"I don't believe in the golden section. At least, I don't believe that it is the single formal principle for our sense of beauty; rather at most one among many, among countless many." - Arnold Schoenberg, Harmonielehre, 1911

"The systematic organization of a musical composition within a pre-determined time span by means of the lower numbers of the Fibonacci and Lucas series, singly or in combination, is common practice indeed. It seems that the more profound the composer, the stricter is his application of these proportions in the musical structure." - Hugo Norden, Proportions in Music, Fibonacci Quarterly, 1964

"Schillinger, inspired by the 'organic' nature of the golden mean, suggests deriving melodic lines from the Fibonacci series (and also from other summation series). He freely makes octave transpositions, so that the proportional and additive properties are lost, as is the Fibonacci source of the resulting melody..., and, if the resulting lines are in fact aesthetically pleasing, I suspect that the Fibonacci series is not the reason." - Jonathan Kramer, The Fibonacci Series in Twentieth-Century Music, Journal of Music Theory, 1973

"Geometry has two great treasures: one is the Theorem of Pythagoras; the other, the division of a line into extreme and mean ratio. The first we may compare to a measure of gold; the second we may name a precious jewel." - Johannes Kepler, Harmonices Mundi, 1609

"The golden ratio, also called by different authors the golden section, golden number, golden mean, divine proportion, and division in extreme and mean ratios, has captured the popular imagination and is discussed in many books and articles. Generally, its mathematical properties are correctly stated, but much of what is presented about it in art, architecture, literature, and esthetics is false or seriously misleading." - George Markowsky, Misconceptions about the Golden Ratio, 1992

"Gustav Ernest's attempt to apply the principle of the golden ratio to the temporal proportions of a piece seems to me to have failed completely indeed. First of all, one cannot carry over the laws of the visible world to the world of sound in this literal way. Second: If Ernest wants to use this principle in the structuring of three-part forms because of the number of measures, one must say that our perception of time cannot comprehend such long time spans in their temporal proportions." - Willy Hess, Die Dynamik der musikalischen Formbildung, 1960

"The golden number is irrational in mathematics. In music it is not possible (exception: electronic music). Providing durational values that a normal musician can count, I formed a [Fibonacci] series as just described." - Iannis Xenakis, Preface to Anastenaria, Le Sacrifice, 1953

"The golden section is one of the biological laws of growth. One finds it in the proportions of the human body; for example, the ratio of a person's height to the location of the solar plexus is the golden section...musical durations are created by muscular discharges which actuate the human limbs. It is clear that the movements of these limbs have a tendency to correspond temporally to the dimensions of these numbers. Thus the consequence: durations corresponding to the golden ratio are more natural for the movements of the human body." - Iannis Xenakis, Preface to Anastenaria, Le Sacrifice, 1953

"[The golden ratio is] that proportion which, as a standard proportion, underlies the arrangement of the human form, the construction of the more beautiful animals, the construction of plants, namely in the form of their leaf-arrangement, the forms of various crystals, the arrangement of the planets, the proportions of architectural and sculptural works recognized as being the most beautiful, the most satisfying chords in musical harmony, as well as many other things in nature and art." - Adolf Zeising, Das Normalverhältniss der chemischen und morphologischen Proportionen, 1856

"Because he [Adolf Zeising] is of the opinion that the same principle of pleasantness must be valid in both the field of vision and sound - and does not want to hear of anything at all that deprives golden ratio of power - he finds himself obligated, in clear contradiction to his general judgment, to declare both [major and minor] sixths to be the most musically pleasant proportions, for which he believes to find reasons in musical compositional proportions. As a result, he would clearly be forced instead of the pure sixth, to declare the impure sixth which corresponds exactly to the golden ratio, to be the musically most pleasant proportion..." - Gustav Theodor Fechner, Zur experimentalen Ästhetik, 1870

"The awareness, if not always clear, that in the most general sense the whole body is an organ and symbol of the spirit and character, and the mysterious feeling that the essence of beauty lies in the appropriate sensory expression of this idea, was established beyond this by artistic physiognomy, from which it branched off as part of the study of proportion, and after its regularity (which was nonetheless already known in ancient times) had been brought to a clearer awareness by Leonardo da Vinci and Albrecht Dürer, it took on a clearer form through the more recent work of Schmidt and Schadow among others, until Zeising finally found the one true key to it." - Friedrich Wilhelm Hagen, Der goldene Schnitt in seiner Anwendung auf Kopf- und Gehirnbau, Psychologie und Pathologie, 1857

"Constructing musical chords according to the scale of the golden ratio would be just as absurd as the apportionment of the human body according to the principles of the harmony of tones." - Constantin Winterberg, Preface to Divina Proportione, Die Lehre vom Goldenen Schnitt, 1889

"The golden section is subject to three conditions: (1) It fulfils its task only if [it] can be perceived. (2) It appears as an organic element of musical dramaturgy (my interest is in the interaction between the building elements, i.e. the balancing force functioning in symmetry, and the attracting and repelling power functioning in positive and negative sections). (3) It represents an idea - being the symbol of 'organic existence'." - Ernõ Lendvai, Comments on Jean-Bernard Condat's Book Review of the Workshop of Bartók and Kodály, 1988

"That golden section is something about which artists talk a lot of nonsense, but what Coxeter tells is quite interesting, and also one of the few portions of his book which I can understand." - Maurits Cornelis Escher, On My Meetings and Correspondence between 1960 and 1971 with the Graphic Artist M. C. Escher, 1962