• 1) "James Tenney reconceived his piece For Ann (rising), which consists of up to twelve computer-generated upwardly glissandoing tones (see Shepard tone), as having each tone start so it is the golden ratio (in between an equal tempered minor and major sixth) below the previous tone, so that the combination tones produced by all consecutive tones are a lower or higher pitch already, or soon to be, produced. ErnÅ‘ Lendvai analyzes Béla Bartók's works as being based on two opposing systems, that of the golden ratio and the acoustic scale, though other music scholars reject that analysis. In Bartok's Music for Strings, Percussion and Celesta the xylophone progression occurs at the intervals 1:2:3:5:8:5:3:2:1. French composer Erik Satie used the golden ratio in several of his pieces, including Sonneries de la Rose+Croix. The golden ratio is also apparent in the organisation of the sections in the music of Debussy's Image, Reflections in Water, in which "the sequence of keys is marked out by the intervals 34, 21, 13 and 8, and the main climax sits at the phi position." The musicologist Roy Howat has observed that the formal boundaries of La Mer correspond exactly to the golden section. Trezise finds the intrinsic evidence "remarkable," but cautions that no written or reported evidence suggests that Debussy consciously sought such proportions. Also, many works of Chopin, mainly Etudes (studies) and Nocturnes, are formally based on the golden ratio. This results in the biggest climax of both musical expression and technical difficulty after about 2/3 of the piece. Pearl Drums positions the air vents on its Masters Premium models based on the golden ratio. The company claims that this arrangement improves bass response and has applied for a patent on this innovation. In the opinion of author Leon Harkleroad, "Some of the most misguided attempts to link music and mathematics have involved Fibonacci numbers and the related golden ratio." Source and further information: 2) "It is not well known that Mozart was fascinated by mathematics as well as music. He even jotted down equations in the margins of some of his compositions. Chances are excellent that he knew of the Golden Section and its reputation for conferring elegance on structures -- even musical compositions. J.F. Putz, a mathematician, has measured some of Mozart's works. Mozart's piano sonatas were convenient targets, because in Mozart's time they were customarily divided into two parts: (1) Exposition; and (2) Development and Recapitulation. Sure enough, the first movement of Mozart's Sonata No. 1 in C Major consists of 100 measures that are divided into the customary two parts; 38 in the first, 62 in the second. This ratio 38/62 (0.613) is as close as one can get to 0.618 in a composition of 100 measures. The second movement of this sonata is also divided according to the Golden Section, but the third movement is not. Many other Mozart piano sonatas seem to employ the Golden section, but some deviate considerably. So Putz could not really claim that Mozart consciously used the Golden Section to "improve" his music (Question #1 above), but there are certainly a lot of "coincidences." (May, Mike; "Did Mozart Use the Golden Section?" American Scientist, 84:118, 1996)" Source and further information: 3) "Expressed as a decimal, the perfect fifth is 2/3 = 0.666… which approximates GS [Golden Section]. No wonder it is the next most consonant-sounding interval after the unison and octave, and is such a common place to modulate." Source and further information: Further information: - "An essay on patterns in musical composition transformations, mathematical groups, and the nature of musical substance.": - "Fibonacci and Music": - "This is a bibliography of sources related to the Golden Section in Music, the Fine Arts, Architecture, Aesthetics, Gestalt Theory, Theory of Proportion etc.":

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