Musical Offerings, Fall 2018

68 True ⦁ Intonation and Modulation done by equating two ratios: four fifths (x 4 ) compared to two octaves and a major third (2x2 x 5/4). Basically, this defines the distance from C3 to E5 via fifths (think C3 to G3 to D4 to A4 to E5) to equal the distance from C3 to E5 via two octaves and a third (C3 to C4 to C5 to E5). This definition is shown below mathematically, where x represents the interval of a fifth. x 4 = 2 2 (5/4) x 4 = 5 x = 5 1/4 = 1.49535… Thus, for 1/4 comma meantone temperament, the fifths are not 3/2 = 1.50000 but are instead slightly less. 2/7 comma meantone and 1/3 comma meantone are defined similarly but with the fifth lowered by slightly different amounts. The exact similarities and differences between these temperaments can be seen below. Figure 4 : Comparison of Comma Temperaments . 23 Aron’s 1/4 Comma Temperament Zarlino’s 2/7 Comma Temperament Salina’s 1/3 Comma Temperament Note Name Distance from Pythagorean Intonation Cents Distance from Pythagorean Intonation Cents Distance from Pythagorean Intonation Cents C 0 0 0 0 0 0 C -7/4 76 -2 70 -7/3 64 D -1/2 193 -4/7 191 -2/3 190 Eb -3/4 310 +6/7 313 +1 316 E -1 386 -8/7 383 -4/3 379 F +1/4 503 +2/7 504 +1/3 505 F -3/2 579 -12/7 574 -2 569 G -1/4 697 -2/7 696 -1/3 695 G -2 773 -16/7 817 -8/3 758 A -3/4 890 -6/7 887 -1 884 Bb +1/2 1007 +4/7 1008 +2/3 1010 B -5/4 1083 -10/7 1078 -5/3 1074 C 0 1200 0 1200 0 1200 23 Barbour, Tuning and Temperament: A Historical Survey , 26.