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 (

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

See also Overused chord progressions; Sonny Bono Copyright Extension Act

Additional Resources

  • "Vaudeville Q & A" by anonymous ( Unfortunately, this one-paragraph summary of the lawsuit is the most detailed description that Google could find.
  • "Melancholy Elephants" by Spider Robinson ( details the dire consequences of literally running out of new ideas.
  • "Columbia Law Library Music Plagiarism Project" by ? ( links to opinions and analysis of several more recent cases.

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

Responses from E2 readers

arrogantsob says I looked for both cases on westlaw, and here's how I see it (as a 1st year law student, so please don't take this without asking a real lawyer: I couldn't find anything on the bananas song, but in general, cases as old as that don't have the same precedential value as newer cases. As for the Harrison case, it's probably better for precedential value, but it's only for new york. my guess is that it's not as dire as you think it is. four notes probably don't equal copyright infringement anymore.

Lometa says My Sweet Lord is all fixed thank you. About Yes! We Have No Bananas. Mind boggling law making. From my POV it would be akin to using the same four colors in a painting or words in a poem and then calling them copyright infringements. But then I don't understand enough about the mechanics of music making to really know what the dispute is about.

Stealth Munchkin says re songwriting Both those songs were plagiarism, one deliberate, the other supposedly unconscious. I once calculated that if every person in the world now living wrote one eight bar melody a second, keeping just to quavers and just to notes within one octave, it would take approximately 10^1000 times the expected life of the sun before one repeated. So I doubt very much that there's any danger of running out of melodies ;)

Stealth Munchkin says The smallest amount you can be sued for is about 2 bars, and usually there won't be any possibility of a lawsuit unless the song is 'substantially similar'. It's impossible to be 100% sure you won't be sued, but the chances of accidentally writing someone else's song are absolutely minimal. Check for every major plagiarism case for popular song in US law... that should give you some idea how unlikely it is you'd get successfully sued...

Stealth Munchkin says Generally speaking you'd only be sued if the song you wrote was successful enough that you *would* have the money - and in the event that it was an unsuccessful suit, the plaintiff would have to pay your costs anyway...

Orpheum says Hmm, well avoiding the mistake George Harrison made will be a difficult thing to actively do - in his case, he was found to have unconsciously plagiarised the original song, so it's fair to say that he had no idea that's what he was doing. And people will get caught out by this in the future, it's certain - they sit down to write, a tune seems to pop into their head...they don't realise that this tune is one they've actually heard elsewhere. To be honest though, if you're throwing together stuff on your home pc, I don't think this will really be too much of an issue. Unless you professionally record and release gets become popular enough to become a blip on the radar screen, actually selling your stuff. If George Harrison had never achieved popular success, I doubt he'd ever have found himself in a court room over that song..

The song reminds me of the movie "Sabrina" (1954) with Bogart, Holden, and Audrey Hepburn. There is a love triangle of sorts in that movie, and each character has a song which represents their view of love.

William Holden's character, the playboy, is associated with the song "Isn't it (love) romantic?" while he seduces various girls on the dancefloor and private tennis court.

Hepburn sings "La vie en rose" ('Looking at the world through rose-colored glasses') to Linus, and she writes of it previously in her letter to her father. It is symbolic of a deeper, romantic love.

Lastly, Humphrey Bogart's record is played with the tune "Yes, We Have No Bananas". The song is obviously not a love song at all, reflecting Linus's mockery and deceitfulness, as it is played against a romantic setting of the two sailing together in a small boat.

The movie has a few contrasting themes, such as the aforementioned songs used in the movie, which add a level of sophistication to the storyline.

For myself, the song evokes an image of Sabrina washing the car after her boatride with Linus, singing the end of the song, "Yes, we have no bananas today".

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