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.Footnote 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 (see Figure 1).
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.
The wave moves along the string in both directions and reflects back inverted at both ends.Footnote 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 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 partialFootnote 3 with frequency f 1 (Figure 1). This frequency is referred to as the fundamental frequency and is often heard as the pitch of the string.Footnote 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 (Figure 2) places nodes at distances 
${L \over b}$ apart: 
$\left( {0,{1 \over b}L,{2 \over b}L,...,{b \over b}L = L} \right)$. These nodes take the form 
${a \over b}L$ and the mode of vibration produces the bth partial with frequency f b = bf 1. Caspar Johannes Walter has simply 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.’Footnote 5
Figure 2: Mode of a bth partial produced by lightly touching the string at 
${a \over b}L$.
Touching the string at 
${a \over b}L$ dampens all modes of vibration that do not have a node at 
${a \over b}$. Only modes that allow the string to be at rest at 
${a \over b}$ may form standing waves and the perceived pitch of the string will change according to the lowest partial possible under these conditions. For example, touching at 
${1 \over 2}L$ dampens the 1st and all other odd partials, each of which has an anti-node at 
${1 \over 2}L$, while allowing the 2nd partial and its multiples to vibrate. The pitch of the string sounds an octave higher because the wavelength has been halved, causing a doubling of f 1.Footnote 6 Likewise, by touching at 
${1 \over 3}L$ or 
${2 \over 3}L$, both the 1st and 2nd partials are dampened while the 3rd partial and its multiples may sound. The sounding pitch is a perfect twelfth higher with frequency 3f 1.
Only integer ratios 
${a \over b}$ that are in lowest terms, i.e. where a and b are coprime, will sound partial b and its harmonics; these are called playable nodes.Footnote 7 If 
${a \over b}$ were not in lowest terms, then a simpler mode of vibration at that node would be possible, producing partial 
${b \over {\gcd \lpar {a,b} \rpar }}$ with frequency 
${b \over {\gcd \lpar {a,b} \rpar }}f_1$. For example, the 4th partial will only sound at 
${1 \over 4}L$ and 
${3 \over 4}L$ but not at 
${2 \over 4}L\equiv {1 \over 2}L \Rightarrow $ 2nd partial.
If the string is divided into b number of parts and b is 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 ϕ(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 
${a \over b}$ 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 harmonic partials called multiphonics – begin to emerge. What form do these take?
In his article ‘Mehrklänge auf dem Klavier. Vom Phänomen zur Theorie und Praxis mikrotonalen Komponierens’, Walter sketches a mathematical model of partials and multiphonics on strings.Footnote 8 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 
${0 \over 1}$ and 
${1 \over 1}$, 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.
For example, the sequence 
$\left\{ {{0 \over 1},{1 \over 1},{1 \over 2},{2 \over 3},{3 \over 5},{5 \over 8}} \right\}$ 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 
${1 \over 8}$, 
${3 \over 8}$, and 
${7 \over 8}$ 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.
$$\left\{ {\displaystyle{0 \over 1},\displaystyle{1 \over 1},\displaystyle{1 \over 2},\displaystyle{1 \over 3},\displaystyle{1 \over 4},\displaystyle{1 \over 5},\displaystyle{1 \over 6},\displaystyle{1 \over 7},\displaystyle{1 \over 8}} \right\}$$
$$\left\{ {\displaystyle{0 \over 1},\displaystyle{1 \over 1},\displaystyle{1 \over 2},\displaystyle{1 \over 3},\displaystyle{2 \over 5},\displaystyle{3 \over 8}} \right\}$$
$$\left\{ {\displaystyle{0 \over 1},\displaystyle{1 \over 1},\displaystyle{1 \over 2},\displaystyle{2 \over 3},\displaystyle{3 \over 4},\displaystyle{4 \over 5},\displaystyle{5 \over 6},\displaystyle{6 \over 7},\displaystyle{7 \over 8}} \right\}$$
The first of these sequences mutates after 
${1 \over 2}$, 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, 
${0 \over 1}$. The second sequence initially follows this same mutation until 
${1 \over 3}$, at which point the usual ‘summing’ of the preceding two terms is resumed. The third sequence mutates after 
${1 \over 2}$, at which point new terms are generated by finding the freshman's sum of the preceding term with 
${1 \over 1}$.
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 F k with degree k is the set of reduced fractions 
${a \over b}$ 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.
$$F_1 = \left\{ {\displaystyle{0 \over 1},\displaystyle{1 \over 1}} \right\}$$
$$F_2 = \left\{ {\displaystyle{0 \over 1},\displaystyle{1 \over 2},\displaystyle{1 \over 1}} \right\}$$
$$F_3 = \left\{ {\displaystyle{0 \over 1},\displaystyle{1 \over 3},\displaystyle{1 \over 2},\displaystyle{2 \over 3},\displaystyle{1 \over 1}} \right\}$$
$$F_4 = \left\{ {\displaystyle{0 \over 1},\displaystyle{1 \over 4},\displaystyle{1 \over 3},\displaystyle{1 \over 2},\displaystyle{2 \over 3},\displaystyle{3 \over 4},\displaystyle{1 \over 1}} \right\}$$Farey sequences have some unique properties:
a) any pair of consecutive fractions
$\left( {{a \over b},{p \over q}} \right)$ 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
$\left( {{a \over b},{p \over q}} \right)$ first appears in F max(b,q); these two fractions remain consecutive in all subsequent F ks until separated by their mediant ${m \over n} = {{a + p} \over {b + q}}$ in the Farey sequence F b+q;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
$\left( {{a \over b},{p \over q}} \right)$ is the fraction with the smallest denominator lying between ${a \over b}$ and ${p \over q}$;f) Farey sequences divide the interval between 0 and 1 symmetrically, i.e. every fraction
${a \over b}$ has a complement $1-{a \over b}$ appearing in the same sequence.
Proofs for these properties are presented in Appendix A.
3. Melodies – intervals – chords
3.1 The Farey sequence Fkmaps the playable nodes for all partials b ≤ k of a vibrating string
Since a Farey sequence F k lists all reduced fractions 
${a \over b}$ between 0 and 1 with denominators k or smaller in ascending numerical order, F k also lists all playable nodes along a string that produce partials up to and including partial k. The playable nodes for a given partial k are first enumerated in the Farey sequence F k. Properties of Farey sequences may therefore be applied to playable nodes:
• any two neighbouringFootnote 9 playable nodes in F k 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.
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 double bass (G2) with vibrating length 1050 mm from nut to bridge. A player may slide from the 2nd partial produced at 
${1 \over 2}L$ to reach the 3rd partial in either direction at 
${1 \over 3}L$ or 
${2 \over 3}L$.Footnote 10 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 F 19 gives an ordered list of all playable nodes up to the 19th partial. The denominators between 
${1 \over 3}$ and 
${1 \over 2}$ or 
${1 \over 2}$ and 
${2 \over 3}$ describe a melody of partials that may be sounded when sliding between the 2nd and 3rd partials. By symmetry, the melody from 
${1 \over 2}$ to 
${2 \over 3}$ is the retrograde of the melody from 
${1 \over 3}$ to 
${1 \over 2}$; each melodic interval is expressed by the ratio between successive denominators.
$$\left\{ {\displaystyle{1 \over 3},\displaystyle{6 \over {17}},\displaystyle{5 \over {14}},\displaystyle{4 \over {11}},\displaystyle{7 \over {19}},\displaystyle{3 \over 8},\displaystyle{5 \over {13}},\displaystyle{7 \over {18}},\displaystyle{2 \over 5},\displaystyle{7 \over {17}},\displaystyle{5 \over {12}},\displaystyle{8 \over {19}},\displaystyle{3 \over 7},\displaystyle{7 \over {16}},\displaystyle{4 \over 9},\displaystyle{5 \over {11}},\displaystyle{6 \over {13}},\displaystyle{7 \over {15}},\displaystyle{8 \over {17}},\displaystyle{9 \over {19}},\displaystyle{1 \over 2}} \right\}$$
$$\left\{ {\displaystyle{1 \over 2},\displaystyle{{10} \over {19}},\displaystyle{9 \over {17}},\displaystyle{8 \over {15}},\displaystyle{7 \over {13}},\displaystyle{6 \over {11}},\displaystyle{5 \over 9},\displaystyle{9 \over {16}},\displaystyle{4 \over 7},\displaystyle{{11} \over {19}},\displaystyle{7 \over {12}},\displaystyle{{10} \over {17}},\displaystyle{3 \over 5},\displaystyle{{11} \over {18}},\displaystyle{8 \over {13}},\displaystyle{5 \over 8},\displaystyle{{12} \over {19}},\displaystyle{7 \over {11}},\displaystyle{9 \over {14}},\displaystyle{{11} \over {17}},\displaystyle{2 \over 3}} \right\}$$ Multiplying each of these fractions by the vibrating length (L = 1050 mm) gives the positions of these playable nodes on a double bass (see Appendix B for a notated example with string lengths). Since each Farey pair 
$\left( {{a \over b},{p \over q}} \right)$ satisfies the property pb − aq = 1, the distance between them is 
${p \over q}-{a \over b} = {{pb-aq} \over {qb}} = {1 \over {qb}}$. Thus, the physical distance on a string between playable nodes with denominators q and b is 
${1 \over {qb}}L$. 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, are separated by 
${{1050\,{\rm mm}} \over {3 \times 17}} = {{1050\,{\rm mm}} \over {51}} = 21\,{\rm mm}$ (see Appendix B).
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 x 1 and x 2.
$$x_1b + x_2q = \gcd \lpar {b,q} \rpar = 1$$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 andq, first appearing in the Farey sequence F max(b,q). This indicates where along a string coprime partials b and q have neighbouring playable nodes.Footnote 11
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 R A.
$$13 = \lpar 2 \rpar 5 + 3$$STEP B. Repeat the process by expressing the smaller number (5) as a multiple of R A plus a new remainder R B.
$$5 = \lpar 1 \rpar 3 + 2$$STEP C. Repeat the process by expressing R A as a multiple of R B plus a new remainder R C.
$$3 = \lpar 1 \rpar 2 + 1$$
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.
$${\bf STEP}\; {\bf A}.\; \lpar 1 \rpar 13 + \lpar {-2} \rpar 5 = 3$$
$${\bf STEP}\; {\bf B}.\; \; \lpar 1 \rpar 5 + \lpar {-1} \rpar 3 = 2$$
$${\bf STEP}\; {\bf C}.\; \; \; \lpar 1 \rpar 3 + \lpar {-1} \rpar 2 = 1$$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 (R C = 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.
$$\lpar 1 \rpar 3 + \lpar {-1} \rpar \lsqb {{\mkern 1mu} {\kern 1pt} 5 + \lpar {-1} \rpar 3{\mkern 1mu} {\kern 1pt}} \rsqb = 1$$
$$\Rightarrow \lpar 2 \rpar 3 + \lpar {-1} \rpar 5 = 1$$Similarly, 3 may be replaced by the expression in STEP A, reducing STEP C to a linear combination of 5 and 13.
$$\lpar 2 \rpar \lsqb {{\mkern 1mu} {\kern 1pt} \lpar 1 \rpar 13 + \lpar {-2} \rpar 5{\mkern 1mu} {\kern 1pt}} \rsqb + \lpar {-1} \rpar 5 = 1$$
$$\Rightarrow \lpar 2 \rpar 13 + \lpar {-5} \rpar 5 = 1$$
$$\Rightarrow \lpar 2 \rpar 13-\lpar 5 \rpar 5 = 1$$ 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 coefficients 2 and 5. For a string with length L, partials 5 and 13 will have neighbouring nodes at 
${5 \over {13}}L$ and 
${2 \over 5}L$, forming the Farey pair 
$\left( {{5 \over {13}},{2 \over 5}} \right)$ in all Farey sequences F k where 13 ≤ k <(5 + 13).Footnote 12 For a guitar with vibrating length c. 635 mm, the distance between these neighbouring playable nodes is approximately 10 mm, while the distance on an average double bass with vibrating length c. 1050 mm is c. 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.Footnote 13 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.
$$a{\rm \; :\;} \lpar {{\rm 2}a+{b}} \rpar {\rm \; :\;} \lpar {{a}+{b}} \rpar {\rm \; :\;} b$$
$$a\,{\rm :\;} \lpar {{a}+{b}} \rpar {\rm \; :\;} \lpar {{a}+{\rm 2}b} \rpar {\rm \; :\;} b$$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 
$f_1 = {v \over {2L}}$. If a playable node at 
${a \over b}$ is stopped, the length of string vibrating is 
${a \over b}L$ with wavelength 
${a \over b} \times 2L$ and with frequency 
$v \div \left( {{a \over b} \times 2L} \right) = {b \over a} \times {v \over {2L}}$. Thus, stopping the string at a playable node will sound a pitch above the open string with frequency ratio equal to the inverse of the Farey fraction, namely 
${b \over a}$. The frequency of this pitch will be 
${b \over a}{\mkern 1mu} {\kern 1pt} f_1$. For example, stopping at 
${5 \over {12}}L$ (measured from the bridge) will produce a pitch with frequency twelve-fifths times that of the open string. If the double bass's G string is tuned to 98 Hz (when A4 is tuned to 441 Hz), this pitch will be tuned to 
${{12} \over 5} \times 98 = 235.2$ Hz.
Moving from one stopped playable node to the next produces a microtonal scale in which every musical interval is epimoric.Footnote 14 Since for Farey neighbours pb − aq = 1,
$$\displaystyle{\,p \over q} \div \displaystyle{a \over b} = \displaystyle{{\,pb} \over {aq}} = \displaystyle{{aq + 1} \over {aq}}.$$ Notice that in Farey sequence F k, 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 F k lies between 
${{k-2} \over {k-1}}$ and 
${{k-1} \over k}$, namely 
${{k\lpar {k-2} \rpar + 1} \over {k\lpar {k-2} \rpar }}$.
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.Footnote 15 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 c. 20 Hz and the common partial below c. 4000 Hz); and size (interval width should be larger than a tone due to critical band effects and generally smaller than c. 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 
${b \over a}$ and, given any two neighbouring ratios, finding the ‘next-simplest’ JI ratio that lies between them: their mediant. For example, consider approximating the equal-tempered major third T with interval size equal to 400 cents.Footnote 16T is nested between 
${2 \over 1}$ (octave, 1200 cents) and 
${1 \over 1}$ (open string (unison), 0 cents). By comparing T with the mediant of these two fractions, it is clear that T lies between 
${3 \over 2}$ (perfect fifth, 702 cents) and 
${1 \over 1}$. The sequence of nested bounds proceeds as follows.
$$\left( {\displaystyle{4 \over 3},\displaystyle{1 \over 1}} \right)\to \left( {\displaystyle{4 \over 3},\displaystyle{5 \over 4}} \right)\to \left( {\displaystyle{9 \over 7},\displaystyle{5 \over 4}} \right)\to \left( {\displaystyle{{14} \over {11}},\displaystyle{5 \over 4}} \right)$$ Since the interval 
${{14} \over {11}}$ is no longer directly tuneableFootnote 17 and all subsequent mediants will be non-tuneable ratios of even larger numbers, the nearest tuneable intervals to T are 
${5 \over 4}$ (386 cents) on the smaller side and 
${9 \over 7}$ (435 cents) on the larger side. The size of this tolerance region between neighbouring tuneable intervals is the microtonal interval 
${{36} \over {35}}$ (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.
$$\left( {\displaystyle{{19} \over {15}},\displaystyle{5 \over 4}} \right)\to \left( {\displaystyle{{24} \over {19}},\displaystyle{5 \over 4}} \right)\to \left( {\displaystyle{{29} \over {23}},\displaystyle{5 \over 4}} \right)\to \left( {\displaystyle{{29} \over {23}},\displaystyle{{34} \over {27}}} \right)$$ In this final range, 
${{29} \over {23}}$ has 401 cents and 
${{34} \over {27}}$ has 399 cents and the size of the region is 
${{783} \over {782}}$ (2 cents), well below experimental thresholds for just noticeable difference, measured at approximately 3 Hz for tones around 500 Hz, equivalent to c. 6–10 cents.Footnote 18
Stopped playable nodes in the range between 
${1 \over 1}$ and 
$\textstyle{1 \over 2}$ may also be considered as a gamut of octave-equivalent, ‘Monophonic’ pitch-classes in the sense introduced by Harry Partch.Footnote 19 Since Farey sequences introduce numerically larger frequency ratios in a systematic way, it could be compositionally useful to consider successive F ks 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.
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,Footnote 20 Graham, Knuth, and Pataschnik,Footnote 21et al.
Given a positive integer k, consider the collection F k of all rational numbers 
${a \over b}$ where
$0 \le {a \over b} \le 1$, b ≤ k, and 
${a \over b}$ 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, F k may be written in strictly increasing order and is called the Farey sequence of order k.
$$F_1 = \left\{ {\displaystyle{0 \over 1},\displaystyle{1 \over 1}} \right\}$$
$$F_2 = \left\{ {\displaystyle{0 \over 1},\displaystyle{1 \over 2},\displaystyle{1 \over 1}} \right\}$$
$$F_3 = \left\{ {\displaystyle{0 \over 1},\displaystyle{1 \over 3},\displaystyle{1 \over 2},\displaystyle{2 \over 3},\displaystyle{1 \over 1}} \right\}$$
$$F_4 = \left\{ {\displaystyle{0 \over 1},\displaystyle{1 \over 4},\displaystyle{1 \over 3},\displaystyle{1 \over 2},\displaystyle{2 \over 3},\displaystyle{3 \over 4},\displaystyle{1 \over 1}} \right\}$$ For k ≤ 4 in the preceding examples, it appears that consecutive fractions in Farey sequences, i.e. any neighbours 
${a \over b}$ and 
${p \over q}$ where 
${a \over b} \lt {p \over q}$, satisfy the property pb − aq = 1.
Definition 1.
Let any two fractions 
${a \over b} \lt {p \over q}$ that satisfy the property pb − aq = 1 be called a Farey pair.
Definition 2.
Define the mediant 
${m \over n}$ of 
${a \over b}$ and 
${p \over q}$ as
$$\displaystyle{m \over n} = \displaystyle{{a + p} \over {b + q}}.$$Theorem.
Any two neighbouring fractions in any Farey sequence F k form a Farey pair. The proof will proceed by induction. Note that 
${0 \over 1}$ and 
${1 \over 1}$ satisfy the definition of a Farey pair in f 1, so the Theorem holds for f 1.
Lemma 1.
Let 
${a \over b}$ and 
${p \over q}$ 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, 
${a \over b}$ and 
${p \over q}$ are in lowest terms.
Proof.
Let c be a positive integer that divides both a and b; then 
$p\left( {{b \over c}} \right)-\left( {{a \over c}} \right)q$ is an integer. Since pb − aq = 1, then 
${{pb-aq} \over c} = \displaystyle{1 \over c} \Rightarrow c = 1$. The same argument holds for each of the other pairs.
Lemma 2.
Let 
${a \over b}$ and 
${p \over q}$ be a Farey pair, and let 
${m \over n}$ be their mediant.
The following properties hold true.
$$\left( {\displaystyle{a \over b},\displaystyle{m \over n}} \right){\rm \; and}\,\left( {\displaystyle{{m} \over {n}}{\rm,} \displaystyle{{p} \over {q}}} \right){\rm \; are\; Farey\; pairs}$$
$$\displaystyle{a \over b} \lt \displaystyle{m \over n} \lt \displaystyle{p \over q}$$
$$\displaystyle{m \over n}{\rm is\; in\; lowest\; terms}$$
$${\rm if\; some\;} \displaystyle{{r} \over {s}}{\rm \; in\; lowest\; terms\; satisfies}\,\displaystyle{{a} \over {b}}\,{\rm \lt} \,\displaystyle{{r} \over {s}}\,{\rm \lt} \,\displaystyle{{p} \over {q}}\,{\rm, \; then\;} s \ge n$$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
$$mb-an = \lpar {a + p} \rpar b-a\lpar {b + q} \rpar = pb-aq = 1$$and
$$pn-mq = p\lpar {b + q} \rpar -\lpar {a + p} \rpar q = pb-aq = 1.$$Proof (2).
Property (2) follows from property (1) because
$$\displaystyle{m \over n}-\displaystyle{a \over b} = \displaystyle{{mb-an} \over {nb}} = \displaystyle{1 \over {nb}} \gt 0$$and
$$\displaystyle{\,p \over q}-\displaystyle{m \over n} = \displaystyle{{\,pn-mq} \over {qn}} = \displaystyle{1 \over {qn}} \gt 0.$$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
$$\displaystyle{\,p \over q}-\displaystyle{a \over b} = \displaystyle{{\,pb-aq} \over {qb}} = \displaystyle{1 \over {qb}}$$
$$\Rightarrow \displaystyle{1 \over {qb}} = \left( {\displaystyle{p \over q}-\displaystyle{r \over s}} \right) + \left( {\displaystyle{r \over s}-\displaystyle{a \over b}} \right) = \displaystyle{{ps-rq} \over {qs}} + \displaystyle{{rb-as} \over {sb}}$$ Since both terms in the sum are positive by the definition of 
${r \over s}$ in Property (4), as are qs and sb, ps − rq and rb − as must also be positive integers, and thus
$$\displaystyle{1 \over {qb}} \ge \displaystyle{1 \over {qs}} + \displaystyle{1 \over {sb}} = \left( {\displaystyle{{b + q} \over s}} \right)\left( {\displaystyle{1 \over {qb}}} \right)$$
$$\Rightarrow s \ge b + q = n.$$Proof of the Theorem.
Proceeding by induction, we assume for some k ≥ 1 that all neighbouring fractions in F k are Farey pairs. Let n = k + 1 and let
$$0 \lt \displaystyle{m \over n} \lt 1$$
be some fraction in lowest terms. Since 
$0 \lt {m \over n}$ and 0 is in F k, let 
${a \over b}$ be the largest element of F k less than 
${m \over n}$. Since 
${m \over n} \lt 1$ and 1 is in F k, then there is some 
${p \over q}$ in F k greater than 
${a \over b}$ that forms a Farey pair with it.
$$\displaystyle{a \over b} \lt \displaystyle{m \over n} \lt \displaystyle{\,p \over q}$$ It was shown in Lemma 2, property (4), that n ≥ b + q. By definition, 
${m \over n}$ is not in F k because its denominator is greater than k. However, if b + q <n, then 
${a \over b}$ and 
${p \over q}$ cannot be neighbours in F k because they would be separated by their mediant 
${{a + p} \over {b + q}}$. Therefore, 
${m \over n} = {{a + p} \over {b + q}}$. By Lemma 2, property (1), 
${m \over n}$ forms Farey pairs with its neighbours 
${a \over b}$ and 
${p \over q}$. The only remaining case in which the induction would not yet be proven is if there were two fractions with denominator n nested between 
${a \over b}$ and 
${p \over q}$. However, as noted above, this is not possible.
Corollary 1.
Any irreducible fractions 
${a \over b}$ and 
${p \over q}$, where 
$0 \le {a \over b} \lt {p \over q} \le 1$, are neighbours in F max(b, q).
Proof.
If 
${a \over b}$ and 
${p \over q}$ are not neighbours in F max(b, q), then there is some irreducible fraction 
${m \over n}$ that lies between them where 
${a \over b} \lt {m \over n} \lt {p \over q}$ and b + q ≤ n ≤ max (b, q). Since b, q > 0, there is no such n.
Corollary 2.
The Farey pair 
$\left( {{a \over b},{p \over q}} \right)$ first occurring in F max (b, q) will remain a Farey pair until it is separated by its mediant 
${{a + p} \over {b + q}}$ in F b+q.
Proof.
This follows from Lemma 2, properties (2) and (4).
B Playable nodes on a double bass
41 playable nodes between 
${1 \over 3}$ and 
${2 \over 3}$ (symmetric around 
${1 \over 2}$) in F 19 notated in the Helmholtz-Ellis JI Pitch Notation. Distances are measured from the bridge along the G string of a double bass with vibrating length (L) c. 1050 mm. Melodic ratios in italics denote the microtonal intervals between the nodes if played as stopped pitches.
