...Roger Mathew Grant The music-theoretical formulations of the early modern mathematician Leonhard Euler (1707-83) are customarily remembered—and often dismissed—for their unconventional rankings of the consonances. A closer examination of Euler’s writings, however, reveals that he intended his...

... described in patents of 1829 and 1844. Like the later tonal spaces of the German dualist theorists, the concertina's button layouts were inspired by the work of eighteenth-century mathematician Leonhard Euler, who used a lattice to show relationships among pitches in just intonation. Wheatstone originally...

... Suzannah Rehding Alexander , 68 – 92 . Cambridge : Cambridge University Press . Damschroder David . 2008 . Thinking about Harmony: Historical Perspectives on Analysis . Cambridge : Cambridge University Press . Euler Leonhard . 1739 . Tentamen novae theoriae musicae ex...

... for horizontal and vertical relationships in the tonnetz] are assigned to acoustically pure intervals (as in Euler, etc or to intervals in pitch-space, then the struc- ture implicitly projects into an infinite plane. The realizations [of the ton- netz] that will hold our focus are generated by equally...

... . Eitz Karl . 1891 . Das mathematisch reine Tonsystem . Leipzig : Breitkopf und Härtel . Euler Leonhard . 1774 . “ De harmoniae veris principiis per speculum musicum repraesentatis .” In Novi Commentarii academiae scientiarum Petropolitanae , vol. 18 , 330 – 53 . St. Petersburg...

.... In particular, it is important to realize that the Tonnetz has a triply ambiguous status as a representation of acoustic, voice-leading, and common-tone relationships. From a historical point of view, the acoustic aspect is primary: Leonhard Euler originally designed...

... is therefore ϕ(N) (Euler’s ϕ-function), the number of integers mod N that are co-prime to N. In the current setting N = 24 and ϕ(N) = 8. See Dummit and Foote 1999, 58–59. 20. It is not generally true that every chain formed by a UTT of order n will cycle through n different triads before repeating...

... of the transforms of its component pitches: 2πipz FP(z) 5 Se . (24) p[P Using Euler’s identity, we substitute cosq 1 i sinq for e iq in Equation 24 to yield the Fourier transform of the characteristic function of P expressed in terms...