Gold is rarely discovered
by one who has not got
the lay of the land.[1]
– Norwood Russell Hanson
In 1916, Ernst Lecher Bacon, at the age of 18, submitted a technical article for publication to The Musical Quarterly. The editor of MQ at the time, Oscar Sonneck, declined to publish Bacon's article, but not because it was untutored or incompetent or poorly written or that it offered nothing original. While he may not have followed the math (the how of the article), Sonneck understood what Bacon had done and knew quite well that it was an astonishing accomplishment.
Still, he rejected the article on the grounds that MQ's primary readership – historians, analysts and others in the musicology orbit – even if they could follow the math (a big leap in the music world, even today), would find it irrelevant (again, even today). But, recognizing the genius of this accomplishment, Sonneck did not simply write a brief rejection. He wrote back an encouraging letter (via Bacon's mentor, Glenn Dillard Gunn) suggesting that Bacon submit the article to the philosophy journal The Monist. It appeared in the October 1917 issue of that journal under the title "Our Musical Idiom."[2] Unlike events in the political world of October 1917, Bacon's groundbreaking work inspired neither a revolution nor a war. Here's what happened:
It was virtually ignored.
What Bacon had done was create a combinatorial algorithm for listing all the sonorities – represented as transposition classes – in the 12-note chromatic scale.[3] Then, using common music notation, he proceeded to list all 350 "prime-position" sonorities (he omitted the empty set and the singleton which he thought of as trivial). It was a bit like snatching the gold ring, when hardly anyone else on the merry-go-round knew there was a gold ring to snatch. Here's how he did it.
First, he stopped thinking in terms of notes or pitches (or pitch-classes) as the primary musical objects and began thinking in terms of the intervals between notes. This is the single move that takes the task from nearly impossible to tractable. So chords or scales were not his quarry, but abstract strings of intervals.
It's the difference between, say, a C-major triad (a concrete structure which consists of the three notes/pitch-clases C and E and G) and an icon representing any triad in the same relationship as the notes in the set {C,E,G}. This unique icon simply names the chromatic intervals between notes in some circular permutation. Using Howard Hanson's notation from Harmonic Materials of Modern Music, we might represent a C-major triad within an octave span as:
C4E3G5(C).
Dropping the referential note-names, the collection of all major triads can be described abstractly by a circular permutation of the interval string 4-3-5. (This relationship must hold (mod12) if you are to respond correctly to the question, "Go to the piano keyboard and play any major triad.") This interval string notation is precisely how Bacon calculates and names all distinct transposition classes mod 12. Here is his table listing all the trichords[4]:
Note that he uses an abbreviated "name" for each unique "harmony." This is a convention that he uses throughout for sonorities of any cardinality. E.g., here the name for C.1 (Combination 1) is given as 1-1 instead of the complete 1-1-10. Since all the intervals in a string in 12tET must sum to 12, the final interval can be left off and still identify the same unique transposition class.[5] Working backward, we now ask, just how did he generate these 19 trichords? Take a look at his chart summarizing his derivation of trichords (ignore the column "Calculations of Harmonies" for the moment):
This chart for trichords represents the solutions to the three equations to the left of the arrows:
(1) 2a + b = 12 → a-a-b (1-1-10, 2-2-8, 3-3-6, 5-5-2)
(2) c + d + e = 12 → c-d-e (1-2-9, 1-3-8, 1-4-7, 1-5-6, 2-3-7, 2-4-6, 3-4-5)
(3) 3f = 12 → f-f-f (4-4-4)
with the stipulation that all the solutions are positive integers less than 12. The mathematician will immediately recognize these equations as simple linear Diophantine equations, and the triples to the right of the arrows (the exact solutions) represent the 3-partitions of 12 (all the triples of integers that add up to 12).[6]
Translating to music, all the integers a,b,c,d,e,f represent intervals less than an octave (measured in number of chromatic steps). To the right of the arrows, the triples represent interval strings. If we stopped here we would have the familiar list of 12 trichord set classes, but like most musicians Bacon wants to distinguish between inversions – e.g., major and minor triads are different animals. So all these results are permuted until all the unique permutations have been discovered. This final process sorts the trichords into symmetric and asymmetric (among other things – see the next post). Permutations of the asymmetric sets (C.2,3,4,5,7,8,11) add the respective inversions, so 1-2-9 is joined by its inversion 2-1-9 (or, what's the same, 1-9-2 using Bacon's preferred ordering). So the complete list of 19 trichords Bacon generated (in integer notation) is:
(1') a-a-b: 1-1-10; 2-2-8; 3-3-6; 5-5-2
(2') c-d-e: 1-2-9, 2-1-9; 1-3-8, 3-1-8; 1-4-7, 4-1-7; 1-5-6, 5-1-6; 2-3-7, 3-2-7;
2-4-6, 4-2-6; 3-4-5, 4-3-5
(3') f-f-f: 4-4-4
which (for those who don't recognize them) appear in abbreviated notation in his list of "harmonies" (see above). Using Bacon's ordering:
1-1 1-2 1-9 1-3 1-8 1-4
1-7 1-5 1-6 2-2 2-3 2-7
2-4 2-6 2-5 3-3 3-4 3-5 4-4
(To be continued)
[1] The context for the quote can be found online (Ch.1 of Patterns of Discovery). But the context is important enough to quote a bit more here: "It is the sense in which Tycho and Kepler do not observe the same thing which must be grasped if one is to understand disagreements within microphysics. Fundamental physics is primarily a search for intelligibility – it is philosophy of matter. Only secondarily is it a search for objects and facts (though the two endeavors are as hand and glove). Microphysicists seek new modes of conceptual organization. If that can be done the finding of new entities will follow. Gold is rarely discovered by one who has not got the lay of the land. . . . It is important to realize ... that sorting out differences about data, evidence, observation, may require more than simply gesturing at observable objects. It may require a comprehensive reappraisal of one's subject matter. This may be difficult, but it should not obscure the fact that nothing less than this may do." [my emphases] Does this apply to music theory? Recall Lewin: "This is the methodological point: We must conceive the formal space of a GIS as a space of theoretical potentialities, rather than as a compendium of musical practicalities." (GMIT 2.3.2)
[2] I'm going from memory in this entire account, having last read the source material 4-5 years ago. I no longer have direct access to the correspondence in the Bacon, Gunn and Sonneck papers in the Music Division at the Library of Congress which are the basis for the story of the publication of "Our Musical Idiom." I should have made private copies or taken extensive notes, but I didn't.
[3] Catherine Nolan has written extensively on the history of this particular aspect of the connection between music and mathematics. Available on line, as one example, is her 2000 Bridges paper "On Musical Space and Combinatorics."
[4] I will henceforth try to remember to use "triad" when referring to the familiar, historically-conceived major, minor, diminished and augmented triads. I will use "trichord" to refer to the (ahistorically-conceived) set of any 3-voice sonority, usually the list of 12 trichord set classes or the list of 19 trichord transposition classes in 12tET. This is exactly backward from how I would prefer to use these terms since trichord suggests sonority and leaves no room for extensions of the concept such as 3-point rhythmic structures; and triad, a more abstract term that could refer to any set of 3 things, has been historically usurped within music to refer to traditional Western 3-voice chords built of superimposed thirds (or however you wish to construe/generate them). So any triad is also a trichord, but not all trichords are triads. No, this does not make any sense, but I didn't create this particular mess.
[5] This shorthand is useful but can easily result in errors such as mistaking a symmetry as an asymmetry leading one to think there is a distinct inversion where there is none. E.g., listing the tetrachords 1-1-5 and 5-1-1 as distinct inversions when citing the "full" name, 1-5-5-1, reveals it is a symmetry & has no distinct abstract inversion. So reader beware.
[6] Readers who have not previously encountered the idea of using numerical partitions to generate chord lists may wish to go to the interactive page for partitions at the Combinatorial Object Server and plug in random values for k and n (ignore m) to quickly get an idea of how important this simple idea is. Start with n=12 then increase n to get partitions (the math term for what Bacon calls "Combinations") for larger musical universes. E.g., for partitions leading to listing hexachords in 12tET set n=12 & k=6 to get 11 partitions to work from; for hexachords in quartertone space set n=24 & k=6 to get 199 partitions to work from.
