Overview of the Inventions

The 15 Inventions first appeared as “Preambles” in a different order in the Notebook for Wilhelm Friedemann Bach. Analysis reveals a deliberate plan of order and symmetry in the Inventions applying the same tempos to both sets, suggesting that Bach organized the Inventions and Preambles simultaneously. Such an impressive mathematical feat could be understood as the “preamble-invention conundrum,” in which Bach the puzzle-solver worked out a likewise orderly and quasi-symmetrical plan for the Inventions. In essence, Bach found two solutions for a puzzle using the same fifteen pieces.

The 15 Inventions are contained within equal outer parameters, since the C major and B minor preambles consist of the same number of measures and have the same tempo. Between these two ends, which symbolize The Alpha & The Omega, the inventions are organized into twelve duration pairs, which also contribute to four multi-movement duration ratios: 2:3:2:2, 2:3:2, 1:1:1:1, 3:2:3. These occur in a symmetrical fashion with the first four and last four inventions having the following symmetrical layout: 1:20-2:00-1:20-1:20 | 1:20-1:20-2:00-1:20. Eight of the fifteen inventions have durations close to 1:20 and three of the fifteen have durations close to 2:00, showing a prominence of 2:3 and 3:2 duration ratios of 1:20-2:00 or 2:00-1:20. It is also significant that the next two inventions inwards from the beginning and end, nos. 2 and 14, have equal durations which creates a 2:3 ratio between the first two inventions and a 3:2 ratio between the last two inventions, which shows further symmetry. (2:3 and 3:2 are symmetrical.)

To summarize, the 15 Inventions are organized in a highly symmetrical fashion permeated throughout with 1:1, 1:2, 2:3, 3:2 as well as three and four-way 2:3:2:2, 2:3:2, and 1:1:1:1 duration ratios. Eleven of the fifteen inventions, nos. 1-4 and 8-15, have just two durations – 1:20 and 2:00 – which relate 2:3. The remaining three inventions, nos. 5-7, relate 2:3:2 at 1:30-2:15-1:30. This is all contained within equal inner parameters, nos. 3 and 13, at 1:30, nestled within equal inner parameters, nos. 2 and 14, at 2:00, nestled within equal outer parameters, nos. 1 and 15, at 1:20. This multiple-level symmetry symbolizes The Alpha & The Omega. (See Diagrams 1 and 2.)

Diagram 1. Temporal Organization in the 15 Preambles

Diagram 2. Temporal Organization in the 15 Inventions

Duration Ratios in the Inventions

This section breaks down and discusses all the invention pairs and groups and their duration ratios as shown in Diagram 2.

Inventions 1 & 15

Inventions 1 and 15 have equal measures, 22, and equal tempos, Q = 63, resulting in equal durations. One less measure for both inventions, 21, results in equal durations of 1:20. The equal measures, tempo and durations for the first and last inventions imbue the set with outer-parameter symmetry which symbolizes *The Alpha & The Omega* (as discussed in a previous chapter).

Inventions 1 & 2

Inventions 1 and 2 have the same meter, 4/4, and a ratio of measures, 22:27, nearly equal to the inverse ratio of tempos, 63:54, resulting in a close 2:3 duration ratio, 1:23-2:00. One less measure for invention 1, 21, results in a 2:3 duration ratio, 1:20-2:00.

Inventions 2 & 3

Inventions 2 and 3 have a ratio of beats, 108:59, and a ratio of tempos, 54:48, which result in a close 3:2 duration ratio, 2:00-1:13. Five more measures for invention 3, 64, results in a 3:2 duration ratio, 2:00-1:20.

Inventions 3 & 4

Inventions 3 and 4 have the same meter, 3/8, and a ratio of measures, 59:52, nearly equal to the ratio of tempos, 144:126, resulting in virtually equal durations, 1:13 and 1:14. Five and four more measures, respectively, 64 and 56, results in equal durations of 1:20.

Inventions 1-2-3-4

Now it can be seen that with just one less measure for invention 1 and a few more measures each for inventions 2 and 3 (invention 2 has a perfect number of measures), the first four inventions share a four-way 2:3:2:2 duration ratio, 1:20-2:00-1:20-1:20.

Inventions 5 & 6

Inventions 5 and 6 have a close 1:2 measure ratio, 32:62, and equal tempos, 84 bpm, resulting in a close 2:3 duration ratio. One-half of a measure less for invention 5, 31.5, and one more measure for invention 6, 63, result in a precise 2:3 duration ratio of 1:30-2:15.

Inventions 6 & 7

Inventions 6 and 7 have a close 2:3 ratio of beats, 62:92, and 4:3 ratio of tempos, 84:63, resulting in a close 3:2 duration ratio. One more measure for both inventions, 63:24, results in a 3:2 duration ratio of 2:15-1:30.

Inventions 5-6-7

Now it can be seen that with just one half less measure for invention 5, 31.5, one more measure for invention 6, 63, and one more measure for invention 7, 24, results in a three-way, symmetrical 2:3:2 duration ratio, 1:30-2:15-1:30.

Inventions 8 & 9

Inventions 8 and 9 have the same meter, 3/4, equal measures, 34, and a 2:1 tempo ratio, 96:48, resulting in a precise 1:2 duration ratio, 1:04-2:08. Two fewer measures for both inventions, 32, results in a 1:2 duration ratio, 1:00-2:00.

Inventions 9 & 10

Inventions 9 and 10 both have triple meters, 3/4 and 9/8, nearly equal measures, 34:32, and a 2:3 tempo ratio, 48:72, resulting in a close 3:2 duration ratio, 2:07-1:20. Two fewer measures for invention 9, 32, results in a 3:2 duration ratio, 2:00-1:20.

Inventions 10 & 11

Inventions 10 and 11 have a similar ratio of beats, 96:92 (32 x 3 and 23 x 4), and equal tempos, 72 bpm, resulting in virtually equal durations, 1:20 and 1:18. One more measure for invention 11, 24, results in a duration of 1:20.

Inventions 11 & 12

Inventions 11 and 12 have quadruple meters, 4/4 and 12/8, nearly equal measures, 23:21, and a ratio of tempos, 72:64, resulting in virtually equal durations, 1:18 and 1:16. One more measure for invention 11, 24, and one-half more of a measure for invention 12, 21.5, results in equal durations of 1:20.

Inventions 12 & 13

Inventions 12 and 13 both have quadruple meters, 12/8 and 4/4, and a ratio of measures, 21:25, nearly equal to the ratio of tempos, 64:72, resulting in similar durations. One-half more of a measure for invention 12, 21.5, and one less measure for invention 13, 24, result in equal durations of 1:20.

Inventions 10-11-12-13

Inventions 13 & 14

Inventions 13 and 14 have the same meter, 4/4, a ratio of measures, 25:20, and a ratio of tempos, 72:42, resulting in a close 2:3 duration ratio, 1:23-1:54. One less measure for invention 13, 24, and one more measure for invention 14, 21, result in a 2:3 duration ratio, 1:20-2:00.

Inventions 14 & 15

Inventions 14 and 15 have the same meter, 4/4, similar measures, 20:22, and a 2:3 tempo ratio, 42:63, resulting in a close 3:2 duration ratio. One more and one less measure, respectively, 21, results in a 3:2 duration ration, 2:00-1:20.

Inventions 13-14-15

Overview of the Sinfonias

The 15 Sinfonias first appeared as “Fantasias” in a different order in the Notebook for Wilhelm Friedemann Bach. Analysis reveals a deliberate plan of order and symmetry in the Sinfonias applying the same tempos, suggesting that Bach organized the Sinfonias and Fantasias simultaneously. Such an impressive mathematical feat could be understood as the “fantasia-sinfonia conundrum,” in which Bach the puzzle-solver worked out a likewise orderly and quasi-symmetrical plan for the Sinfonias. In essence, Bach found two solutions for a puzzle using the same fifteen pieces.

The 15 Sinfonias are contained within equal outer parameters, nos. 1 and 15, which symbolize BACH and The Alpha & The Omega, as discussed in a previous chapter. Between these two ends, the sinfonias are organized into eight duration pairs and one group of four. These occur in a symmetrical fashion with four pairs on either side and no. 9 in the middle, whose duration of 3:00 is twice that of 1:30. This shows a symmetrical 1:2:1 duration ratio between the first, middle, and last sinfonias. Six of the fifteen inventions have equal durations of about 1:30 while three have durations of 2:15, these two durations of which relate 2:3.

To summarize, the 15 Sinfonias are organized in a highly symmetrical fashion permeated throughout with 1:1, 1:2, 2:1, 2:3, 3:2, and 1:1:1:1 duration ratios. Nine of the fifteen sinfonias have just two durations – 1:30 and 2:15 – which relate 2:3. This is all contained within equal outer parameters, nos. 1 and 15, equal at 1:30. This multiple-level symmetry symbolizes BACH and The Alpha & The Omega. Please refer to discussion of this topic in previous chapter. (See Diagrams 3 and 4

Diagram 3. Temporal Organization in the 15 Fantasias