GOLDEN MEANTONE TUNING

Developed by Thorvald Kornerup and advocated by Jacques Dudon, this tuning is based on Phi, the golden mean, equal to (5^1/2+1) / 2 or 1.61803398875. (Phi is a unique constant. It is the ratio of two quantities, so that the smaller is to the greater as the greater is to the sum of the greater and the smaller. It is one larger than its own reciprocal. It is the mean of quotients of adjacent elements in a Fibonacci series. Phi is the ratio of width to height of many Greek temples, because ancient architects considered it pleasant to the eye. Whether it is as pleasing to the ear is left to listeners' taste.)

The whole tone is 2^0.160357456, or 1.117564002. Closure to equal temperament is roughly approached along a Fibonacci series of {5, 7, 12, 19, 31, 50, 81, . . .}. (See definition and comparative data.)

Golden Meantone lends itself to scales of various sizes. After the first few tones, each additional increment divides the existing intervals in a ratio close to 3:2 or 5:3, efficiently filling gaps. (Most regular tunings add new tones close to old ones, leaving uneven gaps.) Golden Meantone may provide advantages in combining different tuning systems into one ensemble.

Interval (above tonic)	Ratio	Value in diapasons	Value in cents	Corresponding just interval	Difference in diapasons from just interval	Difference in cents from just interval
5th	1.49503	0.58018	696.214	3 / 2	-0.00478	-5.741
4th	1.33776	0.41982	503.786	4 / 3	+0.00478	+5.741
Major 2nd	1.11756	0.16036	192.429	9 / 8	-0.00957	-11.481
Minor 7th	1.78961	0.83964	1007.571	16 / 9	+0.00957	+11.481
Major 6th	1.67080	0.74054	888.643	5 / 3	+0.00357	+4.285
Minor 3rd	1.19703	0.25946	311.357	6 / 5	-0.00357	-4.285
Major 3rd	1.24895	0.32071	384.858	5 / 4	-0.00121	-1.456
Minor 6th	1.60135	0.67929	815.142	8 / 5	+0.00121	+1.456
Major 7th	1.86722	0.90089	1081.072	15 / 8	-0.00600	-7.196
Minor 2nd	1.07111	0.09911	118.928	16 / 15	+0.00600	+7.196
Augmented 4th	1.39578	0.48107	577.287	7 / 5	-0.00435	-5.225
(End of 12-note scale)
Diminished 5th	1.43289	0.51893	622.713	10 / 7	+0.00435	+5.225
Augmented 1st	1.04337	0.06125	73.501	25 / 24	+0.00236	+2.829
Diminished 8th	1.91687	0.93875	1126.499	48 / 25	-0.00236	-2.829
Augmented 5th	1.55987	0.64143	769.716	14 / 9	+0.00400	+4.800
Diminished 4th	1.28215	0.35857	430.284	9 / 7	-0.00400	-4.800
Augmented 2nd	1.16603	0.22161	265.930	7 / 6	-0.00078	-0.941
Diminished 7th	1.71522	0.77839	934.070	12 / 7	+0.00078	+0.941
(End of 19-note scale)
Augmented 6th	1.74326	0.80179	962.145	7 / 4	-0.00557	-6.681
Diminished 3rd	1.14728	0.19821	237.855	8 / 7	+0.00557	+6.681
Augmented 3rd	1.30312	0.38197	458.359	64 / 49	-0.00332	-3.989
Diminished 6th	1.53478	0.61803	741.641	49 / 32	+0.00332	+3.989
Augmented 7th	1.94820	0.96214	1154.574	35 / 18	+0.00279	+3.344
Diminished 2nd	1.02659	0.03786	45.426	36 / 35	-0.00279	-3.344
Double-augmented 4th	1.45632	0.54232	650.788	16 / 11	+0.00176	+2.106
Double-diminished 5th	1.37333	0.45768	549.212	11 / 8	-0.00176	-2.106
Neutral 2nd	1.08862	0.12250	147.003	12 / 11	-0.00303	-3.634
Neutral 7th	1.83719	0.87750	1052.997	11 / 6	+0.00303	+3.634
Neutral 6th	1.62753	0.70268	843.217	18 / 11	-0.00781	-9.375
Neutral 3rd	1.22886	0.29732	356.783	11 / 9	+0.00781	+9.375
(End of 31-note scale)
Interval (above tonic)	Ratio	Value in diapasons	Value in cents	Corresponding just interval	Difference in diapasons from just interval	Difference in cents from just interval

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Last revised: 10 September 2006

visitors since 23 February 1999