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Published online by Cambridge University Press: 19 December 2019
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 The frequency of a transversely vibrating string depends on its length L and the speed of propagation of the wave v (related to the tension T and mass per unit length μ) as described by Mersenne's law: 
$f = {v \over {2L}}\equiv {1 \over {2L}}\sqrt {{T \over \mu}} $. 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, the frequency depends merely on the string's length and its physical properties: 
$f = {v \over {2L}}\equiv {1 \over {2L}}\sqrt {{E \over \mu}} $.
2 The amplitude of the vibration diminishes over time due to friction unless energy is continuously reintroduced into the system, by means of bowing.
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 Walter, Caspar Johannes, ‘Mehrklänge auf dem Klavier. Vom Phänomen zur Theorie und Praxis mikrotonalen Komponierens’, in Mikrotonalität – Praxis und Utopie, ed. Pätzold, Cordula and Walter, Caspar Johannes (Mainz: Schott Music, 2014), p. 16Google Scholar. From original: ‘Der Nenner definiert also die Tonhöhe und der Zähler konkretisiert die Position des Schwingungsknotens auf der Saite…’
6 Mersenne's law may be reformulated to relate frequency to the speed of propagation of the wave along the string and wavelength: 
$f = {v \over \lambda} $. 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 Walter, ‘Mehrklänge auf dem Klavier’, pp. 13–40.
9 Neighbouring, here, refers to neighbouring in some Farey sequence F k; as k increases, the physical distance between neighbouring playable nodes reduces.
10 Since Farey sequences are symmetric it may be assumed, for simplicity, that 
${1 \over 3}L$ and 
${2 \over 3}L$ 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 There are no playable nodes isolating partials less than min(b,q) lying between them.
12 Additionally, because Farey sequences are symmetric, there is a complementary Farey pair that also satisfies this condition at 
$\left( {1-{2 \over 5},1-{5 \over {13}}} \right)\,\Rightarrow \left( {{3 \over 5},{8 \over {13}}} \right)$. Note that 
$\left( {{a \over b},{p \over q}} \right)$ and its complement 
$\left( {1-{p \over q},1-{a \over b}} \right)$ are clearly equivalent by symmetry, depending merely on the end of the string from which one measures.
13 The ideal point of contact imparting energy to a given partial b is around the anti-node, i.e. at a distance between 
${1 \over 3}\left( {{1 \over b}} \right)$ and 
${2 \over 3}\left( {{1 \over b}} \right)$ from the bridge.
14 The interval between two frequency ratios is determined by dividing them. An epimoric or superparticular ratio takes the form 
${{n + 1} \over n}$. In musical terms, this is the interval between two successive partials in an harmonic series.
15 Sabat, Marc, 23-Limit Tuneable Intervals Above and Below A (Berlin: Plainsound Music Edition, 2005)Google Scholar.
16 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 
${b \over a}$, its size in cents is defined to be 
$1200 \times \log _2\left( {{b \over a}} \right)$.
17 Sabat, 23-Limit Tuneable Intervals Above and Below A.
18 Kollmeier, Birger, Brand, Thomas, and Meyer, Bernd, ‘Perception of Speech and Sound’, in Springer handbook of speech processing, ed. Benesy, Jacob, Sondhi, M.M., and Huang, Yiteng (New York: Springer, 2008), p. 65Google Scholar.
19 Partch, Harry, Genesis of a Music, 2nd edn (Boston: Da Capo Press, 1979)Google Scholar.
20 Hardy, G.H. and Wright, E.M., ‘The Farey Series and a Theory of Minkowski’, in An Introduction to the Theory of Numbers, 4th edn (London: Oxford University Press, 1975), pp. 23–37Google Scholar.
21 Graham, Ronald L., Knuth, Donald E., and Patashnik, Oren, Concrete Mathematics: A Foundation for Computer Science, 2nd edn (Boston: Addison-Wesley, 1989), pp. 115–139Google Scholar.