Working Notes on the Mathematics of Music
Part 1 – Violin: Preliminary Findings

1.1 Fractions
While it is true that violins can be purchased in partial quantities; i.e. ½ violin, ¾ violin, or full violin (and on occasions, in quantities of even less than ½), it is also worthwhile to note that these fractions do not lend themselves to simple calculations- that is, ½ violin + ½ violin ≠ 1 violin. Likewise, ¾ violin + ¾ violin ≠ 1 ½ violins; 1 violin – ½ violin ≠ ½ violin, ad infinitum.

During experimental algebraic stages, several calculations were attempted by force; all of these calculations resulted in a violin value of 0. A preliminary hypothesis suggests that probable causes for this result may root from the unexplored value of the physical embodiment of the idea of a violin; that is, we contend that the Physical Embodiment of a Violin is equal to 0, (shown as: |PEV| = 0). The implications of these calculations as pertains to the idea of a violin (V) are yet to be determined, and are not considered in this hypothesis.

1.2 Chair A
Sean was adopted from a Russian orphanage when he was 2 yrs. old. Sean is not his birth name, but rather a name selected by Sean's adoptive parents. Sean plays for Chair A in the Washington Manor Elementary School Orchestra (WMESO) (WMESO itself is composed of 5 Violins, 1 Viola¹ and 1 Cello).

Preliminary research (clinical observation of subject; interviews with subject, peers & teachers; computational analyses) indicates that Sean is an excellent violin player.

While interviews with Sean's teachers suggest that Sean's musical proficiency is completely attributable (or nearly completely attributable) to the time he devotes to "practicing his instrument" (estimated at 3.78 hours per weekday and 5.83 hours on Saturdays and Sundays), subject observation indicates another explanation, entirely physiological in nature. The fingers of the Sean's left hand (labeled 1, 2, 3, 4, beginning with index) measure (respectively) .2, .4, .3 and .2 centimeters longer than the composite fingers of the 6 other members of WMESO. Additionally, Sean's wrists boast a circumference 8% (L) and 7.8% (R) slimmer than the composite wrists of the 6 other members of WMESO; despite these considerable differences, Sean met and sometimes exceeded the scores of his fellow WMESO members in strength and endurance testing.

Yet another explanation was provided by Sean himself:

"[I have] a seahorse at home. He is brown with gold spots on his nose [and] on the spikes of his tail. His name is Ark. He helps me to play. My brother says he is dead [because] he doesn't move but he is alive because when I am sleeping he says things [to me] and then I am done."

Following this initial revelation, the item of the seahorse² was returned to at several points over the course of interviews.

1.3 Ark
The mathematical value of the seahorse in relation to the production of music attained between the curve of Sean's wrist and the lacquering peel of 1 bow on 4 strings is inconclusive. We contend that the subject of Sean's seahorse, however, merits further research, as this test case could provide valuable information toward the understanding of the mathematical relations between the idea of a seahorse and the idea of a violin. That is, if the variable of a Seahorse (S) is introduced into the equation of |PEV| = 0, what are the implications? Further, are these implications catastrophized by the introduction of PES into any of the aforementioned equations? Preliminary research, at the very least, suggests that the variable of S responds in an anti-numeric way to the function of a divisive symbol (/).

It remains common folk knowledge that at least one written indication of division (/) is identical to the shape of a scalpel angled toward the growth of an unmade wound. It stands to follow that the application of this divisive symbol should mirror the application of the symbol's shape; that is, the action of the scalpel against flesh.

This folk knowledge was illustrated, quite bravely, by Sean's brother (Sb). During several of the clinical trials which took place in Sean's home, some elementary calculations were introduced which incorporated both a ¼ Violin and a Seahorse. Problems involving both addition and subtraction were executed, but with inconclusive results. At this point, division was introduced as a suitable medium with which to explore the relation between the body of a Violin and the body of a Seahorse.

Although Sean reacted with a series of uncooperative responses³, Sb was quite willing (and even excited) to attempt a solution. Sb, following Sean's rather childlike tantrum, retrieved a Swiss army knife from the left front pocket of his pants, then collected Ark from beneath Sean's bedroom pillow, at which time Sb proceeded to enact division upon the stomach of Ark.

Although the Swiss Army knife operated by Sb admittedly lacked a scalpel, the miniature blade contained therein was determined as appropriate for the initial test case. During the first attempt at penetration, Ark's belly remained for the most part intact, with the exception of some minimal flaking of the outer skin. The second attempt at penetration was successful. Sb submerged the knife at an approximate angle of 45°, then proceeded to divide the flesh of the belly. The impact of the blade on Ark's belly's dried flesh made a noise equivalent to the cry of a smoldering pinecone.

Our initial expectation that this division would create (2)½S proved incorrect, as the blade was not capable of penetrating the full breadth of Ark. Several other factors complicated the solution, with a number of these factors interpreted as purely anti-mathematical responses originating from the body of Ark. These anti-mathematical responses included: a transmogrification of the shape of the wound (toward the shape of a bullet, or nail) which altered the geometry of the slice; the pulsation of a dark red blood, which enveloped the walls of the room, much like a placenta; the cluster of translucent bead-eggs which burst from the belly of Ark, then multiplied, auxetic (several of these eggs burrowed beneath the left eye of Sb and have yet to be retrieved).

1.4 Conclusions
Sean remains in critical condition, and has thus been removed from the position of Chair A in WMESO. This development suggests the strong possibility of a correlation between PES and PEV, although interpretation of this development cannot objectively be extended toward a possible correlation between S and V.

Categorization of S remains in a liminal space; initial attempts by research assistants⁴ to place S within the category of Imaginary Numbers were quickly erased, as S not only exists well outside of the vertical axis of the complex number plane, but in fact rejects geometric interpretation completely.

For the purpose of multiplication, PES and PEV remain incompatible elements.

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¹ Early indications suggest that the Viola was not the first choice of the current Viola player in WMESO. Rather, Violin had been the intended medium. WMESO Viola Player (VaP): “Viola is dumb. I hate it because it’s stupid. It’s not as pretty as the other ones. Why can’t I have a violin?”
² “Ark” was reported by Sean’s mother to be “a souvenier from Somewhere”
³ It was necessary for 2 research assistants to restrain Sean during Sb’s mathematical performance.
⁴ Idiots

Continuity as an Axiomatic Resultant of a Clearly Defined Fragmentation of Lines

Rules:
(1) Didactics are most useful when charted, i.e. maps, algorithms, taxonomic functions.
(2) Taxonomies work best as dialectics.
(3) Gray is almost always more confusing than the worth of the centered bar.
(4) Bars are more useful as dividers than uniters

Illustration:

Derivation:
Here, it is clearly of fundamental importance to note the etymology of both "saffron" and "vultures." One reputable source suggests that the language of saffron as a spice took a trade route originating somewhere amongst the epithelium of the Arabic tongue, later routed through Latinate trail only to incite the papillae of both the French and the English. We are further lead to believe that the vulture sprouts directly from a Latin origin, although etymology and ornithology are admitted to make strange bedfellows.

Use:
The bar can also made a blank horizontal (example follows):

children's naked teeth
children's naked teeth
children's naked teeth
children's naked teeth
children's naked teeth
children's naked teeth

Conclusion:
Proofs made available by both Leonard Schwartz and Zheng Er clearly indicate that while the scientist prefers a vertical form of notation, the poet (lazy by nature) prefers a horizontal delineation: a form made even more conspicuous through the use of an inverted (blank) line.