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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ECONOMICS, MATHEMATICS, POLITICS

How to save the working and middle classes

I’ve been studying more macroeconomics lately, and I figured I would take a shot at proposing a tentative solution to the income-inequality gap—though the concept really emerges from a series of microeconomic shifts.  It is an idea, in more general terms, I’ve suggested to various friends and family members over the last few months: Close all the corporate loopholes, raise capital-gains taxes, and funnel the additional revenue back into companies replete with middle-class workers and laborers in the form of higher (real) wages. Is this a socialist, rob-peter-to-pay-paul strategy?

No, and here’s why.

Increasing aggregate demand (AD) usually translates into increased production costs because wages increase (either in the form of OT or hiring additional workers); this forces companies to raise prices, which means the additional costs are transferred to the consumer. This is essentially what happens in the situation here.

photo [3]

Within the standard LRAS/SRAS model (below), demand then shrinks, prices subsequently fall, and the price-output equilibrium is restored. Increased output under these parameters, then, can only be a temporary phenomenon. But what if the additional costs are covered by the hike in taxes on capital gains (i.e., prices remain sticky)?

photo [1]

The consumer is spared the burden of increased prices—so demand grows (AD to ADs curve) without rebounding—and wages rise for the average worker. As productivity rises (Y* to Yw)—a function of increased real wages—the company actually earns MORE money than it would have at “full” employment (i.e., at Y*). Joe CEO benefits because his company earns more money—the area of P*(Yw – Y*)—stretching his supply-demand curves to a new, better equilibrium, even though he’s initially taxed at a higher rate.  This cycle could, theoretically, continue forever IF we had unlimited stimuli that could shift the AD and LRAS curves to the right ad infinitum.

photo [2]

Such a strategy will help the overall economy: GDP will increase as productivity increases, consumption will increase (as will tax revenue), and unemployment will decrease because a company will eventually have to hire more workers. (There will be an upper bound for the productivity of Y* workers.) This will create a staggered increase in the wage-price ratio; in fact, if adjusted correctly, wages could double the rate of price increases, so real wages will grow. If a company uses price hikes—that is, P* shifts upward to Pw+n—to counterbalance increased labor output (Yw+n above) within a stimulus shift to AD, there will be a surplus of inventory, and the company is at risk for a loss.  (The shift to the new equilibrium through decreased demand is denoted by the shaded right triangle.)  But it is important to note that a company can STILL earn a profit on the surplus labor and inventory at Yw+iff

{Y}w(({P}w+n)-P^*)>{[(({Y}w+n)-{Y}w)(({P}w+n)+{P}^*)]}2^{-1}

Assume such a profit exists.  (If it doesn’t, then there was a miscalculation in trying to stretch output to Yw+n.)  We can use that profit to subsidize the temporary drag of the (Yw+n)-labor surplus, and if another stimulus is introduced, we will shift the demand curve again (from ADw+n to the right) and (1) eliminate the excess inventory at a maximum price—in fact, the additional revenue would be the result of [((Pw+n) – P*)((Yw+n) – Yw)]/2)—(2) allow a maximization of productivity (at Yw+n) by increasing wages while keeping prices sticky, and (3) create another revenue bump, which will equal

[(Pw+n)((Yw+n)-(Yw))]-[Yw((Pw+n)-P^*)]

Of course, we don’t need to manage the costs inherent in the increased Yw+n output; that is, as AD shifts, we could simply raise prices at Yw and maximize revenue that way—without having to deduct the costs of labor drag and inventory surplus.  This would be especially effective if output was maximized at Yw.  (Output would then increase to Yw+n during the next demand shift—with sticky prices at Pw+n—using the revenue gains from raising prices.)  In any case, this general cycle would continue until a new cost-demand equilibrium is reached—say, when total aggregate demand reaches an upper bound—and that could be a significant difference from the original price-output equilibrium.

What does this mean? The basic idea is simple: the LRAS, SRAS (long-run and short-run aggregate supplies), and demand curves shift over time to new (and higher) equilibria, stimulating business growth and minimizing inflationary measures with respect to real wages.  The best part is that it’s a win-win-win-win for everyone involved: the individual worker (higher real wage), the CEO (increased profits), the average citizen (benefiting from GDP growth), and the government (increased income-tax revenue and lower unemployment).  We can begin to close the gap if the rate of real-wage growth outpaces both the modest inflationary shifts and productivity-related profits, a likely possibility if the capital-gains rate is high enough with respect to the number of wages raised.

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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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