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Featured Application: The results obtained in this paper are applicable to the design of new musical instruments intended to facilitate the learning and playing of music. The physical arrangement of pitches in most traditional musical instruments—including the piano and guitar—is non-isomorphic, which means that a given spatial relationship between two keys, buttons, or fretted strings can produce differing musical pitch intervals. Recently, a number of new musical interfaces have been developed with isomorphic pitch layouts where these relationships are consistent. Since the nineteenth century, it has been widely considered that isomorphic pitch layouts facilitate the learnability and playability of instruments, particularly when a piece is transposed into a different key; however, prior to this paper, this has not been experimentally tested. To address this, we investigated four different pitch layouts to examine whether isomorphism facilitates retention and transfer of musical learning within and across keys. Both non-musicians and musicians were tested on two training tasks: two immediate retention tasks and a transfer task. Each participant played every task on two distinct layouts—one being an isomorphic layout (Wicki or Bosanquet), the other being a minimally adjusted non-isomorphic version. For musicians, isomorphism was found to facilitate transfer of learning to a novel task; for non-musicians, the results were mixed. This study provides insight into features that are important to music instrument design. [ABSTRACT FROM AUTHOR]

Our main result is that the Steiner point removal (SPR) problem can always be solved with polylogarithmic distortion, which answers in the affirmative a question posed by Chan, Xia, Konjevod, and Richa in 2006. Specifically, we prove that for every edge-weighted graph G = (V,E,w) and a subset of terminals T ⊆ V , there is a graph G' = (T,E',w') that is isomorphic to a minor of G such that for every two terminals u, v ∊ T, the shortest-path distances between them in G and in G' satisfy dG,w(u, v) ≤ dG',w' (u, v) ≤ O(log5 ∣T∣) . dG,w(u, v). Our existence proof actually gives a randomized polynomial-time algorithm. Our proof features a new variant of metric decomposition. It is well known that every finite metric space (X, d) admits a β-separating decomposition for β = O(log∣X∣), which means that for every Δ > 0 there is a randomized partitioning of X into clusters of diameter at most Δ, satisfying the following separation property: for every x, y ∊ X, the probability that they lie in different clusters of the partition is at most β d(x, y)/Δ. We introduce an additional requirement in the form of a tail bound: for every shortest-path P of length d(P) ≤ Δ/β, the number of clusters of the partition that meet the path P, denoted by ZP , satisfies Pr[ZP > t] ≤ 2e-Ω(t) for all t > 0. [ABSTRACT FROM AUTHOR]

The article discusses how tuning invariance can be a musically useful property. It is stated that there are a variety of ways to tune musical intervals and scales, and that the introduction of computer and software synthesizers makes it possible to realize any sound in any tuning. According to the article, there are several technical, musical and perceptual questions that must be addressed to realize a keyboard that is both transpositionally and tuning invariant. It also discusses how intervals are identified and distinguished, and the criteria by which two numerically different intervals may be said to play analogous roles in different tuning systems.