Comparison of Temperaments

Below is a table comparing various temperaments, including Equal Temperament, Pythagorean, another "natural" tuning, two mean-tone tunings, several "Well" temperaments, two "Indian" tunings, two "Persian" tunings. The discussion is carried out using the key of C as example.
By compare, I mean with respect to using - say - a digital tuner to tune a note. Most comparisons of "Well" temperaments benefit more from comparison of intervals (esp. fifths, major and minor thirds), because they are attempting to achieve greatest consonance, normally using Bach's Well Tempered Clavier as a test.

Equal Temperament
divides the octave into 12 equal semitones, each spaced at a ratio of 1.05946 (being the 12th root of an octave, 2). It was propounded by Mersenne in 1636 and came into use in the late 18th century. It is not the same as Well-Temperament, made famous by Bach, despite what some more superficial texts might say.
Pythagorean
builds ratios on the pure perfect fifth (3:2), scaling back into the appropriate octave by dividing by 2, 4 or so on. So D is 9/8, being 3/2 * 3/2, lowered back in to this octave by dividing by 2. Similarly, E is 81/64. However, for a major third, the ear prefers 5/4 (80/64). This difference 81:80 in ratios is called a (syntonic) comma, and is measured at about 22 cents. A cent is simply a (logarithmic) 1/100th of an equally-tempered semitone, or the 1200th root of two.
Natural
is based on simple ratios and includes the Classic "Just" Diatonic scale. Whereas Pythagorean only allowed 3 as the highest prime, this one allowed up to 17. Variations are achieved by changing the upper limit.
Mean tone 1/4 comma
favoured pure major thirds (5:4), flattening the fifth a bit. This would have worked okay except that it gave rise to ambiguous interpretations for, say, Ab. As G# (two perfect thirds above C), its ratio should be 25:16. As Ab (a perfect third below the higher C) the ratio is 8:5. This led to instrument makers sometimes making keyboards with split black keys for playing each version, or choosing one tuning above the other according to usage, normally C#, Eb, F#, G# and Bb. Others probably became cabinet-makers instead.
Mean tone 1/6 comma
flattens the perfect fifth by sixth of a comma, sharpening the perfect third, but bringing much closer together the enharmonic (black) notes. But you'd still end up with "wolf" notes (intervals that sounded horrible, I guess). This led to ...
Ordinaire
was used in Baroque France for harpsichords and I'm told sounds very nice.
Werckmeister
is a "well" temperament, and achieved 8 pure fifths. The well temperaments would have a different feel for each key, which Bach exploited in his books "The Well Tempered Clavier". It is not known which "well" temperament he intended to be used, though Herbert Anton Kellner believes it to be an adaptation of Werckmeister III, shown below.
Kirnberger
like Werckmeister, is another "well" temperament, and achieved several pure fifths and pure thirds. Kirnberger produced three tunings, called I, II and III.
Young
and Vellotti-Young are yet more Well Temperaments, and the closest to ET that I have seen. Considered very elegant by some, it has lots of perfect fifths.
Persian
tunings for santur (sanduri, santoor, etc.) are included for comparison. They are transcribed from a tonic of E, and flattened by 15 cents so that they can be compared. Their absolute tunings, given in cents from notes derived from A=440, are given here for those people wanting to tune their santurs!
EFGABbCDEb
-15+15+30-25+15+35+20+22
+15+40+50-5+40+50+50+30
Note their predeliction for 1/4 tones!
Shrinivas and Bhatkande
are two Indian musicologists who made great inroads into systematizing Indian music. Their definitions of intervals were expressed as lengths on a string, perhaps in reference to frets on, say, the veena or sitar. It must be noted, however, that both North (Hindustani) and South Indian (Carnatic) musical systems recognise microtonal variations. At least 22 different notes were recognised, each giving rise to different feelings according to their context, but further discussion is beyond the scope of this page!
The table gives the comparison in cents (=100th of an equally tempered semitone). The ratios for Pythagorean and Natural scales are included for comparison. The enharmonic variations for mean tone tunings are also included. Note that the Persian tunings have only 8 notes instead of 12.

"C"
scale		 C Db D Eb E F F# G Ab A Bb B C
Equal temp 0 100 200 300 400 500 600 700 800 900 1000 1100 1200
Pythag 0 114
256/243		 204
9/8		 294
32/27		 408
81/64		 498
4/3		 612
729/512		 702
3/2		 816
128/81		 906
27/16		 996
16/9		 1110
243/128		 1200
2/1
Natural 0 112
16/15		 204
9/8		 316
6/5		 386
5/4		 498
4/3		 603
17/12		 702
3/2		 814
8/5		 884
5/3		 1018
9/5		 1088
15/8		 1200
2/1
Mean tone
1/4 comma 		0 76 C#
112 Db		 193 269 D#
311 Eb		 386 504 580 697 773 G#
814 Ab		 890 1007 1083 1200
Mean tone
1/6 comma		 0 89 C#
108 Db		 197 305 394 502 590 698 787 G#
806 Ab		 895 1003 1092 1200
Ordinaire 0 77 193 290 386 504 580 696 775 890 997 1083 1200
Werckmeister
III		 0 90 192 294 390 498 588 696 792 888 996 1092 1200
Werckmeister
(Kellner)		 0 90 195 294 389 498 588 697 792 892 996 1091 1200
Kirnberger 0 91 192 296 387 498 591 696 792 890 996 1092 1200
Kirnberger III 0 90 193 294 386 498 588 697 792 890 996 1088 1200
Young 0 94 196 298 392 500 592 698 796 894 1000 1092 1200
Persian 1 0 130 !! 345 !! 490 630 !! 850 !! 1035 1137 1200
Persian 2 0 125 !! 335 !! 480 625 !! 835 !! 1035 1115 1200
Shrinivas 0 133 204 316 394 498 624 702 835 906 1018 1096 1200
Bhatkhande 0 99 204 316 394 498 597 702 801 906 1018 1096 1200
"Indian"
Scale		 SA re RE ga GA MA ma PA dha DHA ni NI SA

The only thing they agree on is the octave!

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