Farey Sequences Map Playable Nodes on a String∗ Thomas Nicholson and Marc Sabat Universität der Künste Berlin Studio für Intonationsforschung und mikrotonale Komposition Abstract Natural harmonics, i.e. partials and their harmonic series, may be isolated on a vibrating string by lightly touching specific points along its length. In addition to the two endpoints, stationary nodes for a given partial n present themselves at n − 1 locations along the string, dividing it into n parts of equal length. It is not the case, however, that touching any one of these nodes will necessarily isolate the nth partial and its integer multiples. The subset of nodes that will activate the nth partial (termed playable nodes by the authors) may be derived by following a mathematically predictable pattern described by so-called Farey sequences. The authors derive properties of these sequences and connect them to physical phenomena. This article describes various musical applications: locating single natural harmonics, forming melodies of neighbouring harmonics, sounding multiphonic aggregates, as well as predicting the relative tuneability of just intervals. 1 Playable nodes isolate partials of a vibrating string Consider a string fixed at both ends that is relatively thin with respect to its length L. Bowing or plucking disturbs the string by introducing a transverse wave, which causes the string to vibrate.1 Since the endpoints of the string are fixed, they remain at rest. A point of rest along a vibrating string is called a node while the point of maximum displacement exactly halfway between two nodes is called an anti-node. The wave moves along the string in both directions and reflects back inverted at both ends.2 Reflected waves that are in phase with waves travelling in the opposite direction establish a waveform that is constant in time. This stable superposition of waves travelling Published in TEMPO 74, no. 291 (2020): 86–97. doi:10.1017/S0040298219001001. (Note, this is the authors’ original manuscript). 1 The frequency of a transversely vibrating string depends on its length L and the speed of propagation of the wave v, q related to the tension T and mass per unit length µ as described by Mersenne’s law: 1 v f = 2L ≡ 2L Tµ . Note it is also possible to induce longitudinal vibration in a string by stroking (i.e. ∗ bowing parallel to the string), a technique Ellen Fullman uses to play her Long String Instrument. In the case of longitudinal vibration, propagation depends rather on small differences in local tension (stiffness) related to the string’s modulus E and not the mean tension T ; thus, inqthis case the frequency depends 1 E v ≡ 2L merely on the string’s length and its physical properties: f = 2L µ. 2 The amplitude of the vibration diminishes over time due to friction unless energy is continuously reintroduced into the system, e.g. by means of bowing. 1 Nicholson/Sabat : Farey Sequences Map Playable Nodes on a String 2 0 L Figure 1: The ends on the string form nodes while the point of maximum displacement in the middle of the string forms an anti-node. in opposite directions is called a standing wave because the string appears to be ‘standing still’. If the only nodes present in the standing wave are the endpoints, then the string is vibrating in its fundamental mode, producing the 1st partial3 with frequency f1 (Figure 1). This frequency is referred to as the fundamental frequency and is often heard as the pitch of the string.4 In the fundamental mode, a single propagation along the string only produces a half period of the sinusoidal waveform, so the wavelength of the 1st partial is twice the length of the string (2L). Any division of the string into a whole number of parts produces a possible mode of vibration. For example, dividing the string into b equal parts places nodes at distances Lb apart (0, 1b L, 2b L, ..., bb L = L). These nodes take the form ab L and the mode of vibration produces the bth partial with frequency fb = bf1 . Caspar Johannes Walter has summarised the function of these fractions in practical terms, writing ‘[t]he denominator defines ... the [resulting] pitch and the numerator makes the position of the node along the string concrete.’5 0 1 L b a L b b−1 L b L Figure 2: Mode of a the bth partial produced by lightly touching the string at ab L. Touching the string at ab L dampens all modes of vibration that do not have a node at a . Only modes that allow the string to be at rest at ab may form standing waves and, in b most cases, the perceived pitch of the string will change according to the lowest partial possible under these conditions. For example, touching at 12 L dampens the 1st and all other odd partials, each of which has an anti-node at 12 L, while allowing the 2nd partial and its multiples to vibrate. The pitch of the string sounds an octave higher because the 3 In this article, when the authors refer to producing or sounding “partials” of a string, the term is understood as referring to the fundamental frequency perceived. These sounds are actually aggregates consisting of a partial along with its integer multiples, forming an harmonic series. 4 Partials are individual sinusoidal frequency components of a non-sinusoidal sound. Aggregates that consist of whole number multiples of a single, generating fundamental frequency (present or not) are perceived as harmonic sounds because they form part of one harmonic series. Partials of a vibrating string are, for the most part, harmonic. 5 Caspar Johannes Walter, ‘Mehrklänge auf dem Klavier. Vom Phänomen zur Theorie und Praxis mikrotonalen Komponierens’, in Pätzold and Walther (eds.), Mikrotonalität – Praxis und Utopie, SMS Band 3 (Mainz: Schott Music, 2014), p. 16. From original: ‘Der Nenner definiert also die Tonhöhe und der Zähler konkretisiert die Position des Schwingungsknotens auf der Saite...’ Nicholson/Sabat : Farey Sequences Map Playable Nodes on a String 3 wavelength has been halved, causing a doubling of f1 .6 Likewise, by touching at 13 L or 2 L, both the 1st and 2nd partials are dampened while the 3rd partial and its multiples 3 may sound. The sounding pitch is a perfect twelfth higher with frequency 3f1 . Only integer ratios ab that are in lowest terms, i.e. where a and b are coprime, will isolate partial b and its harmonics; these are called playable nodes.7 If ab were not in lowest terms, then a simpler mode of vibration at that node would be possible, producing partial b b f1 . For example, the 4th partial will only sound at 14 L and with frequency gcd(a,b) gcd(a,b) 3 L but not at 42 L ≡ 12 L ⇒ 2nd partial. 4 If the string is divided into b parts and b is a prime, then lightly touching each of the b − 1 nodes (i.e. excluding endpoints) will sound partial b. For any other (nonprime) values of b, all nodes that coincide with simpler modes of vibration must be subtracted. The exact number of playable nodes that produce partial b is calculated by Euler’s totient function (b), which counts the number of positive integers a less than b that are relatively prime with b (or, equivalently, the number of positive integers a less than b such that ab is in lowest terms). As successively higher partials are activated, playable nodes fall increasingly close together when compared to the width of a finger producing harmonics and to the thickness of the string in question. Gradually, it becomes more and more difficult to play clear partials, especially on shorter, thicker strings. Instead, aggregates – clusters of neighbouring partials called multiphonics – begin to emerge. What form do these take? In ‘Mehrklänge auf dem Klavier. Vom Phänomen zur Theorie und Praxis mikrotonalen Komponierens’,8 Walter sketches a possible mathematical model of partials and multiphonics on strings.9 He bases his theory on empirical observations from which he has inferred a procedure that invokes the Fibonacci sequence and a further process of transformative branching that he calls mutation. Walter correctly recognises a pattern that emerges when ordering playable nodes. Beginning with the boundaries of a string at 01 and 11 , progressive divisions may be listed systematically by calculating new divisions, called mediants, which would lie between each pair of already existing fractions. The mediant is equal to the sum of the two numerators divided by the sum of the two denominators, sometimes called the ‘freshman’s sum’ of the two fractions. Because new terms of the Fibonacci sequence are also generated by summing the previous two terms, Walter associates this ‘addition’ of fractions with an expansion of Fibonacci terms in the numerators and denominators. This method produces sequences of fractions in which each new term lies between the preceding two, a back-and-forth ‘zig-zag’ of nested intervals narrowing around a particular location where a multiphonic may possibly be produced. 6 Mersenne’s law may be reformulated to relate frequency to the speed of propagation of the wave along the string and wavelength: f = λv . Since v is constant for a given string (depending on its tension and density), frequency is affected only by wavelength. 7 Here the term playable is used to mean potentially playable on an ideal string. Clearly, higher partials become increasingly difficult to play on a real string; in practice, the limit of playability depends on physical characteristics of the string, e.g. its length and material properties. 8 ‘Multiphonics on the Piano. From Phenomenon to Theory and Practice of Microtonal Composing’ 9 Walter, ‘Mehrklänge auf dem Klavier’, pp. 13–40. Nicholson/Sabat : Farey Sequences Map Playable Nodes on a String 4  For example, the sequence 01 , 11 , 21 , 23 , 35 , 85 follows the ‘Fibonacci rule’ to calculate numerators and denominators and leads to a particular multiphonic combination. However, to complete the set of all possible multiphonic combinations of playable nodes up to and including 8 requires three additional sequences: those leading to 18 , 38 , and 78 respectively. Generating each of these sequences follows a different mutation in Walter’s framework, where mediants are generated from two previous, but non-successive terms in the Fibonacci sequence.   0 1 1 1 1 1 1 1 1 , , , , , , , , 1 1 2 3 4 5 6 7 8   0 1 1 1 2 3 , , , , , 1 1 2 3 5 8   0 1 1 2 3 4 5 6 7 , , , , , , , , 1 1 2 3 4 5 6 7 8 The first of these sequences mutates after 12 , at which point each new term is generated by calculating the freshman’s sum of the previous term with the first term in the sequence, 0 . The second sequence initially follows this same mutation until 13 , at which point the 1 usual ‘summing’ of the preceding two terms is resumed. The third sequence mutates after 1 , at which point new terms are generated by finding the freshman’s sum of the preceding 2 term with 11 . Reformulating Walter’s observations through properties of Farey sequences, which systematically organise the fractions and their mediants in increasing order, makes it possible to more easily develop the theory of playable nodes in a rigorous and transparent way. This article discusses a theoretical mapping of string divisions based on these structures and derives various musical applications. 2 Properties of Farey sequences A Farey sequence Fk with of degree k is the set of reduced fractions ab between 0 and 1 whose denominators b do not exceed k, listed in strictly increasing order. The first four Farey sequences may be expressed in the following form.   0 1 F1 = , 1 1   0 1 1 F2 = , , 1 2 1   0 1 1 2 1 , , , , F3 = 1 3 2 3 1   0 1 1 1 2 3 1 , , , , , , F4 = 1 4 3 2 3 4 1 Nicholson/Sabat : Farey Sequences Map Playable Nodes on a String 5 Farey sequences have some unique properties:   a) any pair of consecutive fractions ab , pq in a Farey sequence is called a ‘Farey pair’; all Farey pairs have the property that pb − aq = 1, implying that the denominators b and q as well as the numerators a and p are relatively prime, i.e. share no common factors;   b) any Farey pair ab , pq first appears in Fmax(b,q) ; these two fractions remain consecutive in all subsequent Fk s until separated by their mediant Fb+q ; m n = a+p b+q in the Farey sequence c) once a Farey pair is separated by its mediant, the mediant forms a new Farey pair with each of the ‘parent’ fractions; d) the mediant of a Farey pair is in lowest terms;   e) the mediant of a Farey pair ab , pq is the fraction with the smallest denominator lying between ab and pq ; f) Farey sequences divide the interval between 0 and 1 symmetrically, i.e. every fraction a has a complement 1 − ab appearing in the same sequence. b Proofs for these properties are presented in Appendix A. 3 Melodies – intervals – chords 3.1 The Farey sequence Fk maps the playable nodes for all partials b ≤ k of a vibrating string Since a Farey sequence Fk lists all reduced fractions ab between 0 and 1 with denominators k or smaller in ascending numerical order, Fk also lists all playable nodes along a string that produce partials up to and including k. The playable nodes for a given partial k are first enumerated in the Farey sequence Fk . Properties of Farey sequences may therefore be applied to playable nodes: • any two neighbouring10 playable nodes in Fk produce partials that are coprime; conversely, given two coprime partials, it is possible to determine where along the string they have neighbouring playable nodes (see Subsection 3.2 below); • between any two neighbouring playable nodes with coprime partials b and q, the lowest partial with a playable node between them is b + q; • every pair of neighbouring playable nodes appears symmetrically from either end of the string. 10 I.e. neighbouring in some Farey sequence Fk ; as k increases, the physical distance between neighbouring playable nodes reduces. Nicholson/Sabat : Farey Sequences Map Playable Nodes on a String 6 Playable nodes may be interpreted musically in a number of ways. For example, while lightly gliding along a vibrating string, it is possible to produce melodies traversing neighbouring partials. Consider the highest string of a contrabass (G2) with vibrating length 1050 mm from nut to bridge. A player may slide from the 2nd partial produced at 12 L in either direction, reaching the 3rd partial at 13 L or 23 L respectively.11 The lowest partial occurring between the 2nd and 3rd is their sum, partial 5, though many other higher partials may also be sounded along the way. Choosing some arbitrary limit, e.g. 19, the Farey sequence F19 gives an ordered list of all playable nodes up to the 19th partial. The denominators between 13 and 12 or 12 and 23 describe a melody of partials that may be sounded when sliding between the 2nd and 3rd partials. By symmetry, the melody from 12 to 23 is the retrograde of the melody from 13 to 12 ; each melodic interval is expressed by the ratio between successive denominators.   1 6 5 4 7 3 5 7 2 7 5 8 3 7 4 5 6 7 8 9 1 , , , , , , , , , , , , , , , , , , , , 3 17 14 11 19 8 13 18 5 17 12 19 7 16 9 11 13 15 17 19 2   1 10 9 8 7 6 5 9 4 11 7 10 3 11 8 5 12 7 9 11 2 , , , , , , , , , , , , , , , , , , , , 2 19 17 15 13 11 9 16 7 19 12 17 5 18 13 8 19 11 14 17 3 Multiplying each of these fractions by the vibrating length (L = 1050 mm) gives the positions of these playable nodes on a contrabass  (see  Appendix B for a notated example a p with string lengths). Since each Farey pair b , q satisfies the property pb − aq = 1, = qb1 . Thus, the physical distance on a the distance between them is pq − ab = pb−aq qb string between playable nodes with denominators q and b is qb1 L. In other words, the distance between two neighbouring (relatively prime) partials is the string length divided by their product. In the example above, partials 3 and 17, producing the melody 3:17, mm are separated by 1050 = 105051mm = 21 mm (see Appendix B). 3×17 3.2 The extended Euclidean algorithm locates neighbouring nodal positions of any two coprime partials The definition of a Farey pair pb − aq = 1 has the shape of Bézout’s identity for coprime integers b and q, which expresses the greatest common divisor of b and q as a linear combination of b and q multiplied by integer coefficients x1 and x2 . x1 b + x2 q = gcd(b, q) = 1 The extended Euclidean algorithm may be used to calculate the coefficients of Bézout’s identity. The definition of a Farey pair ‘converts’ the coefficients into the numerators of a Farey pair with denominators b and q, first appearing in the Farey sequence Fmax(b,q) . Since Farey sequences are symmetric, it may be assumed, for simplicity, that 13 L and 23 L refer to fractions of the string measured from the bridge. Thus, when a node is stopped, the fraction will refer to the sounding length. 11 Nicholson/Sabat : Farey Sequences Map Playable Nodes on a String 7 This indicates where along a string coprime partials b and q have neighbouring playable nodes.12 As an illustration, consider partials 5 and 13. The extended Euclidean algorithm is employed to calculate Bézout’s identity. STEP A. Express the larger number (13) as a multiple of the smaller number (5) plus a remainder RA . 13 = (2)5 + 3 STEP B. Repeat the process by expressing the smaller number (5) as a multiple of RA plus a new remainder RB . 5 = (1)3 + 2 STEP C. Repeat the process by expressing RA as a multiple of RB plus a new remainder RC . 3 = (1)2 + 1 Once the remainder is equal to 1, which eventually happens whenever the two initial values are coprime, the algorithm stops. Each step above may be rearranged to express the remainder as a sum of the other terms. The coefficient ‘1’ is added where needed such that each term has a coefficient. STEP A. (1)13 + (−2)5 = 3 STEP B. (1)5 + (−1)3 = 2 STEP C. (1)3 + (−1)2 = 1 This rearranged form allows for the construction of Bézout’s identity. Working backward, earlier lines of the algorithm may be successively substituted to express the final remainder (RC = 1) in terms of a difference of the original integers (5 and 13) multiplied by coefficients. In STEP C, 2 may be replaced by the expression in STEP B. (1)3 + (−1)[ 5 + (−1)3 ] = 1 ⇒ (2)3 + (−1)5 = 1 Similarly, 3 may be replaced by the expression in STEP A, reducing STEP C to a linear combination of 5 and 13. (2)[ (1)13 + (−2)5 ] + (−1)5 = 1 ⇒ (2)13 + (−5)5 = 1 ⇒ (2)13 − (5)5 = 1 The definition of a Farey pair is pb − aq = 1. In this example, denominators b and q are assigned to partials 13 and 5 respectively and numerators p and a are assigned to the 12 That is, there are no playable nodes isolating partials less than min(b,q) lying between them. Nicholson/Sabat : Farey Sequences Map Playable Nodes on a String 8 coefficients 2 and 5. For a string with length L, partials 5 and 13 will have neighbouring
5 2 5 L and 25 L, forming the Farey pair 13 , 5 in all Farey sequences Fk where nodes at 13 13 ≤ k < (5 + 13).13 For a guitar with vibrating length ca. 635 mm, the distance between these neighbouring playable nodes is approximately 10 mm, while the distance on an average contrabass with vibrating length ca. 1050 mm is ca. 16 mm. 3.3 Branches of mediants in Farey sequences describe the components of multiphonics A multiphonic is an activation of several neighbouring playable nodes. It is facilitated by making a broader point of contact with the string in the region around these nodes and by plucking or bowing the string at a point maximising energy of the desired partials.14 The balance of components in the sound complex – partials, noise – is affected by both where and how the string is activated (finger position, plucking or bowing position, pressure, velocity). Production of multiphonics is also influenced by physical resistances within the string caused by its material qualities, e.g. stiffness, thickness, etc. These may also cause a certain degree of distortion in the harmonic spectrum of the string, slightly raising (‘stretching’) the actual frequencies of (some) higher partials. Despite these variations, the Farey sequence ordering holds and, therefore, may be used to predict the structure of possible multiphonics. In the simplest case, these take the form a : (a + b) : b where a and b are coprime partials. Erring on either side of the mediant (a + b) will favour either (2a + b) or (a + 2b) respectively, suggesting two possible four-pitch aggregates for any coprime a and b. a : (2a + b) : (a + b) : b a : (a + b) : (a + 2b) : b It would be useful in further studies to determine how various real-world parameters affect these predictions. Which positions on what kinds of strings favour the production of multiphonics? What is the relationship between overall string length, the size of the point of contact (fingertip or other object), and highest partials isolated? How do the frequency components of multiphonics produced at similar positions on a variety of strings and instruments vary? 3.4 Farey sequences define an harmonic space of stopped pitches By Mersenne’s law, a string of length L vibrates with wavelength 2L and frequency v f1 = 2L . If a playable node at ab is stopped, the length of string vibrating is ab L with v wavelength ab × 2L and with frequency v ÷ ( ab × 2L) = ab × 2L . Thus, stopping the string at a playable node will sound a pitch above the open string with frequency ratio equal to 13 Additionally, because Farey sequences there is a complementary Farey pair that also
are 3symmetric,
5 8 satisfies this condition at 1 − 52 , 1 − 13 . Note that ( ab , pq ) and its complement (1 − pq , 1 − ab ) ⇒ 5 , 13 are clearly equivalent by symmetry, depending merely on the end of the string from which one measures. 14 The ideal point of contact imparting energy to a given partial b is around the antinode, i.e. at a distance between 13 ( 1b ) and 23 ( 1b ) from the bridge. Nicholson/Sabat : Farey Sequences Map Playable Nodes on a String 9 the inverse of the Farey fraction, namely ab . The frequency of this pitch will be ab f1 . For 5 L (measured from the bridge) will produce a pitch with frequency example, stopping at 12 twelve-fifths times that of the open string. If the contrabass’ G string is tuned to 98 Hz (when A4 is tuned to 441 Hz), this pitch will be tuned to 12 × 98 = 235.2 Hz. 5 Moving from one stopped playable node to the next produces a microtonal scale in which every musical interval is epimoric.15 Since for Farey neighbours pb − aq = 1, p a pb aq + 1 ÷ = = . q b aq aq Notice that in Farey sequence Fk , the largest denominator is k and the largest numerator is k − 1. But, if one fraction has denominator k, then its Farey neighbour cannot have numerator k − 1 since its denominator would then have to be k as well and k is not coprime with itself. The largest possible a is therefore k − 2 with aq = k(k − 2). The smallest melodic step between stopped playable nodes in Fk lies between k−2 and k−1 , k−1 k k(k−2)+1 namely k(k−2) . In the early 2000s, inspired by the experience of playing James Tenney’s Koan, Marc Sabat undertook an empirical investigation of microtonal intervals on string instruments, comparing the sound of all lowest terms ratios of numbers up to and including 28.16 The intention was to establish a set of just intonation intervals tuneable directly by ear. Such intervals allow musicians playing non-fixed-pitch instruments to reliably establish microtonal pitches and provide a possible psychoacoustic basis for just intonation (JI) composition. Tuneable intervals are constrained by various factors affecting the perception of harmonic sound: timbre (harmonic, spectrally rich sonorities are easier to tune); register (the periodicity pitch or common fundamental should be generally above ca. 20 Hz and the common partial below ca. 4000 Hz); and size (interval width should be larger than a tone due to critical band effects and generally smaller than ca. 3 octaves due to segregation of the frequencies). Tuneability of intervals correlates, in part, with the timbral sound of ‘spectral fusion’. Farey sequences offer a systematic method of ordering candidate tuneable intervals at stopped playable nodes ab and, given any two neighbouring ratios, finding the ‘next-simplest’ JI ratio that lies between them: their mediant. For example, consider the equal-tempered major third T with interval size equal to 400 cents.17 T is nested between 21 (octave, 1200 cents) and 11 (open string or unison, 0 cents). By comparing T with the mediant of these two fractions, it is clear that T lies between 32 15 The interval between two frequency ratios is determined by dividing them. An epimoric or superparticular ratio takes the form n+1 n . In musical terms, this is the interval between two successive partials in an harmonic series. 16 Marc Sabat, 23-Limit Tuneable Intervals above and below A (Berlin: Plainsound Music Edition, 2005). 17 Mathematician Alexander J. Ellis (1814–1890) proposed the division of each equal tempered semitone into very small, equal units of measure. By dividing each semitone into 100 units called ‘cents’, the octave is pixelated into an equal division scale of 1200 parts, which may be used as a kind of ruler to measure and compare the absolute sizes of intervals. For a stopped pitch on a string at division ab , its size in cents is defined as 1200 × log2 ( ab ). Nicholson/Sabat : Farey Sequences Map Playable Nodes on a String 10 (perfect fifth, 702 cents) and 11 . The sequence of nested bounds proceeds as follows. ( 43 , 11 ) → ( 43 , 45 ) → ( 97 , 54 ) → ( 1411 , 45 ) Since the interval 14 is no longer directly tuneable18 and all subsequent mediants will 11 be non-tuneable ratios of even larger numbers, the nearest tuneable intervals to T are 5 (386 cents) on the smaller side and 97 (435 cents) on the larger side. The size of this 4 tolerance region between neighbouring tuneable intervals is the microtonal interval 36 35 (49 cents, approximately a quartertone). To find the ‘simplest’ frequency ratios nesting 400 cents within a practically imperceptible tolerance of 2 cents, the process may be continued. 19 5 24 5 29 5 29 34 , , , , → → → 15 4 19 4 23 4 23 27 ( ) ( ) ( ) ( ) Comparing the size of these last two ratios, 29 has 401 cents and 34 has 399 cents and 23 27 783 the size of the region they enclose is 782 (2 cents), well below experimental thresholds for just noticeable difference, measured at approximately 3 Hz for tones around 500 Hz, equivalent to ca. 6–10 cents.19 Stopped playable nodes in the range between 11 and 12 may also be considered as a gamut of octave-equivalent, ‘Monophonic’ pitch-classes in the sense introduced by Harry Partch.20 Since Farey sequences introduce numerically larger frequency ratios in a systematic way, it could be compositionally fruitful to consider successive Fk s filtered by prime factors (or up to a certain prime limit, like Partch’s 11-limit scale) to generate subsets suggesting musically interesting proximities and enharmonic moves within harmonic space. 18 Sabat, 23-Limit Tuneable Intervals. B. Kollmeier, T. Brand, and B. Meyer, ‘Perception of Speech and Sound’, in Benesty, Sondhi, and Huang (eds.), Springer handbook of speech processing, (New York: Springer, 2008), p. 65. 20 Harry Partch, Genesis of a Music, 2nd Edition (Boston: Da Capo Press, 1979). 19 Nicholson/Sabat : Farey Sequences Map Playable Nodes on a String 11 Appendices A Mathematical proofs: Farey Sequences Though properties of Farey sequences have already been derived in a number of elegant and rigorous ways, the authors devised the following proofs to clarify specific results referred to in this text. We wish to acknowledge the excellent and comprehensive mathematical contributions to this domain in Hardy and Wright,21 Graham, Knuth, and Pataschnik,22 et al. Given a positive integer k, consider the collection Fk of all rational numbers ab where 0 ≤ ab ≤ 1, b ≤ k, and ab is in lowest terms. Only a finite number of fractions satisfies these conditions for each k and, being in lowest terms, they are unique. Therefore, Fk may be written in strictly increasing order and is called the Farey sequence of order k.   0 1 F1 = , 1 1   0 1 1 , , F2 = 1 2 1   0 1 1 2 1 F3 = , , , , 1 3 2 3 1   0 1 1 1 2 3 1 , , , , , , F4 = 1 4 3 2 3 4 1 For k ≤ 4 in the preceding examples, it appears that consecutive fractions in Farey sequences, i.e. any neighbours ab and pq where ab < pq , satisfy the property pb − aq = 1. Definition 1. Let any two fractions ab < pq that satisfy the property pb − aq = 1 be called a Farey pair. of ab and pq as Definition 2. Define the mediant m n m a+p = . n b+q Theorem. Any two neighbouring fractions in any Farey sequence Fk form a Farey pair. The proof will proceed by induction. Note that 01 and 11 satisfy the definition of a Farey pair in F1 , so the Theorem holds for F1 . Lemma 1. Let ab and pq be a Farey pair. It follows that all of the following pairs of integers are coprime: (a, b), (p, q), (a, p), and (b, q). In particular, ab and pq are in lowest terms. 21 G.H. Hardy and E.M. Wright, ‘The Farey Series and a Theory of Minkowski’, An Introduction to the Theory of Numbers, 4th Edition (London: Oxford University Press, 1975), pp. 23–37. 22 R.L. Graham, D.E. Knuth, and O. Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd Edition (Boston: Addison-Wesley, 1989), pp. 115–139. Nicholson/Sabat : Farey Sequences Map Playable Nodes on a String 12

Proof. Let c be a positive integer that divides both a and b; then p cb − ac q is an = 1c ⇒ c = 1. The same argument holds for each integer. Since pb − aq = 1, then pb−aq c of the other pairs. Lemma 2. Let ab and pq be a Farey pair, and let m be their mediant. n The following properties hold true. a m m p ( b , n ) and ( n , q ) are Farey pairs (1) m p a < < b n q (2) m is in lowest terms n r a r p if some in lowest terms satisfies < < , then s ≥ n s b s q (3) (4) An obvious consequence of Properties (1) and (4) is that there cannot be two lowest terms fractions with the same denominator nested inside a Farey pair. Proof (1). Property (1) follows from the Farey pair definition because mb − an = (a + p)b − a(b + q) = pb − aq = 1 and pn − mq = p(b + q) − (a + p)q = pb − aq = 1. Proof (2). Property (2) follows from property (1) because m a mb − an 1 − = = >0 n b nb nb and pn − mq 1 p m − = = > 0. q n qn qn Proof (3). Property (3) follows from Lemma 1 and property (1) because all fractions that form Farey pairs are in lowest terms. Proof (4). Property (4) follows because pb − aq 1 p a − = = q b qb qb ⇒ 1 p r r a ps − rq rb − as = − + − = + qb q s s b qs sb ( ) ( ) . Since both terms in the sum are positive by the definition of rs in Property (4), as are qs and sb, ps − rq and rb − as must also be positive integers, and thus 1 1 1 b+q ≥ + = qb qs sb s ( ⇒ s ≥ b + q = n. )( qb1 ) Nicholson/Sabat : Farey Sequences Map Playable Nodes on a String 13 Proof of the Theorem. Proceeding by induction, we assume for some k ≥ 1 that all neighbouring fractions in Fk are Farey pairs. Let n = k + 1 and let 0< m <1 n be some fraction in lowest terms. Since 0 < m and 0 is in Fk , let ab be the largest element n < 1 and 1 is in Fk , then there is some pq in Fk greater than ab of Fk less than m . Since m n n that forms a Farey pair with it. a m p < < b n q It was shown in Lemma 2, property (4), that n ≥ b + q. By definition, m is not in Fk n a because its denominator is greater than k. However, if b + q < n, then b and pq cannot be neighbours in Fk because they would be separated by their mediant a+p . Therefore, b+q a+p m m = b+q . By Lemma 2, property (1), n forms Farey pairs with its neighbours ab and pq . n The only remaining case in which the induction would not yet be proven is if there were two fractions with denominator n nested between ab and pq . However, as noted above, this is not possible. Corollary 1. Any irreducible fractions ab and pq , where 0 ≤ ab < pq ≤ 1, are neighbours in Fmax(b,q) . Proof. If ab and pq are not neighbours in Fmax(b,q) , then there is some irreducible fraction m that lies between them where ab < m < pq and b + q ≤ n ≤ max(b, q). Since b, q > 0, n n there is no such n.   Corollary 2. The Farey pair ab , pq first occurring in Fmax(b,q) will remain a Farey pair in Fb+q . until it is separated by its mediant a+p b+q Proof. This follows from Lemma 2, properties (2) and (4). B Playable nodes on a contrabass Overleaf: 41 playable nodes between 13 and 23 (symmetric around 12 ) in F19 notated in the Helmholtz-Ellis JI Pitch Notation. Distances are measured from the bridge along the G string of a contrabass with vibrating length (L) ca. 1050 mm. Melodic ratios in italics denote the microtonal intervals between the nodes if played as stopped pitches. Nicholson/Sabat : Farey Sequences Map Playable Nodes on a String :f  17° sounding contrabass (G string) snd cb (I) snd cb (I) snd cb (I) snd cb (I)    3°  n 17° -18 :>f 2 5 420mm   >u +16 120 : 119 4 11°     573mm n18°  5 9 583mm -10 497mm n -8 n 1 2 -4 525mm 7° 9 16 < -35 4 7 600mm /e  19° 0 n8°  144 : 143 642mm 0 10 19 553mm 65 : 64 96 : 95 E+42 /5 64 : 63 7 16 n :f -3 608mm 9 17 > +29 0 5 11 u 91 : 6 13 Ab-35 485mm 477mm 013°  105 : 104 0>v 8 15 78 : 7 13 -32 -16 560mm 565mm :f  17° 5° m 120 : 119 7 12 613mm +15 :g 51 : 50 10 17 m -20 55 : 3 5 630mm 618mm :f  17° < 14° 4 4f 4 9 136 : 135 556mm 133 : 132 66 : 65 A-39 u15°  n 11 19 45 : 44 467mm 12° 11° 3° n 133 : 132 Eb+37 11 18 /f 77 : 76 64 : 63  013° 171 : 170 /e  < n  411° 17° +7 -50 7 18 408mm :f  20 : 19 > +31 5 13 404mm 459mm 19° 591mm 13° Eb+49  5 2° /e  9 19 81 : 80 o 450mm 19 : 18  n16° +14 6 11 442mm 19° /e 494mm 55 : 54 +45 : 54 8 17  n9° : 77  4 :f > +27 3 7 -37 n -6 9° 49 : 48 <e 8 19 -6 153 : 152 +1 490mm /e 5 12 19° 17° 7 15 f 57 : 56 /e  :f  : 90 < 0o 3 8 36 : 394mm  n16° 7° 96 : 95 438mm 432mm u15°  19° +12 n 7 19 387mm 382mm /e  n 7 17 /> 4 11 91 : 90 40 : 39 57 : 56 +25 n18°  013°  n8°  77 : 76 4 5 14 375mm 85 : 84 +32 4 +47 -21 12° 35 : 34 19° 56 : 55 <f 6 17 371mm :f  : 35  m :f 1 3 /e  11° 85 : 84 +3 350mm 5° 14° 18 : 17 n  -2  m < 14 99 : 98 154 : 153 34 : 33 D#-21 8 13 646mm f +10 5 8 656mm /e 12 19 -8 663mm 4> 7 11 668mm <e 9 14 -39 675mm :5f 11 17 D+50 679mm n 2 3 -2 700mm