
|^M7
|
|
|-
|-
| vD
|vD
|D
|D
| -29¢
| -29¢
Line 379: Line 393:
|-
|-
|D
|D
| D
|D
| +0¢
| +0¢
|P8
|P8
|2/1
| 2/1
|}
|}
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 [https://kiteguitar.com/what-it-is/ "What it is -- long explanation"] page for an overview.  
See also: [[Ups and downs notation]]
===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.


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 5/4 of 386¢. So one simply flattens the 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.
TO DO: complete the table
{| class="wikitable"
|+
!
!C
!up-C
!down-C
!A
!up-A
!down-A
|-
!plain major
|C D E F G A B C
|^C ^D ^E ^F ^G ^A ^B ^C
|
|
|
|
|-
!plain minor
|C D Eb F G Ab Bb C
|^C ^D ^Eb ^F ^G ^Ab ^Bb ^C
|
|
|
|
|-
!upmajor
|C D ^E F G ^A ^B C
|^C ^D ^^E ^F ^G ^^A ^^B ^C
|
|
|
|
|-
!upminor
|C D ^Eb F G ^Ab ^Bb C
|^C ^D ^^Eb ^F ^G ^^Ab ^^Bb ^C
|
|
|
|
|-
!downmajor
|C D vE F G vA vB C
|^C ^D E ^F ^G A B ^C
|
|
|
|[[File:VA vMajor keysig.png]]


Of course, not all composers use simple JI ratios in their harmonies!  
vA vB vvC# vD vE vvF# vvG#
 
|-
See also: [[Ups and downs notation]]
!downminor
 
|C D vEb F G vAb vBb C
===Key signatures===
|^C ^D Eb ^F ^G Ab Bb ^C
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. Down-A downmajor looks like this:
|
 
|
[[File:VA vMajor keysig.png]]
|
|
|}
Occasionally key signatures will have double-sharps or double-flats, e.g. Db upminor.


The scale is vA vB vvC# vD vE vvF# vvG# vA. Down-A upmajor would omit the triple stack. Down-A plain major would have a triple stack with downs instead of duds.
See also: [[Kite's Thoughts on 41edo Note Names and Key Signatures]]


=== Cancelling rules===
=== Cancelling rules===
Line 404: Line 472:
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 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 simplest way is to explicitly spell any new notes with both accidentals and arrows. Thus every 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.  
{| class="wikitable"
{| class="wikitable"
|+
|+
Line 431: Line 499:
|^^<big>♮</big>
|^^<big>♮</big>
|#
|#
|^#
| ^#
|^^#
|^^#
|-
|-
Line 445: Line 513:
|<big>♮</big>
|<big>♮</big>
|^<big>♮</big>
|^<big>♮</big>
|^^<big>♮</big>
| ^^<big>♮</big>
|
|
|^#
|^#
Line 466: Line 534:
|
|
|}
|}
Another approach seeks to reduce clutter by having arrows and accidentals behave independently of each other, and not cancel each other. Consider these 15 possible D's:
Arrows never appear alone. But when upping or downing a natural note, this rule can be relaxed, and the natural sign can be omitted.  
{| class="wikitable"
{| class="wikitable"
|+
|+
|^^Db
! rowspan="2" |start with
this
! colspan="3" |turn it into this
|-
!C
!^C
!^^C
|-
!C
|
|^
|^^
|-
!^C
|<big>♮</big>
|
|^^
|-
!^^C
|<big>♮</big>
|^
|
|}
Another approach seeks to reduce clutter by having 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"
|+
| ^^Db
|^^D
|^^D
|^^D#
| ^^D#
|-
|-
|^Db
| ^Db
|^D
|^D
|^D#
|^D#
Line 489: Line 583:
|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.
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.
{| class="wikitable"
{| class="wikitable"
|+
|+
Line 512: Line 606:
|-
|-
!^C
!^C
|◇
| ◇
|
|
|^^
|^^
Line 524: Line 618:
|
|
|◇#
|◇#
|^#
| ^#
|#
|#
|-
|-
!C#
! C#
|<big>♮</big>
|<big>♮</big>
|^<big>♮</big>
| ^<big>♮</big>
|^^<big>♮</big>
|^^<big>♮</big>
|
|
|^
|^
|^^
| ^^
|-
|-
!^C#
!^C#
Line 539: Line 633:
|<big>♮</big>
|<big>♮</big>
|^^<big>♮</big>
|^^<big>♮</big>
|◇
| ◇
|
|
|^^
|^^
Line 551: Line 645:
|
|
|}
|}
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, the most likely one is vC.
=== Chord names===
 
See also: [[Kite's Thoughts on 41edo Note Names and Key Signatures]]
===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 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.  


Line 560: Line 656:


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.


See also:
See also:
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