Linear Intervallic Patterns (LIPs)

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Linear intervallic patterns (LIPs) are the series of intervals that govern what are commonly called "sequences" (= repeated melodic passages that, at each instance, move up or down a step, and usually described according to the underlying harmonic progression, e.g. falling-fifth sequence). A LIP is the subsurface pairs of intervals that guide such melodic passages. The subsurface LIP may remain constant while the surface melodic patterning may change.

Example 1: 10-7 LIP
Listen to and examine Inventio 14, mm. 4-5. On hearing the passage, most will quickly recognize it as a falling-fifth sequence (Bb: I-IV-vii-iii-vi-ii-V). Just beyond the surface, we will discover the governing LIP.

Assuming that the left-hand 16th rest at the beginning of m. 4 stands for a Bb, the interval between right and left hands on beat 1 is a 10th (Bb-D). As the harmony changes from I to IV at beat 2, an Eb enters in the left hand and forms a 7th with what we may assume to be a right-hand D at the 16th rest at the beginning of beat 2, as though the right-hand D from beat one were imaginarily held over to beat 2. That D, a dissonant 7th against the bass Eb, must resolve down by step to C, which it does on beat 3 as the harmony shifts to vii and the bass moves to A, forming another 10th (A-C), a step lower than the previous one (Bb-D). As the harmony falls another fifth on beat 4, to iii, the bass moves to D and forms another 7th with an imaginary right-hand C at the 16th rest on beat 4. The emerging, subsurface LIP is 10-7, 10-7, etc. That intervallic pattern guides and governs mm. 4-5. [Bach's Inventio 12 in A major, mm. 5-8, and Sinfonia 15 in B minor, mm. 7-11, provide two further, clear examples of 10-7 LIPs.]

Example 2: 10-5 LIP
In the 10-7 LIP described above, every chord was in root position. When the second chord of each pair outlining a falling fifth is in first inversion (every other chord is a six-five), the resulting LIP is 10-5, a variation on 10-7. An example occurs in Inventio 4, mm. 22-26. Bass and treble on beat 1 in m. 22 form a 10th (A-C), which becomes a 5 (diminished) when the bass introduces F# on the downbeat of m. 23. That diminished 5th resolves at the beginning of m. 24 to another 10th (G-Bb), which yields to another diminished 5th (E-Bb) when the bass moves to E in m. 25. It resolves to the last 10th (F-A) of this 10-5 LIP at m. 25.

Post-exposition LIPs in Inventions
In Inventions, a LIP often follows immediately after the exposition (the introduction of the motive in each voice or register). Its function is modulatory, i.e. to carry the music to the next key in the tonal plan--generally to the key most closely related to the tonic (e.g. the dominant in a major-key piece, the relative major in a minor-key piece). We see such a post-exposition, modulatory LIP clearly in Inventio 4 (mm. 7-18), 7 (mm. 3-5), and 13 (mm. 3-6). The melodic material in LIPs derives from the motive (the "inventio," or thematic idea), as does all melodic material in an Invention.

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