A Cosine-derived Well Temperament

Ever since I read about Brad Lehman's "Bach" tuning (some time in 2006), I have been interested in experimenting with unequal "well" temperaments for keyboard instruments.  But, it was only recently that I got a hardware synthesizer with reasonable capabilities for custom tunings.

While there seems to be insufficient proof that Lehman's tuning truly recreates anything used by J. S. Bach, it does adhere to the general goal that's implied in other historical well temperaments I've since read about:  A gradual change in "coloration" of the tuning as you go around the circle of fifths, with the most "just" major third (i.e. flat of equal-tempered) occurring in the key of C and the most "Pyhagorean" major third (i.e. sharp of ET) at the other side of the circle at F♯.  The width of the fifths in these temperaments appear to have been derived by trial and error to close the circle while achieving the desired thirds.  Precision was of course limited by the mechanical precision of the piano and the capabilities of the human tuner.

I decided to develop a well temperament mathematically, using a cosine function to adjust the width of the fifths gradually.  I used a spreadsheet to derive the thirds, and below are links (to Google Sheets) and a copy of the output, with two variations. The amount of stretch was chosen so that the widest fifths would be just (i.e. 3:2, about 701.955¢). For each of version, the flattest third (C-E) ends up about 7¢ sharp of just, and the sharpest third is about 20¢ sharp of just (but not quite Pythagorean).

Centered on C, with just fifths at C♯-G♯ and G♯-D♯:

-->

Note	Index	Cosine	Fifth	Degree	M3		∆ Just
C	0	0.7071	698.568840	0.000000	C-E:	393.227678	6.913964
G	1	0.9659	698.044999	698.568840	G-B:	394.134997	7.821284
D	2	0.9659	698.044999	196.613839	D-F#:	396.613839	10.300125
A	3	0.7071	698.568840	894.658838	A-C#:	400.000000	13.686286
E	4	0.2588	699.476159	393.227678	E-G#:	403.386161	17.072447
B	5	-0.2588	700.523841	1092.703837	B-D#:	405.865003	19.551289
F#	6	-0.7071	701.431160	593.227678	F#-A#:	406.772322	20.458608
C#	7	-0.9659	701.955001	94.658838	C#-F:	405.865003	19.551289
G#	8	-0.9659	701.955001	796.613839	G#-C:	403.386161	17.072447
D#	9	-0.7071	701.431160	298.568840	D#-G:	400.000000	13.686286
A#	10	-0.2588	700.523841	1000.000000	A#-D:	396.613839	10.300125
F	11	0.2588	699.476159	500.523841	F-A:	394.134997	7.821284

Centered between C and G, with a just fifth at G♯-D♯:

-->

Note	Index	Cosine	Fifth	Degree	M3		∆ Just
C	0	0.5000	699.022500	0.000000	C-E:	393.681338	7.367624
G	1	0.8660	698.306920	699.022500	G-B:	393.681338	7.367624
D	2	1.0000	698.044999	197.329419	D-F#:	395.374418	9.060704
A	3	0.8660	698.306920	895.374418	A-C#:	398.306920	11.993206
E	4	0.5000	699.022500	393.681338	E-G#:	401.693080	15.379367
B	5	0.0000	700.000000	1092.703837	B-D#:	404.625582	18.311868
F#	6	-0.5000	700.977500	592.703837	F#-A#:	406.318662	20.004948
C#	7	-0.8660	701.693080	93.681338	C#-F:	406.318662	20.004948
G#	8	-1.0000	701.955001	795.374418	G#-C:	404.625582	18.311868
D#	9	-0.8660	701.693080	297.329419	D#-G:	401.693080	15.379367
A#	10	-0.5000	700.977500	999.022500	A#-D:	398.306920	11.993206
F	11	0.0000	700.000000	500.000000	F-A:	395.374418	9.060704

Here are the tunings in Scala format if you would like to use them:

! Cosine C Well.scl

!

C-Centered Cosine Well

 12

!

 94.65884

 196.61384

 298.56884

 393.22768

 500.52384

 593.22768

 698.56884

 796.61384

 894.65884

 1000.00000

 1092.70384

 2/1

! Cosine CG Well.scl

!

C/G-Centered Cosine Well

 12

!

 93.68134

 197.32942

 297.32942

 393.68134

 500.00000

 592.70384

 699.02250

 795.37442

 895.37442

 999.02250

 1092.70384

 2/1

Incidentally, these are rather nice-looking when viewed through Scala's temperament radar:

C-Centered Cosine Well Temperament - Radar

C/G-Centered Cosine Well Temperament - Radar