The Well-Tempered Clavier, Book 1 (BWV 846-869)
The prelude and fugue have a close 4:3 beat ratio, 140:108, and virtually precise 4:3 tempo ratio, 63:48, resulting in a 1:1 duration ratio. In other words, the beat ratio equals the tempo ratio resulting in a 1:1 duration ratio (4:3 beat ratio ÷ 4:3 tempo ratio = 1:1 duration ratio).
Prelude and Fugue in C Major — 1:1 duration ratio 2:15-2:15
It is also possible that Bach planned both prelude and fugue to last two minutes, in which case both tempos speed up to the next fastest possibility in the tempo matrix. At 72 bpm, the prelude has a duration of 1:56, while at 54 bpm the fugue has a duration of 2:00. If correct, then Bach planned 36 measures for the prelude. In other words, the beat ratio equals the tempo ratio resulting in a 1:1 duration ratio (4:3 beat ratio ÷ 4:3 tempo ratio = 1:1 duration ratio). Initially, this possibility seems like the correct choice because of the presence of whole number durations; however, on repeated hearings and judging with scrutiny, the tempos seem a little too fast.
Prelude and Fugue in C Major — 1:1 duration ratio — 2:00-2:00
The prelude consists of four sections that total 1:58. At its natural tempo, the fugue also has a duration of 1:58, showing a 1:1 duration ratio, 2:00-2:00.
Prelude and Fugue in C Minor — 1:1 duration ratio — 2:00:2:00
The prelude and fugue have a close 1:2 beat ratio, 104:220, and 1:1 tempo ratio, 72:72, resulting in a 1:1 duration ratio. Four more measures for the prelude, 108, and one less measure for the fugue, 54, result in a 2:1 measure ratio, 108:54, a 1:2 beat ratio, and a 1:2 duration ratio, 1:30-3:00. In other words, the beat ratio equals half the tempo ratio resulting in a 1:1 duration ratio (1:2 beat ratio ÷ 1:1 tempo ratio = 1:2 duration ratio).
Prelude and Fugue in C-sharp Major — 1:2 duration ratio — 1:30-3:00
The prelude and fugue have a close 1:1 beat ratio, 234:230, and 2:1 tempo ratio, 96:48, resulting in a close 1:2 duration ratio, 2:26-4:48. One more measure for the prelude, 40, and five more measures for the fugue, 120, result in a 1:1 beat ratio, 240:240, and a 1:2 duration ratio, 2:30-5:00. In other words, the beat ratio equals half the tempo ratio resulting in a 1:1 duration ratio (1:1 beat ratio ÷ 2:1 tempo ratio = 1:2 duration ratio).
Prelude and Fugue in C-sharp Minor — 1:2 duration ratio — 2:30-5:00
The prelude and fugue have a close 4:3 beat ratio, 140:108, and 4:3 tempo ratio, 72:54, resulting in a 1:1 duration ratio. One more measure for the prelude, 36, results in a 4:3 beat ratio, 144:108, and a 1:1 duration ratio, 2:00-2:00. In other words, the beat ratio equals the tempo ratio resulting in a 1:1 duration ratio (4:3 beat ratio ÷ 4:3 tempo ratio = 1:1 duration ratio).
Prelude and Fugue in D Minor — 1:1 duration ratio — 2:00-2:00
The prelude and fugue have a close 3:4 beat ratio, 104:132, and virtually precise 3:4 tempo ratio, 48:63, resulting in similar durations, 2:10 and 2:05. One and a half more measures for the fugue, 45.5, results in a 3:4 beat ratio, 104:138, and a 1:1 duration ratio, 2:10-2:10. In other words, the beat ratio equals the tempo ratio resulting in a 1:1 duration ratio (3:4 beat ratio ÷ 3:4 tempo ratio = 1:1 duration ratio). In this pair, Bach apparently originally planned 24 and 42 measures to achieve 2:00 for both prelude and fugue, but after overshooting the measures in the prelude by two, 26, which amounts to ten seconds, Bach then intentionally overshot the measures in the fugue by two, 44, to retain the planned 1:1 duration ratio accurately.
Prelude and Fugue in D Minor — 1:1 duration ratio — 2:10-2:10
The prelude and fugue have a close 2:1 beat ratio, 280:148, and 2:3 tempo ratio, 48:72, resulting in a close 3:1 duration ratio, 5:50-2:03. Two more measures for the prelude, 72, and one less measure for the fugue, 36, result in a 2:1 measure ratio, 72:36, a 2:1 beat ratio, 288:144, and a 1:1 duration ratio, 2:00-2:00. In other words, the beat ratio equals three times the tempo ratio resulting in a 3:1 duration ratio (2:1 beat ratio ÷ 2:3 tempo ratio = 3:1 duration ratio).
Prelude and Fugue in E-flat Major — 3:1 duration ratio — 6:00-2:00
The prelude and fugue have a close 2:3 beat ratio, 240:348, and 4:3 tempo ratio, 72:54, resulting in a close 1:2 duration ratio, 3:20-6:26. One less measure for the prelude, 39, and less three-quarters of a measure more for the fugue, 87.75, result in a 2:3 beat ratio, 234:351, and a 1:2 duration ratio, 3:15-6:30. In other words, the beat ratio equals half the tempo ratio resulting in a 1:2 duration ratio (2:3 beat ratio ÷ 4:3 tempo ratio = 1:2 duration ratio).
Prelude and Fugue in E-flat Minor — 1:2 duration ratio — 3:15-6:30
The prelude and fugue have a close 8:9 beat ratio, 96:116, and 8:9 tempo ratio, 64:72, resulting in similar durations, 1:30 and 1:36. Two fewer measures for the fugue, 27, results in an 8:9 measure ratio, 24:27, an 8:9 beat ratio, 96:108, and a 1:1 duration ratio, 1:30-1:30. In other words, the beat ratio equals the tempo ratio resulting in a 1:1 duration ratio (8:9 beat ratio ÷ 8:9 tempo ratio = 1:1 duration ratio).
Prelude and Fugue in E Major — 1:1 duration ratio — 1:30-1:30
The prelude consists of two sections in which the natural tempos result in a close 2:1 ratio between the sections that add to precisely 3:00. At its natural tempo, the fugue has a duration of 1:30, showing a 2:1 duration ratio between prelude and fugue, 3:00-1:30. Bach achieved this relationship with no margin of error, which indicates that he probably planned both sections of the prelude to have 21 measures to facilitate the 2:1 division with a total duration of three minutes: 2:00 + 1:00 = 3:00. In actual practice, Bach achieved one measure over in the first section, 22, and two measures under in the second section, 21, which average out and do not affect the planned duration of 3:00. We may learn from this that Bach’s use of Presto in the second section of the prelude does not mean “as fast as possible” (as the word is often interpreted today), but rather, “twice the speed” of the first section.
Prelude and Fugue in E Minor — 2:1 duration ratio — 3:00-1:30
The prelude and fugue have a 1:4 measure ratio, 18:72, a 1:1 beat ratio, 72:72, and 1:1 tempo ratio, 48:48, resulting in a 1:1 duration ratio, 1:30-1:30. In other words, the beat ratio equals the tempo ratio resulting in a 1:1 duration ratio (1:1 beat ratio ÷ 1:1 tempo ratio = 1:1 duration ratio).
Prelude and Fugue in F Major — 1:1 duration ratio — 1:30-1:30
The prelude and fugue have a virtually precise 3:8 beat ratio, 88:232, and 3:4 tempo ratio, 36:48, resulting in a virtually precise 1:2 duration ratio, 2:26-4:50. Half a measure more for the prelude, 22.5, and two moeasures more for the fugue, 60, result in a 3:8 beat ratio, 90:240, and a 1:2 duration ratio, 2:30-5:00. In other words, the beat ratio equals half the tempo ratio resulting in a 1:2 duration ratio (3:8 beat ratio ÷ 3:4 tempo ratio = 1:2 duration ratio).
Prelude and Fugue in F Minor — 1:2 duration ratio — 2:30-5:00
The prelude and fugue have a 7:8 beat ratio, 120:140, and 7:4 tempo ratio, 84:48, resulting in a virtually precise 1:2 duration ratio, 1:25-2:55. One and a half more measures for the prelude, 31.5, and less than half a measure more for the fugue, 37.4, result in a 1:2 duration ratio, 1:30-3:00. In other words, the beat ratio equals half the tempo ratio resulting in a 1:2 duration ratio (7:8 beat ratio ÷ 7:4 tempo ratio = 1:2 duration ratio).
Prelude and Fugue in F-sharp Major — 1:2 duration ratio — 1:30-3:00
The prelude and fugue have a close 3:8 beat ratio, 96:240, and 3:4 tempo ratio, 63:84, resulting in a close 1:2 duration ratio, 1:31-2:51. Half a measure less for the prelude, 23.5, and two more measures for the fugue, 42, result in a 3:8 beat ratio, 95:252, and a 1:2 duration ratio, 1:30-3:00. In other words, the beat ratio equals half the tempo ratio resulting in a 1:2 duration ratio (3:8 beat ratio ÷ 3:4 tempo ratio = 1:2 duration ratio).
Prelude and Fugue in F-sharp Minor — 1:2 duration ratio — 1:30-3:00
The prelude and fugue have a virtually precise 4:9 beat ratio, 76:172, and 4:3 tempo ratio, 64:48, resulting in a close 1:3 duration ratio, 1:11-3:35. One-third measure less for the prelude, 18.66, and two fewer measures for the fugue, 84, result in a 4:9 beat ratio, 75:168, and a 1:3 duration ratio, 1:10-3:30. In other words, the beat ratio equals one-third the tempo ratio resulting in a 1:3 duration ratio (4:9 beat ratio ÷ 4:3 tempo ratio = 1:3 duration ratio).
Prelude and Fugue in G Major — 1:3 duration ratio — 1:10-3:30
The prelude and fugue have a close 8:7 beat ratio, 152:136, and 8:7 tempo ratio, 72:63, resulting in virtually equal durations, 2:06 and 2:09. Three-quarters less of a measure for the fugue, 32.25, results in an 8:7 beat ratio, 152:133, and a 1:1 duration ratio, 2:06-2:06. In other words, the beat ratio equals the tempo ratio resulting in a 1:1 duration ratio (8:7 beat ratio ÷ 8:7 tempo ratio = 1:1 duration ratio).
Prelude and Fugue in G Minor — 1:1 duration ratio — 2:06-2:06
The prelude and fugue have a close 7:8 beat ratio, 132:140, and 7:4 tempo ratio, 84:48, resulting in a 1:1 duration ratio. Two more measures for the prelude, 42, and one more measure for the fugue, 36, result in a 7:8 beat ratio, 126:144, a 1:2 duration ratio, 1:30-3:00. In other words, the beat ratio equals half the tempo ratio resulting in a 1:1 duration ratio (7:8 beat ratio ÷ 7:4 tempo ratio = 1:2 duration ratio).
Prelude and Fugue in A-flat Major — 1:2 duration ratio — 1:30-3:00
The prelude and fugue have a close 1:1 beat ratio, 174:164, and 2:1 tempo ratio, 108:54, resulting in a close 1:2 duration ratio, 1:36-3:02. One less measure for the prelude, 28, and one more measure for the fugue, 42, result in a 1:1 beat ratio, 168:168, and a 1:2 duration ratio, 1:30-3:00. In other words, the beat ratio equals half the tempo ratio resulting in a 1:2 duration ratio (1:1 beat ratio ÷ 2:1 tempo ratio = 1:2 duration ratio).
Prelude and Fugue in G-sharp Minor — 1:2 duration ratio — 1:30-3:00
The prelude and fugue have a close 9:16 beat ratio, 96:162, and 9:8 tempo ratio, 63:56, resulting in a close 1:2 duration ratio, 1:31-2:53. Half a measure less for the prelude, 23.5, and two measures more for the fugue, 56, result in a 9:16 beat ratio, 94:168, and a 1:2 duration ratio, 1:30-3:00. In other words, the beat ratio equals half the tempo ratio resulting in a 1:2 duration ratio (9:16 beat ratio ÷ 9:8 tempo ratio = 1:2 duration ratio).
Prelude and Fugue in A Major — 1:2 duration ratio — 1:30-3:00
The prelude and fugue have a close 1:4 beat ratio, 84:348, and 1:1 tempo ratio, 48:48, resulting in a close 1:4 duration ratio, 1:45 and 7:15. Three fewer measures for the fugue, 84, results in a 1:4 duration ratio, 1:45-7:00. In other words, the beat ratio equals 1/4 the tempo ratio resulting in a 1:4 duration ratio (1:4 beat ratio ÷ 1:1 tempo ratio = 1:4 duration ratio).
Prelude and Fugue in A Minor — 1:4 duration ratio — 1:45-7:00
The prelude and fugue have a close 7:12 beat ratio, 80:144, and 7:12 tempo ratio, 63:48, resulting in a 1:1 duration ratio. One more measure for the prelude, 21, results in a 7:12 beat ratio, 84:144, and a 1:1 duration ratio, 2:00-2:00. In other words, the beat ratio equals the tempo ratio resulting in a 1:1 duration ratio (7:12 beat ratio ÷ 7:12 tempo ratio = 1:1 duration ratio).
Prelude and Fugue in B-flat Major — 1:1 duration ratio — 2:00-2:00
The prelude (which resembles a molto Adagio style, like the opening movement of BWV 106, and thus requires a very slow subdivided tempo with the beat on the eighth note) and fugue have a close 4:3 beat ratio, 192:150, and virtually precise 4:3 tempo ratio, 63:48, resulting in similar durations, 3:02 and 3:07. Three fewer measures for the fugue, 72, results in a 1:3 measure ratio, 24:72, a 4:3 beat ratio, 192:144, and a 1:1 duration ratio, 3:00-3:00. In other words, the beat ratio equals the tempo ratio resulting in a 1:1 duration ratio (4:3 beat ratio ÷ 4:3 tempo ratio = 1:1 duration ratio).
Prelude and Fugue in B-flat Minor — 1:1 duration ratio — 3:00-3:00
The prelude and fugue have a close 9:16 beat ratio, 76:136, and a 9:8 tempo ratio, 54:48, resulting in a precise 1:2 duration ratio. In other words, the beat ratio equals half the tempo ratio resulting in a 1:2 duration ratio (9:16 beat ratio ÷ 9:8 tempo ratio = 1:2 duration ratio).
Prelude and Fugue in B Major — 1:2 duration ratio — 1:25-2:50
The prelude and fugue have a close 2:3 beat ratio, 188:304, and virtually precise 4:3 tempo ratio, 63:48, resulting in a close 1:2 duration ratio, 2:59-6:20. Four fewer measures for the fugue, 72, results in a 1:2 duration ratio, 3:00-6:00. In other words, the beat ratio equals half the tempo ratio resulting in a 1:2 duration ratio (2:3 beat ratio ÷ 4:3 tempo ratio = 1:2 duration ratio). Bach overshot the measures of the fugue by four, which in this case amounts to twenty seconds. This example represents one of the largest margins of error, 6.7%., to be found in Bach’s works as the average for all margins of error is 2% or less.
Prelude and Fugue in B Minor — 1:2 duration ratio — 3:00-6:00 (without repeats) or 1:1 duration ratio, 6:00-6:00 (with repeats)
