Preludes & Fugues and Other Works for Organ (BWV 531-590)
The prelude and fugue have a close 7:12 beat ratio, 40:74, and 7:8 tempo ratio, 63:72, resulting in a close 2:3 duration ratio, 2:32-4:06. Two more measures for the prelude, 42, and two fewer measures for the fugue, 72, result in a 7:12 beat ratio, 168:288, and a 2:3 duration ratio, 2:40-4:00. In other words, the beat ratio equals 2/3 times the tempo ratio resulting in a 2:3 duration ratio (7:12 beat ratio ÷ 7:8 tempo ratio = 2:3 duration ratio).
Prelude and Fugue in C Major — 2:3 duration ratio — 2:40-4:00
The prelude consists of three sections with different variants of the same tempo, 72 bpm, resulting in a total duration of 4:20. The fugue has 137 measures of 4/4 and a tempo of 84 bpm, resulting in a duration of 6:31. One-half less measure for the fugue, 136.5, results in a 2:3 duration ratio, 4:20-6:30.
Prelude and Fugue in D Major — 2:3 duration ratio — 4:20-6:30
The prelude and fugue have a virtually precise 6:7 measure ratio, 124:144, and 6:7 tempo ratio, 54:63, resulting in equal durations of 2:17. In other words, the beat ratio equals 2/3 times the tempo ratio resulting in a 2:3 duration ratio (6:7 beat ratio ÷ 6:7 tempo ratio = 1:1 duration ratio).
Prelude and Fugue in E Minor — 1:1 duration ratio — 2:17-2:17
The prelude and fugue have a close 8:9 beat ratio, 228:276, and 4:3 tempo ratio, 72:54, resulting in a close 2:3 duration ratio, 3:10-5:06. Four more measures for the prelude, 80, and three fewer measures for the fugue, 135, result in an 8:9 beat ratio, 240:270, and a 2:3 duration ratio, 3:20-5:00. In other words, the beat ratio equals 2/3 times the tempo ratio resulting in a 2:3 duration ratio (8:9 beat ratio ÷ 4:3 tempo ratio = 2:3 duration ratio).
Prelude and Fugue in F Minor — 2:3 duration ratio — 3:20-5:00
The prelude and fugue have a close 7:12 beat ratio, 168:308, and 7:8 tempo ratio, 63:72, resulting in a close 2:3 duration ratio, 2:40-4:16. Five fewer measures for the fugue, 72, results in a 2:3 duration ratio. 2:40-4:00. In other words, the beat ratio equals 2/3 times the tempo ratio resulting in a 2:3 duration ratio (7:12 beat ratio ÷ 7:8 tempo ratio = 2:3 duration ratio).
Prelude and Fugue in G Minor — 2:3 duration ratio — 2:40-4:00
The prelude and fugue have a close 2:9 beat ratio, 128:546, and 2:3 tempo ratio, 72:108, resulting in a close 1:3 duration ratio, 1:46-5:03. Two fewer measures for both prelude and fugue, 30 and 180, results in a 2:9 beat ratio, 120:540, 1:3 duration ratio, 1:40-5:00. (Compare these durations to those in BWV 572.) In other words, the beat ratio equals 1/3 the tempo ratio resulting in a 1:3 duration ratio (2:9 beat ratio ÷ 2:3 tempo ratio = 1:3 duration ratio).
Prelude and Fugue in A Major — 1:3 duration ratio — 1:40-5:00
The fantasia and fugue have a close 8:7 beat ratio, 288:260, and 8:7 tempo ratio, 72:63, resulting in similar durations, 4:00-4:07. Four fewer measures for the fugue, 126, results in a an 8:7 beat ratio, 288:252, 1:1 duration ratio, 4:00-4:00. 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).
Fantasia and Fugue in C Minor — 1:1 duration ratio — 4:00-4:00
The toccata and fugue have a close 8:9 beat ratio, 396:444, and 4:3 tempo ratio, 84:63, resulting in a virtually precise 2:3 duration ratio, 4:42-7:02. One less measure for the prelude, 98, and one and a half fewer measures for the fugue, 220.5, results in a an 8:9 beat ratio, 392:441, and a 2:3 duration ratio, 4:40-7:00. In other words, the beat ratio equals 2/3 the tempo ratio resulting in a 2:3 duration ratio (8:9 beat ratio ÷ 4:3 tempo ratio = 2:3 duration ratio).
Toccata and Fugue in D Minor — 2:3 duration ratio — 4:40-7:00
The fugue consists of two sections, 92 measures of 4/4 followed by a brief Adagio for 2 measures, which add to a total duration of 5:24. The prelude consists of 43 measures of 2/2 at 54 bpm, resulting in a duration of 1:48, showing a precise 1:3 duration ratio between prelude and fugue, 1:48-5:24.
Prelude and Fugue in D Minor — 1:3 duration ratio — 1:48-5:24
The toccata and fugue have a close 4:3 beat ratio, 438:340, and 8:9 tempo ratio, 56:63, resulting in a close 3:2 duration ratio, 7:49-5:23. Ten more measures for the toccata, 448, and two more measures for the fugue, 168, result in a 4:3 beat ratio, 448:336, 3:2 duration ratio, 8:00-5:20. In other words, the beat ratio equals 3/2 the tempo ratio resulting in a 3:2 duration ratio (4:3 beat ratio ÷ 8:9 tempo ratio = 3:2 duration ratio).
Toccata and Fugue in F Major — 3:2 duration ratio — 8:00-5:20
The prelude (marked Vivace) and fugue have a close 16:21 beat ratio, 246:332, and 8:7 tempo ratio, 96:84, resulting in a close 2:3 duration ratio, 2:33-3:57. Three more measures for the prelude, 85, and one more measure for the fugue, 84, result in a 16:21 beat ratio, 255:336, and a 2:3 duration ratio, 2:40-4:00. In other words, the beat ratio equals 2/3 the tempo ratio resulting in a 2:3 duration ratio (16:21 beat ratio ÷ 8:7 tempo ratio = 2:3 duration ratio).
Prelude and Fugue in G Major — 2:3 duration ratio — 2:40-4:00
The fantasia and fugue have a close 7:8 beat ratio, 392:460, and 7:8 tempo ratio, 63:72, resulting in similar durations, 6:13 and 6:23. One more measure for the fantasia, 50, and one less measure for the fugue, 114, result in a virtually precise 7:8 beat ratio, 400:456, and a 1;1 duration ratio, 6:20-6:20. In other words, the beat ratio equals the tempo ratio resulting in a 1:1 duration ratio (7:8 beat ratio ÷ 7:8 tempo ratio = 1:1 duration ratio).
Fantasia and Fugue in G Minor — 1:1 duration ratio — 6:20-6:20
The prelude and fugue have a close 2:3 beat ratio, 212:302, and a virtually precise 4:3 tempo ratio, 64:48, resulting in a close 1:2 duration ratio, 3:21-6:17. One less measure for the prelude, 52, and five more measures for the fugue, 156, result in a 1:3 measure ratio, 52:156, a 2:3 beat ratio, 208:312, 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 A Minor — 1:2 duration ratio — 3:15-6:30
The prelude and fugue have a close 3:2 beat ratio, 510:352, and a virtually precise 3:2 tempo ratio, 96:63, resulting in similar durations, 5:18 and 5:35. Three more measures for the prelude, 88, and a little over one and a half measures less for the fugue, 86.63, result in a 3:2 beat ratio, 528:347, and a 1:1 duration ratio, 5:30-5:30. In other words, the beat ratio equals the tempo ratio resulting in a 1:1 duration ratio (3:2 beat ratio ÷ 3:2 tempo ratio = 1:1 duration ratio).
Prelude and Fugue in B Minor — 1:1 duration ratio — 5:30-5:30
The prelude and fugue have a close 7:12 beat ratio, 124:222, and 7:6 tempo ratio, 63:54, resulting in a close 1:2 duration ratio, 1:58-4:06. Half a measure less for the prelude, 31.5, and three fewer measures for the fugue, 108, result in a 7:12 beat ratio, 126:216, and a 1:2 duration ratio, 2:00-4:00. In other words, the beat ratio equals half the tempo ratio resulting in a 1:2 duration ratio (7:12 beat ratio ÷ 7:6 tempo ratio = 1:2 duration ratio).
Prelude and Fugue in C Major — 1:2 duration ratio — 2:00-4:00
The prelude and fugue have a close 8:9 beat ratio, 288:318, and 8:9 tempo ratio, 48:54, resulting in similar durations, 6:00 and 5:53. Three more measure for the fugue, 162, results in an 8:9 beat ratio, 288:324, and a 1:1 duration ratio, 6:00-6:00. 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 C Minor — 1:1 duration ratio — 6:00-6:00
The prelude and fugue have a close 8:9 beat ratio, 264:288, and 8:9 tempo ratio, 56:63, resulting in similar durations, 4:42 and 4:34. One less measure for the prelude, 87, and one and a half more measures for the fugue, 73.5, results in an 8:9 beat ratio, 261:294, and a 1:1 duration ratio, 4:40-4:40. 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 C Major — 1:1 duration ratio — 4:40-4:40
The prelude and fugue have a close 7:8 beat ratio, 411:462, and 7:8 tempo ratio, 63:72, resulting in similar durations, 6:31 and 6:25. Half a measure less for the prelude, 136.5, and three more measures for the fugue, 234, result in a 7:8 beat ratio, 410:468, and a 1:1 duration ratio, 6:30-6:30. In other words, the beat ratio equals the tempo ratio resulting in a 1:1 duration ratio (7:8 beat ratio ÷ 7:8 tempo ratio = 1:1 duration ratio).
Prelude and Fugue in E Minor — 1:1 duration ratio — 6:30-6:30
The prelude and fugue have a close 1:2 beat ratio, 116:236, and 3:4 tempo ratio, 54:72, resulting in a virtually precise 2:3 duration ratio, 2:08-3:16. One less measure for the fugue, 58, results in a 1:2 beat ratio, 116:232, and a 2:3 duration ratio, 2:08-3:12. In other words, the beat ratio equals 2/3 the tempo ratio resulting in a 2:3 duration ratio (1:2 beat ratio ÷ 3:4 tempo ratio = 2:3 duration ratio).
Prelude and Fugue in E Minor — 2:3 duration ratio — 2:08-3:12
The prelude consists of two sections, a longer main section of 58 measures in 3/2 and a brief concluding section of 3 measures in 4/4, which add to 3:26. The fugue, marked Alla breve e staccato, has 149 measures of 2/2 and a tempo of 84 bpm, resulting in a duration of 3:32. This shows a 1:1 duration ratio, 3:30-3:30.
Prelude and Fugue in G Major — 1:1 duration ratio — 3:30-3:30
The prelude and fugue have a close 3:4 beat ratio, 152:204, and 8:7 tempo ratio, 72:63, resulting in a virtually precise 2:3 duration ratio, 2:06-3:14. One less measure for the fugue, 50, results in a 16:21 beat ratio, 152:200, 2:3 duration ratio, 2:08-3:12. In other words, the beat ratio equals 2/3 the tempo ratio resulting in a 2:3 duration ratio (16:21 beat ratio ÷ 8:7 tempo ratio = 2:3 duration ratio).
Prelude and Fugue in A Minor — 2:3 duration ratio — 2:06-3:10
The prelude consists of 205 measures of 2/2 at 54 bpm, resulting in a duration of 7:35. The fugue consists of three sections each with a different meter and natural tempo, as well as a 36:45:36 measure ratio and 2:3:4 tempo ratio, 72:108:144, resulting in a total duration of 7:30 divided as 2:00-2:30-3:00. Two and a half fewer measures for the prelude, 202.5, result in a duration of 7:30, suggesting that Bach originally intended 203 measures.
Prelude and Fugue in E-flat Major — 1:1 duration ratio — 7:30-7:30
Fugue consists of three sections — 2:00 + 2:30 + 3:00 = 7:30
The fantasia consists of 81 measures of 4/4 at 72 bpm, resulting in a duration of 4:30. The fugue is unfinished after 27 completed measures, which allows us to speculate as to how many measures Bach planned. Zoltán Göncz (please refer to the YouTube video on his completion of the fugue) concluded that Bach planned a double fugue with one subject and four countersubjects. Göncz’s completion of the fugue with a total of 89 measures is impressive. If correct, then Bach probably intended a duration of 6:45 for the fugue to create a 2:3 duration ratio with the fantasia, 4:30-6:45. Assuming a slightly faster tempo of 84 bpm for the fugue (Bach almost never intended the same tempo for preludes and fugues, but rather, one is usually a little slower or faster than the other), this means that the fugue should have a total of 95 measures to achieve 6:45. To hear how 84 bpm sounds for the fugue, listen to Göncz’s completion and adjust the playback speed to .75.
Fantasia in C Minor — duration of 6:00 — Bach probably planned the unfinished fugue to also have a duration of 6:00.
The fantasia and imitation have a close 16:27 beat ratio, 192:336, and 8:9 tempo ratio, 96:108, resulting in a close 2:3 duration ratio, 2:00-3:06. Four fewer measures in the imitation, 108, results in a 16:27 beat ratio, 192:324, and a 2:3 duration ratio, 2:00-3:00. In other words, the beat ratio equals 2/3 the tempo ratio resulting in a 2:3 duration ratio (16:27 beat ratio ÷ 8:9 tempo ratio = 2:3 duration ratio).
Fantasia con Imitatione in B Minor — 2:3 duration ratio — 2:00-3:00
The toccata and fugue have a beat ratio, 336:282, and 9:8 tempo ratio, 63:56, resulting in similar durations, 5:20 and 5:02. The Adagio section of the second movement has a duration of 2:40, which is half of the duration of the toccata, 5:20, showing a 2:1 duration ratio, 5:20-2:40. In addition, when the Grave section is added on, the second movement has a total duration of 3:24. Three fewer measures for the toccata, 81, and three more measures for the fugue, 144, result in a 9:8 beat ratio, 324:288, and a 1:1 duration ratio, 5:08-5:08, showing this three-movement work to have a symmetrical 3:2:3 duration ratio, 5:08-3:24-5:08.
Toccata, Adagio, and Fugue in C Major — 3:2:3 duration ratio — 5:08-3:24-5:08
The toccata, consisting of two sections, has a total duration of 2:03. The recitativo has a duration of 1:02. The fugue proper consists of 97 measures of 4/4 at 96 bpm (one of if not the fastest of all Bach’s fugues), resulting in a duration just over 4:02. One less measure for the fugue, 96, results in a duration of 4:00, showing an overall plan of 2:00-4:00-1:00 which adds to seven minutes, 7:00.
Toccata and Fugue in D Minor — three sections — 2:00 + 1:00 + 4:00 = 7:00
This four-section prelude and fugue are like two preludes and fugues combined. The first and second movements have a virtually precise 3:8 beat ratio, 132:356, and 3:4 tempo ratio, 63:84, resulting in a virtually precise 1:2 duration ratio, 2:05-4:14. Three-quarters of a measure less for the fugue, 88.25, results in a 1:2 duration ratio of 2:06-4:12. 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). The fourth movement, the final fugue section, consists of 96 measures of 3/4 at 96 bpm, resulting in a duration of three minutes, 3:00. Assuming 54 bpm for the remaining third movement, a brief prelude of 11 measures in 4/4, results in a duration of 0:48 which when rounded off to 0:45 is one-fourth the length of the concluding fugue section, 3:00. In sum, the four sections are divided into two duration pairs, 1:2 and 1:4.
Prelude and Fugue in E Major — 1:2 duration ratio, 2:06-4:12 — 1:4 duration ratio, 0:45-3:00.
The first and third movements, Trés vitement and Lentement, have a close 6:7 beat ratio, 112:136, and 6:7 tempo ratio, 72:84, resulting in virtually equal durations, 1:33 and 1:37. In other words, the beat ratio equals the tempo ratio resulting in a 1:2 duration ratio (6:7 beat ratio ÷ 6:7 tempo ratio = 1:1 duration ratio). The second movement, Gravement, consists of 157 measures of 2/2 at 63 bpm, resulting in a duration of 4:59. Two more measures for the first movement, 30, and half a measure more for the second and third movements, 157.5 and 17.5, result in a symmetrical 1:3:1 duration ratio, 1:40-5:00-1:40. (Compare these durations to those in BWV 536.)
Fantasia in G Major — three sections have a 1:3:1 duration ratio — 1:40-5:00-1:40
The passacaglia and fugue have a close 21:16 ratio of beats, 504:372, and 7:8 tempo ratio, 63:72, resulting in a close 3:2 duration ratio, 8:00-5:10. Four more measures for the fugue, 128, result in a 21:16 duration ratio, 504:384, and a 3:2 duration ratio of 8:00-5:20. In other words, the beat ratio equals 3/2 the tempo ratio resulting in a 3:2 duration ratio (21:16 beat ratio ÷ 7:8 tempo ratio = 3:2 duration ratio). Having fallen four measures short of the 128-measure ideal in the fugue and needing to extend it by about ten seconds provides the best explanation for Bach's Adagio indication in the penultimate measure.
Passacaglia and Fugue in C Minor — 3:2 duration ratio — 8:00-5:20
The first and fourth movements have a close 8:7 beat ratio, 148:124, and 8:7 tempo ratio, 64:56, resulting in similar durations, 2:18 and 2:12. One less measure for the first movement, 36, and one more measure for the fourth movement, 63, result in a 1:1 duration ratio, 2:15-2:15. 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). The second and third movements, (Andante) and (Adagio), have a close 7:16 beat ratio, 88:192, and 7:8 tempo ratio, 63:72, resulting in a close 1:2 duration ratio of 1:23-2:40. One less measure for the second movement, 21, results in a 7:16 beat ratio, 84:192, and a 1:2 duration ratio, 1:20-2:40. In other words, the beat ratio equals half the tempo ratio resulting in a 1:2 duration ratio (7:16 beat ratio ÷ 7:8 tempo ratio = 1:2 duration ratio).
Pastorale in F Major — four movements organized into two duration pairs — 1:1 duration ratio between the first and fourth movements, 2:15-2:15 — 1:2 duration ratio between the second and third movements, 1:20-2:40 (without repeats), or a 1:1 duration ratio, 2:40-2:40 (with repeats)
