Definitions of tuning terms

© 1998 by Joseph L. Monzo

All definitions by Joe Monzo unless otherwise cited


1/4-comma meantone

The most frequently encountered version of meantone tuning. This tuning narrows or flattens each '5th' in a series of '5ths' by 1/4 of a syntonic comma, hence its name. This results in 'perfect 5ths' of approximately 697 cents or 6.97 Semitones.

Meantone tunings can be and have been extended to as many notes per 'octave' as desired, but when used for fretted strings or, especially, keyboards, they are usually limited to 12 discrete pitches per 'octave'.

Regardless of how many notes the meantone has, comparing the two notes at either end results in a 'wolf 5th' and a 'wolf 4th', as can be seen in the following two tables. The 'wolves' occur between the two '5ths' or '4ths' which bound the notes at the end of the series of '5ths'.

In the tables below, the reference pitch (1/1) is 'A', and the series of '5ths' runs arbitrarily from 3-6 (= 'Eb') to 35 (= G#). The first table is an interval matrix of all possible intervals between any two pitches in this tuning. The second table lists all available intervals and the notes between which they can be found.


Interval Matrix of 12-tone 1/4-comma meantone

interval sizes given in Semitones

      G# G F# F E Eb D C# C B Bb A
    3^x 5 -2 3 -4 1 -6 -1 4 -3 2 -5 0
G#   5 0.00 11.24 10.07 9.31 8.14 7.38 6.21 5.03 4.27 3.10 2.34 1.17
G   -2 0.76 0.00 10.83 10.07 8.90 8.14 6.97 5.79 5.03 3.86 3.10 1.93
F#   3 1.93 1.17 0.00 11.24 10.07 9.31 8.14 6.97 6.21 5.03 4.27 3.10
F   -4 2.69 1.93 0.76 0.00 10.83 10.07 8.90 7.73 6.97 5.79 5.03 3.86
E   1 3.86 3.10 1.93 1.17 0.00 11.24 10.07 8.90 8.14 6.97 6.21 5.03
Eb   -6 4.62 3.86 2.69 1.93 0.76 0.00 10.83 9.66 8.90 7.73 6.97 5.79
D   -1 5.79 5.03 3.86 3.10 1.93 1.17 0.00 10.83 10.07 8.90 8.14 6.97
C#   4 6.97 6.21 5.03 4.27 3.10 2.34 1.17 0.00 11.24 10.07 9.31 8.14
C   -3 7.73 6.97 5.79 5.03 3.86 3.10 1.93 0.76 0.00 10.83 10.07 8.90
B   2 8.90 8.14 6.97 6.21 5.03 4.27 3.10 1.93 1.17 0.00 11.24 10.07
Bb   -5 9.66 8.90 7.73 6.97 5.79 5.03 3.86 2.69 1.93 0.76 0.00 10.83
A   0 10.83 10.07 8.90 8.14 6.97 6.21 5.03 3.86 3.10 1.93 1.17 0.00


List of Intervals of 12-tone 1/4-comma meantone

diminished 8ve 11.24 G#:G F#:F E:Eb C#:C B:Bb              
major 7th 10.83 G:F# F:E Eb:D D:C# C:B Bb:A A:G#          
minor 7th 10.07 G#:F# G:F F#:E F:Eb E:D D:C C#:B C:Bb B:A A:G    
augmented 6th 9.66 Eb:C# Bb:G#                    
diminshed 7th 9.31 G#:F F#:Eb C#:Bb                  
major 6th 8.90 G:E F:D E:C# Eb:C D:B C:A B:G# Bb:G A:F#      
minor 6th 8.14 G#:E G:Eb F#:D E:C D:B C#:A B:G A:F        
augmented 5th 7.73 F:C# Eb:B C:G# Bb:F#                
wolf 5th 7.38 G#:Eb                      
'perfect' 5th 6.97 G:D F#:C# F:C E:B Eb:Bb D:A C#:G# C:G B:F# Bb:F A:E  
diminished 5th 6.21 G#:D F#:C E:Bb C#:G B:F A:Eb            
augmented 4th 5.79 G:C# F:B Eb:A D:G# C:F# Bb:E            
'perfect' 4th 5.03 G#:C# G:C F#:B F:Bb E:A D:G C#:F# C:F B:E Bb:Eb A:D  
wolf 4th 4.62 Eb:G#                      
diminished 4th 4.27 G#:C F#:Bb C#:F B:Eb                
major 3rd 3.86 G:B F:A E:G# Eb:G D:F# C:E Bb:D A:C#        
minor 3rd 3.10 G#:B G:Bb F#:A E:G D:F C#:E C:Eb B:D A:C      
augmented 2nd 2.69 F:G# Eb:F# Bb:C#                  
diminished 3rd 2.34 G#:Bb C#:Eb                    
major 2nd 1.93 G:A F#:G# F:G E:F# Eb:F D:E C:D B:C# Bb:C A:B    
minor 2nd 1.17 G#:A F#:G E:F D:Eb C#:D B:C A:Bb          
augmented unison 0.76 G:G# F:F# Eb:E C:C# Bb:B              
unison / 8ve 0 or 12 G#:G# G:G F#:F# F:F E:E Eb:Eb D:D C#:C# C:C B:B Bb:Bb A:A

Last Updated 2000.8.10
by Joe Monzo

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