LEGAL DISCLAIMER: I MAKE NO WARRANTY that my statistical measure of "substantial similarity" will in any way predict how a judge will rule. I CANNOT BE HELD LIABLE for damages resulting from any defect in the model.

This writeup is about to be rewritten using a further refinement of the assumptions.

Abstract: The standard for what constitutes copying or creating a derivative work under American copyright law is so broad that songwriters run a significant risk of accidentally infringing a copyright. An approximate mathematical model of the legal standard is presented, which results in fewer than 50,000 possible melodies given a few assumptions.

## Yes! We have a chilling effect on songwriting!

The song "Yes! We have no bananas!", first published on March 23, 1923 (putting it under perpetual copyright on the installment plan), has a quite interesting legal history. As soon as it was published, George Frideric Handel's American publisher filed a copyright infringement lawsuit against songwriters Frank Silver et al. for copying four notes from the "Hallelujah Chorus" of Messiah and won, taking some of the profits. (At least it wasn't as bad as the "Bitter Sweet Symphony" case, where the Rolling Stones' music publisher assumed full control of the Verve's derivative work.)

The important part about that case is that it set a legal precedent that matching even four notes can easily be enough to establish that two songs are substantially similar. Such music copyright infringement cases raise some interesting questions about copyrights and combinatorics, challenging the notion that it is still possible to compose a song that is considered original under copyright law.

### Counting melodies

A fellow may wonder, how many melodies are there in the universe? A melody is determined by the distances from each note to the next in frequency (intervals) and in time (note duration). A melody of n notes will have n - 1 distance vectors, which can be expressed as an ordered pair (interval, duration). (The last note doesn't count because there is no "next note".)

Because most judges are not musicians, and because they are looking for substantial similarity rather than exact identity between melodies, they will probably ignore some of the "embellishments" that belong to a particular performance rather than to the underlying musical work. In general, a typical melody will make the most use of three note lengths (such as half, quarter, eighth or quarter, eighth, sixteenth); a judge may ignore slight variations of duration such as triplets or change a dotted quarter note into a half note. To a judge, eighth note + eighth rest + rest of melody equals quarter note + rest of melody because staccato is an aspect of the performance and does not diminish similarity of the underlying songs. Most judges will not be familiar with the Eastern scales used in some later Beatles music, and a judge has the right to round each note to the closest note in a Western scale because just the fact that two melodies belong to different keys or scales doesn't diminish their similarity. Thus, I feel justified in using a simplified model for discussion.

Assume that all songs use a Western musical scale and that such a scale contains twelve distinct intervals. Assume that a judge (not a musician but a judge) will distinguish three distinct note durations (which roughly correspond to eighth, quarter, and half notes, or through a trivial change in time signature, to quarter, half, and whole notes, or to sixteenth, eighth, and quarter notes). Thus, there are 36 possible distance vectors from one note to the next, and 36(n - 1) melodies of n notes.

Now, 36 to the third power equals 46,656 distinct melodies. No other melodies are possible in the Western musical scale.

How many is 46,656? If only one hundred songwriters in the world were to write one song each week with a unique melodic hook, they would run out of new melodies within nine years. Information theory states that using the preceding model, a melodic hook can be encoded in a number from 1 to 46,656, and that number can be encoded within 16 bits. Applying an estimate of the entropy of English (about 8 bits per 3 letters) gives six letters. Trademarkable? Yes. Copyrightable? Only if entrenched songwriters can buy the judge.

On the other hand, the four notes threshold is from a quite old case. More recent cases may have expanded this threshold. Allen v. Walt Disney (1941) shows an analysis typical of more recent analyses, finding for the defendant even in the face of several chance resemblances. Campbell v. Acuff-Rose (1993) stated that a small appropriation from "Oh, Pretty Woman" was fair because it was used in a parody of the copyrighted work itself. However, we still have Bright Tunes v. Harrisongs (1976), which shows that chance similarities to a published song may be assumed to be "unconscious copying", but here, at least eight notes (367 = 78,364,164,096 possibilities by the model presented above) were involved.

### There is no originality

From an amicus curiae brief in support of the Sonny Bono Copyright Extension Act (http://llr.lls.edu/eldred/martin-original1.pdf):

The fact that creators of new works cannot merely re-use the expression contained in copyrighted work of others without permission forces them to be creative. Composers cannot rehash the melodies created by earlier composers, they must create their own new original melodies.

How is this possible? As noted above, case law states that copying four notes of another song's "hook" is enough to get a songwriter in trouble with copyright law, and that the standard for copying is not an exact match but merely substantial similarity. Another case (citation omitted) seems to state that there is no unprotected "idea" in music, only "expression".

Applying such a low standard of originality to melodies can in theory create a situation where nobody can create a new song without using a copyrighted melody because previous songwriters have exhausted the space of melodies. It almost looks as bad as the situation with software patents, which have such a high probability of reinventing a patented invention so as to create a chilling effect on "the progress of science and useful arts" (U.S. Const., Art. 1, Sect. 8, Cl. 8).

I suggest that the U.S. Congress set down some more reasonable guidelines as to how much of a melody a songwriter can reuse without accidentally creating a derivative work and stepping on some government-granted monopoly. Will future lawmakers use algorithmic information theory to construct an even more precise model?

### Related Articles on E2

• "Vaudeville Q & A" by anonymous (http://members.rogers.com/vaudeville/vaudeville2.htm#Bananas). Unfortunately, this one-paragraph summary of the lawsuit is the most detailed description that Google could find.
• "Melancholy Elephants" by Spider Robinson (http://www.baen.com/chapters/W200011/0671319744___1.htm) details the dire consequences of literally running out of new ideas.
• "Columbia Law Library Music Plagiarism Project" by ? (http://www.ccnmtl.columbia.edu/projects/law/library/entrance.html) links to opinions and analysis of several more recent cases.

© 2001-2003 Damian Yerrick. Verbatim copying and redistribution are permitted.