Proving musical borrowings is similar to the process of proving a geometry theorem. In geometry, one must establish a hypothesis and come up with justified points that prove that hypothesis. The hypothesis must be proven to be unshakable by attempting to find examples which counter it. If there is a counter-example then the hypothesis must be amended and reproven. A hypothesis which has been proven and widely accepted then becomes known as a theorem.
In a proof of the borrowing of a musical passage the process is similar. A musical passage may be defined as the statement of a musical idea which fits into the larger work, much like a theorem or postulate within an axiomatic system. And borrowing, in this case, may be defined as the use of one composer s musical ideas within another composer s complete work.
To prove a musical borrowing, a hypothesis must be established. For example: the music of Ives, contemporary American composer and insurance salesman, uses embellished, or modified, versions of musical passages from Beethoven s already existing work. A specific case of such is the hypothesis that Ives borrowed passages from Beethoven s 5th Symphony and changed them for use in his Concord Sonata. This hypothesis can then be backed by different pieces of evidence taken from the works of several music history scholars including: Kirkpatrick, Lambert, Swaford, Burkholder and Block. Musical examples, composer motivation and the concurrent unanimous conclusion of known scholars are all pieces of evidence used to prove this hypothesis.
In the same manner, mathematical models and the use of previously proven theorems in an axiomatic system can be used to justify a mathematical hypothesis. For instance, when proving, in Euclidian geometry, that corresponding parts of congruent triangles are congruent one may utilize any of the previously accepted angle or congruence postulates to give evidence to the proof. One may also create and compare mathematical models in an attempt to prove of disprove that the hypothesis is true.
To illustrate the first type of evidence in this claim about Ives borrowings, musical examples must be taken from the scores of pieces by Ives and by Beethoven and analyzed separately. Then by comparison one may show that a particular passage is completely identical excepting a small embellished or modified factor such as key or voicing. Therefore the rhythm and the melody, or tune, in a Beethoven passage would be the same rhythm and melody seen quoted in an Ives passage. However, the key may be different; which is to say that the melody may start and end on a different pitch though the intervals creating the melody remain the same. Or, the voicing may be modified, having the melody played by a different instrument or group of instruments. Additionally, in the development of the piece the melody may also be played in retrograde (backward), or in inversion (similar musical intervals moving up or down in pitch opposite to the original melody), or even in retrograde-inversion.
In Geoffrey Block's book, Ives, Concord Sonata: Piano Sonata no.2, this point is illustrated by example. This process can be compared to drawing diagrams to illustrate a geometrical idea. The specific passage of Beethoven s which is widely known to be used in Ives Concord Sonata is the Human Faith Melody, the classic opening to Beethoven s 5th Symphony. This excerpt is modified in all of the above mentioned ways and quoted directly. Even a person with no knowledge of music theory can hear the famous melody amidst the other musical ideas of Ives in the Concord Sonata.
One may also look for motivational clues by the composer, like using evidence of previously proven, and relevant, theorems. Ives wrote program music, or music which told a story most easily understood when accompanied by a written explanation of the story. Ives often quoted other composers for programmatic reasons. According to Kirkpatrick, many of Ives works previous to the Concord Sonata included musical borrowings. The Concord Sonata is a tribute to the great literary artists from Concord: Emerson, Hawthorne, The Alcotts and Thoreau. In an attempt to tell the story of these great artists Ives borrowed from the works of several great composers, the musical language which he understood, in order to convey his impressions through music. These borrowings include excerpts from composers such as: Stravinsky, Debussy, Wagner and, of course, Beethoven (Block, Ives Concord Sonata).
In his program, for the Concord Sonata, Ives explains his reasons for quoting the aforementioned composers which leaves little room for doubt that these passages were borrowed. Since Ives believed the authors of Concord to be great literary figures he compared each to a great composer. He then assigned a passage from the work of a given composer to be the musical description of one of the Concord writers. For example, Emerson may be depicted musically by a passage from one of Wager s works. This passage would therefore appear in the music each time Emerson appears in the story. The human faith melody alludes to the Concord Transcendentalists, as a group, and is therefore used throughout the Concord Sonata (Block&Burkholder, Charles Ives and the Classical Tradition).
Other motivational clues provide further evidence for the hypothesis that Ives borrowed from Beethoven specifically. According to Swaford, Charles Ives had reportedly said a number of times that he believed himself to be the natural successor to Beethoven and that he was improving upon the music of the late composer. And, when analyzing his own work alongside Beethoven s he had stated that if Beethoven had had the tools of contemporary music during the period in which he was composing that Beethoven s music would sound like Ives . Of course, some of these examples carry more weight than others. However, much of this information is on record in the form of letters written from Ives to his contemporaries.
Finally, the example of the Human Faith Melody, from Beethoven, in Ives Concord Sonata is one which has been proven many times by known musical scholars, including those cited in this essay. Counter-examples would include Ives denial of the borrowing, proof that Ives could not have had any influence from Beethoven, or proof that the work bears no musical resemblance to any of Beethoven s work. Since there are no known counter-examples to this hypothesis, and because it is widely accepted and cannot yet be disproven, it may be considered a theorem.
I think this method for proving borrowings in music is a relatively accurate one. Certainly it is as accurate as a mathematical proof. Proving the relation to melody, harmony and rhythm is similar to generating mathematical models to show the validity of one s hypothesis. And, proof in the form of testimony would be similar to the process of searching for and discrediting counter-examples in mathematics.
Therefore, by generating and proving, through musical example, composer motivation and scholarly analysis, a hypothesis of musical borrowing one can generate a musical proof. Like geometry, music history is a series of proofs worked and reworked by scholars internationally in an attempt both to form