MATHEMATICS, MUSIC

Simple formula for determining musical royalties

If you’re looking for a useful guide to determine royalty percentages when collaborating with other songwriters, here’s a simple way to calculate a general royalty percentage with respect to musical changes: R=(100e)(bdn)^{-1} where e = total events changed, d = quantity of the unit of metric measurement per bar, b = total number of measures, n = the smallest rhythmic value changed during alteration, and R = the total royalties earned (as a percentage).  Remember, the value for n must be based on the shortest rhythmic value that is changed.

Example:  John writes “Song X,” a 120-measure ballad using a 3/4 meter. Emily collaborates with John and changes 176 events at the nth-note level—in this case, say, sixteenth notes. (This could be 176 sixteenth notes, 88 eighth notes [n = 2], 44 quarter notes [n = 1], 704 thirty-second notes [n = 8], or any combination of such note values, but if we assume the shortest note-value Emily changed was (at least) one sixteenth note, the total amount changed should be based on the sixteenth-note value for n.

Therefore, the total number of musical events—based on the shortest rhythmic value of Emily’s changes—can be calculated as follows: bdn=(120)(3)(4)=1,440. That is, there are 1,440 sixteenth-note events in “Song X,” (bd gives us the total number of musical events with respect to the unit of metric measurement), and Emily will earn the following songwriter’s royalty if she changes 176 (sixteenth-note) musical events: R=100(176)(1,440)^{-1}=.122.

Simple—and pretty convenient if Emily begins clamoring for a 40-60 royalty split based on her efforts.  We could also create a similar algorithm for “chord changes” (based on harmonic rhythm) and even lyrics, but we’ll leave those projects as exercises for the reader.

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MUSIC, PHILOSOPHY

Schubert, Schenker, and propositional calculus…oh, my.

The complexity of Schubert’s lied Der Musensohn lies in its simple harmonic syntax within the deep-middleground Schicht.  There the oscillation between G major and B major creates an ostensible problem for fleshing out an orthodox Urlinie: The B major prolongations do not act as either third dividers or applied dominants to the submediant; thus, they cannot be used as evidence for such voice-leading paradigms (i.e., neither D major nor E minor arrives after the B major prolongation).  Well, how do the  B major prolongations fit into the large-scale voice-leading infrastructure—if it is possible, into an orthodox Schenkerian background?

I’ve written a paper I hope to publish that reveals a very tenable and (hermeneutically) satisfying solution to this problem, but I’ve always been bothered by the above logical argument concerning the role of B major.  For example, is it logically valid—formally speaking—to say “if B major is a third-divider, then D major follows”?  The short answer is yes: If the antecedent is false, the entire conditional is always true.  But I’ve assumed B major isn’t a third-divider from the beginning, so I’ve established a situation where my conditional is guaranteed to be true.  No good.  What about the negation of the consequent (i.e., modus tollens)?  This tells us that because D major does not materialize, B major is not a third-divider.  But, again, this is derived from the conditional as designed.  Circumventing these issues involves a more complicated proof that does not derive the identity of B major with such immediacy:

R = D major arrives (after B major), S = E minor arrives (after B major), P = B major is a third-divider, Q = B major is an applied dominant

Assumptions and proof:

¬R, ¬S, [((P → R) ˅ (Q → S)) ˄ ¬((P → R) ˄ (Q → S))]  ˫  ¬P ˄ ¬Q

__________________________

**1. ((P → R) ˅ (Q → S)) ˄ ¬((P → R) ˄ (Q → S))
A
**2. ¬R
A
**3. ¬S
A
**4. (P → R) ˅ (Q → S)
1 (˄E)
**5. ¬((P → R) ˄ (Q → S))
1 (˄E)
**6. | P → R
H
**7. | ¬R
2 RE
**8. | ¬P
6, 7 MT
**9. (P → R) → ¬P
6, 8 (→I)
**10. P → ¬(P → R)
9 (TRANS)
**11. | P
H
**12. | ¬(P → R)
10, 11 (→E)
**13. | ¬(¬P ˅ R)
12 MI
**14. | P ˄ ¬R
13 DM
**15. P → (P ˄ ¬R)
11, 14 (→I)
**16. | P
H
**17. | P ˄ ¬R
15, 16 (→E)
**18. | ¬R
17 (˄E)
**19. P → ¬R
16, 18 (→I)
**20. ¬P ˅ ¬R
19 MI
**21. | ¬ ¬P
H
**22. | P
21 DN
**23. | ¬R
20, 22 DS
**24. ¬ ¬P → R
21, 23 (→I)
**25. P → R
24 DN
**26. ¬P
9, 25 (→E)
**27. | Q → S
H
**28. | ¬S
3, RE
**29. | ¬Q
27, 28 MT
**30. (Q → S) → ¬Q
27, 29 (→I)
**31. Q → ¬(S → Q)
30 TRANS
**32. | Q
H
**33. | ¬(S → Q)
31, 32 (→E)
**34. | ¬(¬S ˅ Q)
33 MI
**35. | S ˄ ¬Q
34 DM
**36. Q → (S ˄ ¬Q)
32, 35 (→I)
**37. | Q
H
**38. | S ˄ ¬Q
36, 37 (→E)
**39. | ¬S
38 (˄E)
**40. Q → ¬S
37, 39 (→I)
**41. ¬Q ˅ ¬S
40 MI
**42. | ¬ ¬Q
H
**43. | Q
42 DN
**44. | ¬S
41, 43 DS
**45. ¬ ¬Q → S
42, 44 (→I)
**46. Q → S
45 DN
**47. ¬Q
30, 46 (→E)
**48. ¬P ˄ ¬Q
26, 47 (˄E)

Q.E.D.

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MUSIC

Happy birthday, Rachel…

I was browsing one of my favorite independent bookstores—O’gara & Wilson’s in Hyde Park, Chicago—when I stumbled upon an old copy of Emotion and Meaning in Music by Leonard Meyer.  As I was flipping through it, I noticed a musical inscription on the inside cover—by Meyer himself—to one “Rachel” dated February 19, 1957.  As the photo reveals, the first phrase (including the anacrusis) is clearly written in C major, though Meyer notates a G major key signature.  I began to wonder if Meyer really heard “Happy Birthday” as a reification of a Riemannian S function.

Rachel

I forced myself to hear the opening of the phrase as tonic prolongation (in G major), but it seemed quite difficult to overcome the middleground connection (via “reaching-over”) between the B (at the end of the first phrase) and the C that concludes the second phrase, a short and effective voice-leading connection that anchors the auxiliary cadence—and not, as Meyer’s notation would suggest, a transformation from T to (D7)S moving to S.  Meyer’s analysis suggests a phrase-level progression that involves an appreciable plagal analysis, a hearing that belies the phenomenological resolutions supported by the voice leading.  The entire second half of the song would be heard as an expansion of subdominant function supported locally by T = (D7)S, and there would be no tonal closure—the piece ending on scale-degree 4 in the final move to S.

No, Meyer must have simply notated the inscription incorrectly—perhaps confusing the prominence of the anacrusis (or the unfolded G-B third of the first phrase) with tonic function.

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