The unfortunate term Meantone seems to be derived from the traditional integer frequency ratio logic, as producing a spiral interval mapping from average deviations from integer ratios and expressed as commas.

Eg Quarter comma meantone.

Some of the systems are listed below:

Meantone tuning derived from Value of Fifth

(in cents)







1/3 comma 6/5 694.77 189.58 126.15
LucyTuning 2(^(1/(2*pi)) 695.49 190.99 122.54
Wilson's Meta-Meantone many values 695.63 191.26 121.85
Kornerup Phi phi={(5^(1/2)+1)/2}


696.21 192.43 118.93
1/4 comma 5/4 696.58 193.16 117.10
1/5 comma 15/8 697.65 195.30 111.75
12tET 1200/12 700 200 100
Pythagorean 3/2 and 4/3 701.96 203.91 90.23

for more information on Pythagorean tuning and the anomalies and lemmas of this system and

Just Intonation and other historical integer frequency ratio tunings

Many equal temperaments and circular tunings could also be considered meantones, yet after a whole number of steps of fourths or fifths, they arrive at the same position at which they began.

For independent mathematical and graphic analyses of these tuning systems

Go to for mathematical comparison of LucyTuning to 1/4 comma meantone and 12tET

Link to Tantrum (Mac program for temperament and tuning analysis and comparison)
More comparisons of LucyTuning to other meantone and equal temperaments.

Meantones are usually considered to be neutral (i.e. 700 cents for the fifth in 12tET); negative (less than 700 cents), or positive (more than 700 cents).

All meantones can be notated in conventional music notation. In negative meantones (like LucyTuning) the sharp of a note will be closer to the starting note than to the Large interval above; whereas in positive meantones it will be nearer to the Larger interval above. i.e in negative meantones L< 2s; in positive meantone L> 2s and in 12tET L=2s.

Return to Chapter One of Pitch, Pi, .....

Get Microsoft Works 4.0 adjustable spreadsheet of fretting positions, and cent + ratio values for any meantone and most xtET tunings. (

Return to LucyTuning homepage