# [MATH] [2021. II] Smooth numbers in music and architecture Abstract. Besides their natural setting within mathematics, *smooth numbers* also appear in art, particularly in music and architecture. ## Introduction >“...thou hast ordered all things in measure and number and weight.” - Solomon A natural number n ∈ N is *B-smooth* if none of its prime factors is greater than B. For example, 72 has prime factorization 2^3*3^2 and is hence 3-smooth. Study of smooth number has long history. In particular, 5-smooth numbers, also called *regular numbers*, have a prominent role in Babylonian mathematics. In modern mathematics, smooth numbers feature prominently in cryptography and various fast Fourier transform algorithms such as Cooley-Tukey FFT algorithm. Smooth numbers also appear in art, most notably, in music. In this note we cover their connection to musical scales and look how those scale come up, under a different guise, in architecture as well. ## Pythagorean and Ptolemy’s intense diatonic scales >“Music is the pleasure the human mind experiences from counting without being aware that it is counting.” - Leibniz Pythagoras and his school have laid the foundations not only of European mathematics but of music theory as well. In particular, their experimentation on monochord lead to the construction of a diatonic major scale generated by the progression of *perfect fifths*. Recall that a perfect fifth is a musical interval (i.e. difference in pitch between two sounds) corresponding to a pair of pitches with a frequency ratio of 3/2. Using the progression of perfect fifths we can obtain the following seven frequencies: 1 → 3/2 → 9/4 → 27/8 → 81/16 → 243/32 → 729/64. =>gemini://tilde.club/~filip/images/smooth_numbers_in_music_and_architecture/pythagoras.png Figure 1. Woodcut from Franchino Gaffurio’s Theorica musicae (1492) showing Pythagoras and his successor Philolaus doing musical experiments. Choosing the corresponding frequencies within the base octave (i.e. dividing with the appropriate power of 2) and assuming, without loss of generality, that the base pitch corresponds to be C, we obtain the C diatonic major scale. Pythagorean scale is an example of *just tuning*, i.e. tuning such that the frequencies of notes are rational numbers. ```pythagorean scale +---------------------------------------------------+ | C | D | E | F | G | A | B | C | | 1 | 9/8 | 81/64 | 4/3 | 3/2 | 27/16 | 243/128 | 2 | +---------------------------------------------------+ Table 1. Pythagorean scale. ``` Pythagorean scale is an example of *just tuning*, i.e. tuning such that the frequencies of notes are rational numbers. A B-limit tuning is a tuning that uses the rational numbers whose numerator and denominator are $B$-smooth. Pythagorean tuning is, therefore, an example of 3-limit tuning. While exceptionally elegant in its generation, Pythagorean scale has several drawbacks. For example, third, sixth, and seventh (notes of the scale) are difficult to tune. To overcame this, Ptolemy proposed a variation, called Ptolemy’s intense diatonic scale, by lowering the third, sixth, and seventh (E, A, and B in our example) by 81/80 (called syntonic comma). ```ptolemy’s intense diatonic scale. +--------------------------------------------+ | C | D | E | F | G | A | B | C | | 1 | 9/8 | 5/4 | 4/3 | 3/2 | 5/3 | 15/8 | 2 | +--------------------------------------------+ Table 2. Ptolemy’s intense diatonic scale. ``` Ptolemy’s scale is an example of 5-limit tuning. Notice that a lot of the ratios appearing in it are of the form (n+1)/n. Such ratios are called *superparticular ratios*. Moreover, the superparticular ratios in Ptolemy’s scale are such that both the numerator and denominator are 5-smooth. Interestingly, there are only finitely many such pairs (n,n+1) exist, a consequence of the famous Størmer’s theorem[1]. Theorem (Størmer’s theorem). Let P={p_1,...,p_k} be a set of distinct primes and let N_P={p^{m_1}_1,...,p^{m_k}_k | m_1,...,m_k ∈ N_0} be a set of all P-smooth numbers. Then, N_P contains finitely consecutive pairs, i.e. the set { n ∈ N | n,n+1 ∈ N_P} is finite. Moreover, Størmer proof is constructive and allows to obtain all pairs of consecutive B-smooth numbers. In particular, 5-smooth pairs (n,n+1) with n<100 are: (1,2), (2,3), (3,4), (4,5), (5,6), (8,9), (9,10), (14,15), (15,16), (20,21), (24,25), (27,28), (35,36), (63,64), (80,81). Represented as superparticular rations, many of them musically meaningful. ```some superparticular ratios and their musical interpretation +----------------------------+ | 2/1 | the octave | | 3/2 | the perfect fifth | | 4/3 | the perfect fourth | | 5/4 | the major third | | 6/5 | the minor third | | 9/8 | the whole step | | 10/9 | the minor tone | | 16/15 | the minor second | | 25/24 | the minor semitone | | 81/80 | the syntonic comma | +----------------------------+ Table 3: Some superparticular ratios and their musical interpretation. ``` ## Floor plans in Palladian architecture > “Music is liquid architecture; architecture is frozen music.” - Goethe Andrea Palladio, one of the foremost architects of the Italian Renascence, left a large body of work both practical, large part of it protected as UNESCO World Heritage Site `Palladian villas of the Veneto,` and theoretical - “The Four Books on Architecture”[2] being his most well-known treatise. Zeitgeist of his time emphasized rationalism and revival and reinterpretation of Gecko-Roman antiquity which naturally lead to aesthetic objectivism and the quest to explain the world in terms of ideal proportions. This is evident in the architecture of the time and in Palladio’s work in particular. Especially interesting are Palladio’s thoughts regarding floor plans. In the aforementioned treaties he lists seven best shapes for the floor plan: * circular * rectangular with a length l and width w that satisfy w/l ∈ {1, 4/3, sqrt(2), 3/2, 5/3, 2}. Preference for circular and square plans can be easily explained due to their pleasing symmetry. The choice of non-square rectangular plans, however, requires elaboration. Rudolf Wittkower was first to propose that the ratios of Palladio’s rectangular plans can be explained via musical scales ([3]; see [4,5] for more detailed study of this question). =>gemini://tilde.club/~filip/images/smooth_numbers_in_music_and_architecture/rotonda.jpg Figure 2: Andrea Palladio’s famous Villa Rotunda: (a) Picture of Villa Rotunda. =>gemini://tilde.club/~filip/images/smooth_numbers_in_music_and_architecture/rotonda_floor_plan.jpg Figure 2: Andrea Palladio’s famous Villa Rotunda: (b) Floor plan of Villa Rotunda. Indeed, if we compare the ratios suggested by Palladio with the ratios of the Ptolemy’s intense diatonic scale we see that, with the exception of 2, they correspond to unison (1) C, subdominant (4/3) F, dominant (3/2) G, (5/3) A, and octave (2) C. The connection with Ptolemy’s scale is not unexpected as it was widely used during Renascence. Famous musical theorist and composer Gioseffo Zarlino, Palladio’s contemporary, even declared it as the only reasonable scale with regards to singing. To give the musical interpretation of the remaining ratio sqrt(2)/1, it is necessary to move from Ptolomy’s scale to *equal tempered scale*. To overcome the practical difficulty in tuning the instrument so as to easily accommodate the change of key, by end of Baroque equal temperament tuning came to dominate European music. The beginning of this shift is seen in Renascence. For example, one of the early proponents of equal temperament was musical theorist and composers Vincenzo Galilei (father of Galileo Galilei). It equal temperament the octave is divided in 12 equal semisteps, each pitch therefore having frequency of a power of $\sqrt[12]{2}$. In particular, $\sqrt{2}=\sqrt[12]{2}^6$ which means that the ratio sqrt(2)/1 corresponds to a *tritone* (augmented fourth i.e. diminished fifth). It is interesting to note that while other ratios Palladio suggest correspond to consonant intervals the tritone is not only dissonant but possibly the most dissonant (it was historically referred as `diabolus in musica' and there were even attempts to officially ban its use in composition). # Back matter ## References [1] Størmer C., Quelques théoremes sur l’équation de Pell x^2-Dy^2=±1 et leurs applications, Skrifter Videnskabs-selskabet (Christiania), Mat.-Naturv. Kl. I (2). 1897. [2] Palladio A., I quattro libri dell'architettura, Venice, 1570. [3] Wittkower R., Architectural Principles in the Age of Humanism, London, 1949. [4] Mitrović B. and Djordjević I, Palladio’s Theory of Proportions and the Second Book of the “Quattro Libri dell'Architettura”, Journal of the Society of Architectural Historians Vol. 49, No. 3 (Sep., 1990), pp. 279-292. DOI: 10.2307/990519. [5] Mitrović B., A Palladian Palinode: Reassessing Rudolf Wittkower’s Architectural Principles in the Age of Humanism, Architectura, 31(2001), 113-131. ## Other information Gemtext version of a 2019. draft for a survey article.