Dissonance, sound spectrum and musical scale for ancient idiophones and aerophones

Journal of Mathematics and Music,2025 https://doi.org/10.1080/17459737.2025.2560922 Dissonance, sound spectrum and musical scale for ancient idiophones and aerophones Victor Etxebarria Ecenarro ∗ Department of Electricity and Electronics, Faculty of Science and Technology, University of the Basque Country UPV/EHU, Leioa, Spain (Received 22 May 2025; accepted 10 September 2025) We do not know how the earliest musical instruments—such as idiophones and aerophones—were played, but their acoustic properties can provide valuable clues. As a first step, we present here the concept of dissonance curves for a sound of a given spectrum. These curves show the relative dissonance that results for all intervals of a given musical instrument. This idea leads to the association of spectra and scales, which are related because the dissonance curve has minima in the intervals that define the scale. A computational method for calculating dissonance curves is presented and several examples of its use in practical cases, both for Western and Eastern musical instruments, are given and interpreted. These results allow us to explain from a physical point of view the reason of existence of well-known modern 12-note scales, as well as some uncommon but documented scales for various instruments in early musical history. Keywords: Musical instruments; ethnomusicology; dissonance; musical scale; spectrum 1. Introduction Idiophones for rhythm and aerophones for melody are probably what our ancestors played over 50,000 years ago (Morley 2013). The evolution of ancient musicality is unknown, but we can try to deduce how early musical instruments sounded by analyzing their acoustic properties and combining this with the importance of their tuning and temperament (Barbour 1951). Archaeological sites have revealed many musical instruments made of stone, bone, wood or metal. All civilizations have developed such instruments simultaneously, and although Western and Eastern music partly developed independently, we can find common ground in both musical areas by comparing ancient instruments developed in both geographical ends. Documented styles and scales of early idiophones such as the Javanese gambang or the Thai renat (Morton 1976)intheEast,andtheGambianbala or the Basque txalaparta in the West (Jones 1971;Beltran 2013), seem to be absolutely different. Although their respective cultures have evolved separately, we may find similar instruments whose acoustic principles we can analyze. Thai culture has been in contact with other civilizations for centuries, and Thai music and musical instruments have been influenced by China, Indonesia and India, among others. Some of the musical instruments used in Thai classical music are a type of xylophone (the renat ek and its ∗Email: victor[email protected] © 2025 Informa UK Limited, trading as Taylor & Francis Group This is an Open Access article distributed under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives License (http://creativecommons.org/licenses/by-nc-nd/4.0/), which permits non-commercial re-use, distribution, and reproduction in any medium, provided the original work is properly cited, and is not altered, transformed, or built upon in any way. The terms on which this article has been published allow the posting of the Accepted Manuscript in a repository by the author(s) or with their consent. 2V. Etxebarria Ecenarro low-pitched version, the renat thum), and melodic aerophones such as the pi and other melodic instruments such as the jakeh (a type of zither). The txalaparta is an ancient idiophone, originally from the Basque Country (European region in the Western Pyrenees) whose peculiarities as a percussion instrument and its mysterious history have aroused great interest among musicologists and historians. The first historical reference to the txalaparta appears in 1882 in a book on cider production in the Basque Country (Aguirre-Miramon 1882, 129), although there are earlier mentions of toberas (a metal variant of the txalaparta). The first of these is in a legal document from 1688 (Lekuona 1920,52–53). Melodic instruments such as the alboka (an ancient type of clarinet) were mentioned in 1443 in the Basque Country (Donostia 1952). Very little is known about the practice of the txalaparta and alboka before the twentieth century, but anthropologists, historians, musicians and other scholars have placed the instrument on a new path of growth, use and cultural renewal of great international interest. Music is usually considered to be a purely social science, belonging only to the humanities, since it deals with knowledge considered to have been invented by and for mankind. The fact that certain musical scales exist is still an important study for musicologists, anthropologists or psychologists (Gill and Purves 2009). Other social scientists also study related musical properties, such as consonance or dissonance of sound, also approaching the study of human musicality (Bowling and Purves 2015)andthetuningofmusicalinstrumentsasanaspectmainly related to human preference (Friedman et al. 2021). The study of cultural preference or musical psychoacoustics (Eerola and Lahdelma 2021)isanareathatstillrequiresmuchresearch. Important advances in the perception of familiar or unfamiliar music (Smit and Milne 2021)and human preferences for musical harmony (Stolzenburg 2015)havebeenmadebyresearchersin multidisciplinary fields. Large scale analysis of the perception of consonance and dissonance, such as the important Harrison and Pearce psychoacoustic model (Harrison and Pearce 2020)hasidentifiedthree main categories (roughness, harmonicity, familiarity), to evaluate or predict dissonance. Other researchers (Eerola and Lahdelma 2021)haveincludednewelementssuchasspectralenvelope as an additional category. The order of importance of these features on the perception of consonance and dissonance remains an important field not completely known which requires further research, especially in the context of cross-cultural studies of tuning. The main objective of this study is to provide physical reasons to explain the existence of specific musical scales, instead of using the classical mathematical temperaments. Based on the calculation of certain dissonance curves, the ordinary Western 12 equal temperament and the special Eastern seven equal temperament scales appear naturally. In this context, the non-harmonic spectrum of ancient idiophones is a key component of the proposed acoustic application. The existence of certain musical scales and the associated tuning of musical instruments can be treated from the point of view of acoustics as a branch of applied physics. Our approach here is based on the basic acoustic properties of both modern idiophones as well as some ancient musical instruments, which can give us some insight into how they might have been played. The study is structured as follows: section 2 introduces materials and methods, including abriefreviewoftheclassicalidentificationofdissonanceappliedtomusicalsounds.Then, section 2.1 presents a basic method for calculating dissonance in sounds of any spectrum, and we propose a computational technique for easy calculation of dissonance curves. In section 3, results of the method applied to both harmonic and non-harmonic instruments are given. In particular, it is shown how the well-known modern 12 semitone scale, approximately equal tempered, is obtained directly. Also, by applying the dissonance curves to modern xylophones, valuable interpretations of their musical characteristics are derived. In section 3.3 the proposed technique is applied to the non-harmonic renat and txalaparta and their related traditional Thai and Basque instruments, respectively. Interesting features of these ancient instruments as well as Journal of Mathematics and Music 3 Figure 1. Areas of roughness calculated and plotted by H. Helmholtz in 1870 when two violin notes are played simultaneously (Helmholtz 1954). appropriate but uncommon scales for their use are found. In the final section, the obtained results are discussed, and the main conclusions of the study are summarized. 2. Materials and methods One of the most famous studies of the perception of consonance and dissonance in music was by Helmholtz (1954), who proposed a model of dissonance based on the phenomenon of beats. When two pure tones of close frequencies sound simultaneously, the interference of the two tones produces beats. The beats become slower as the frequencies of the tones become more similar, and disappear when the frequencies coincide. Slow beats are typically perceived as smooth waves, but fast beats tend to be rough and unpleasant, with maximum roughness observed when the beats occur around 32 times per second. Considering that every sound can be broken down into sinusoidal partials, Helmholtz concluded that the dissonance perceived when listening to two tones simultaneously is caused by the rapid beating of the partials. Thus, according to Helmholtz, consonance is the absence of such dissonant beats. Assuming that the roughness of all interacting partials of two tones add up, the dissonance of any interval can be calculated simply by considering all possible combinations of pairs of partials and summing their contributions. Helmholtz performed this type of calculation for harmonic sounds such as those produced by violins (Etxebarria and Riera 2002), and presented the results graphically in diagrams such as the one shown in Figure 1, taken directly from Helmholtz (1954). The horizontal axis represents the interval between the two tones. One remains at a constant frequency (labelled c’), and the other one moves from c’ to the upper octave (labelled c’’). The height of the curves (vertical axis) is proportional to the roughness produced by the partials whose frequency ratios are labelled on the plot, using the maximum roughness criterion for 32 Hz beats. The result is a plot that has minima (intervals where minimum roughness occurs) near many of the intervals of the major scale, suggesting a relationship between the phenomenon of beats and the musical notions of consonance and dissonance. Helmholtz’s method for calculating roughness is narrated in his book (Helmholtz 1954)on pages 192–194, where he admits he was “forced to assume a somewhat arbitrary law” and chose “the simplest mathematical formula,” which he does not specify analytically but on his graph. Instead of mathematically modelling dissonance, he counted beats when musical intervals appear (c’–g fifth; c–f fourth; c–c octave, etc.) and judged the maximum of roughness with his 32 Hz beats criterion. This remarkable idea is one of the first psychoacoustic experiments in 4V. Etxebarria Ecenarro history. In terms of auditorium acoustics, when two simultaneous sine waves are very close in frequency, a single pleasant tone with slow variation in loudness (beats) is heard. Further apart in frequency, beats become faster (and dissonant). Further, the two tones are perceived individually and dissonance decreases. These three steps—single sound pleasant, single sound rough, two independent, non-interfering sounds—are perceived when one of the two simultaneous tones changes frequency and the border between consonance and dissonance is defined. Helmholtz’s work was of enormous importance and opened up many avenues in psychoacoustic research, many of which are still being developed. One of the most famous refinements of Helmholtz’s work on consonance and dissonance is due to Plomp and Levelt (1965), who carried out a series of experiments on the perception of consonance and dissonance sensations on volunteers with no musical knowledge, using two pure tones whose relative dissonance was judged by the listeners. These experiments with pure tones made it possible to refine Helmholtz’s 32 Hz criterion and to use more closely the concept of critical bandwidth, which is not independent of the frequency. Based on their results, Plomp and Levelt were able to calculate dissonance curves for non-pure harmonic tones. According to the theoretical procedure described in Sethares (1993), which in turn builds on the work of Plomp and Levelt (1965), we present a computational method for calculating general dissonance curves. The method allows the calculation of dissonance curves for both harmonic and non-harmonic sounds. This makes it possible to relate a sound spectrum (of an instrument) to a musical scale (defined by intervals that have dissonance minima). 2.1. Calculation of dissonance curves Next, we show the calculation method for obtaining dissonance curves. The first step in obtaining aclosedformforcalculatingdissonanceasafunctionofintervalistoencapsulatePlompand Levelt’s pure tone curve in a mathematical formula. The dissonance curve for two simultaneous pure tones obtained experimentally by Plomp and Levelt (Figure 2)canbeconvenientlyparameterizedbyamodeloftheform: d(x)=e−b1x−e−b2x(1) where xrepresents the frequency difference between the two sinusoids, and b1and b2determine how quickly the curves rise and fall. Using a least squares fit, the respective values for b1=3.5 and b2=5.75 are obtained. On the other hand, the Plomp–Levelt curves depend on the absolute frequency (and not only on the difference between the frequencies of the two tones), as shown in Figure 3.Thisfamilyof curves can be expressed in a single functional as described in (Sethares 1993): d(f1,f2,a1,a2)=a1a2[e−b1s(f2−f1)−e−b2s(f2−f1)](2) where f1and f2are the frequencies (f1≤f2) of the sinusoidal tones and a1and a2are the respective amplitudes. The parameter shas the form: s=x∗ s1f1+s2 (3) where x∗is the maximum of (1). For the above values for b1and b2this results in x∗=0.24. The parameters sin (3) allow the functional to interpolate between the different curves in Figure 3, by moving the dissonance curve along the frequency axis so that it starts at f1and that the maximum dissonance occurs at the corresponding frequency. Using the Plomp-Levelt curves, the parameters can be adjusted to the values of s1=0.0207 and s2=18.96. Journal of Mathematics and Music 5 Figure 2. Parametrization model for the dissonances observed by Plomp and Levelt (1965), in a series of experiments of two simultaneous pure tones sounding at close frequencies, to refine the phenomenon of beats observed by Helmholtz. Figure 3. Plomp-Levelt dissonance curves in experiments for different base frequencies (Plomp and Levelt 1965). It is observed that the dissonance depends not only on the difference between the frequencies of the two pure tones, but on the absolute frequency as well. In general, a sound Fof fundamental frequency f1is a collection of nsine waves of frequencies f1<f2<... <fnand amplitudes aj, so that the intrinsic dissonance of F can be calculated as the sum of the dissonances of all the pairs of partials: DF= n ! i=1 n ! j=1 d(fi,fj,ai,aj)(4) Finally, if two tones of F sound simultaneously in an interval of ratio α(i.e. Fand αFsound, where αFcontains the frequencies αf1,αf2,... ,αfn,withamplitudesaj), then the dissonance of Fin the interval αwill be the sum of the two intrinsic dissonances of the two tones, plus the sum of the dissonances of the pairs of partials taken one at each tone: DF(α)=DF+DαF+ n ! i=1 n ! j=1 d(fi,αfj,ai,aj)(5) 6V. Etxebarria Ecenarro Thus, the dissonance curve generated by Fis defined as the function DF(α)forallintervalsof interest α. 3. Results In the Appendices A and B we list the programmes provided by Sethares on his website: https:// sethares.engr.wisc.edu/comprog.html.ThesehavebeentranslatedtoPythonandwillbeusedto generate the dissonance curves in this Results section. These appendices are included in this article so that it remains as a complete and self-contained text, but note that the mathematical computation is based on (Plomp and Levelt 1965)and(Sethares 1993,2004). The main calculation is obtained through the code shown in A. This code takes a vector of frequencies and amplitudes as arguments and calculates the dissonance according to the method described above. The main programme – shown in B – calls repeatedly the previous function for the parameters defining the corresponding sound spectrum we want to consider. Equivalent computer adaptation of Sethares’ dissonance measurement functions can also be found as an electronic supplement of this article for the reader. A good source is in the Github site: https://gist.github.com/endolith/3066664, where there is a programme called sethares.py that implements the same functionality that can be used directly in any standard computer. These compact codes allow very fast and efficient calculation of dissonance curves for arbitrary sounds, and will be used in the following to obtain the results. The dissonance curves can be calculated to describe the expected characteristics of general kind of musical instruments taking into account their corresponding spectra. Note that phase differences between partials may also occur in a musical instrument depending on the mode of sound production. Playing style may modify the coherence between component frequencies and thereby reduce the fluctuations in dissonance values. In our computations we will assume a single playing style and phase difference for each instrument. 3.1. Dissonance curves for harmonic sounds Using the proposed computational model listed in the Appendix, Figure 4is calculated, which is the dissonance curve corresponding to a harmonic sound with a fundamental frequency of 500 Hz and six additional harmonics, each of them of equal amplitude. The result is very illustrative. As it can be seen, the curve has minima where the frequencies are related by simple ratios. In addition, the consonant intervals of the octave (2/1), the perfect fifth (3/2), the major sixth (5/3), the perfect fourth (4/3), and the major and minor thirds (5/4and6/5, respectively) stand out. Other intervals that can be approximately identified are the augmented sixth (about 7/4), the augmented fourth (about 7/5) and the major second (about 7/6). Finally, the dissonant intervals of the minor second (approximately 8/7), minor sixth (8/5) and seventh (approximately 9/5) are also identified. Overall, the dissonance curve for harmonic sounds allows the definition of the 12-semitone scale in common use, simply by identifying the intervals of the scale of minimum dissonance. In this sense, it can be said that the harmonic spectrum and the 12-semitone scale (typical of most modern Western musical instruments) are closely related. 3.2. Dissonance curve for xylophone bars Having applied the method of dissonance curves to harmonic sounds, we can now ask what consonances and dissonances are to be expected when non-harmonic spectrum instruments are Journal of Mathematics and Music 7 Figure 4. Dissonance curve calculated using the proposed computational model for a harmonic sound with a fundamental frequency and six additional harmonics. From this result, it is very important to note that the dissonance minima coincide very closely with the usual 12 musical intervals. This reveals the clear association between approximate musical scales and dissonance. used. In this sense, it is quite possible that intervals that are commonly considered to be consonant or dissonant will change their characteristics when played on non-harmonic instruments. It is also possible that, given the dissonance curve of a non-harmonic instrument being studied, a more appropriate associated scale can be defined, better than the usual one of 12 approximately equal tempered tones. The experimental setup consists of a personal computer with Python 3.11 or later installed in the operating system, which can be either Windows, macOS or Linux. The programme code for the computation of dissonance curves shown in Appendix A should be installed in the corresponding Python environment. The experiments are carried out using the calling code (Appendix B)tocomputeandploteachdissonancecurve,definingforeachexamplethecorresponding spectrum vector freq and the amplitude vector for each partial amp shown in Appendix B. The following application result studies the experimental response spectrum of xylophone bars presented in Bretos, Santamaria, and Alonso Moral (1997). According to these experimental data, the response of a “Royal Percussion” Studio-49 (Germany) xylophone bar tuned to an A4 is not harmonic and has a spectrum: F=[f,4.0f,9.1f,14.8f,19.9f,25.5f]; f=437.1 Hz (6) Using these data, and assuming that all partials have the same amplitude for simplicity, the dissonance curve shown in Figure 5is calculated. The dissonance predicted by the curve is generally lower than that of the harmonic tones (compare Figures 5and 4). Perhaps the most notable differences between the two curves are the relative dissonance expected for the major sixth interval (interval 1.67) and the predicted consonance for the seventh interval (interval 1.89) in the xylophone bars, both of which are the opposite for the harmonic sounds. Using the data from Bretos, Santamaria, and Alonso Moral (1997)againtoweighttherelative contribution of the partials, their relative amplitudes in the spectrum of the measured bars are approximately as follows: A=[1, 0.631, 0.3162, 0.2818, 0.1585, 0.1] (7) 8V. Etxebarria Ecenarro Figure 5. Dissonance curve calculated using the proposed computational model for the non-harmonic spectrum sound of the modern Royal Percussion Studio-49 (Germany) xylophone. Figure 6. Dissonance curve calculated using the proposed computational model for the non-harmonic spectrum sound of the modern Royal Percussion Studio-49 (Germany) xylophone, using the measured real decreasing amplitudes. Taking (6) and (7), a new dissonance curve can be computed, as shown in Figure 6.Asit can be seen, the dissonances calculated for this last curve are much lower than those given in Figure 5, which is not surprising since the high frequency partials are now much less important in contributing to dissonance. 3.3. Application to East and West traditional instruments The ordinary Basque txalaparta consists of wooden planks laid horizontally on two supports and struck with wooden sticks by four hands. Due to the rural origins of the instrument, it is very common to use planks from local trees such as oak, chestnut or alder, and ash sticks, similar to the short handles of rural tools. The metal bar used in the old smithies of the Basque Country was a bronze tube weighing several kilos, with a slightly flattened conical shape, and as such it Journal of Mathematics and Music 9 Figure 7. Dissonance curve calculated using the proposed computational model for the non-harmonic spectrum sound of the Basque txalaparta and alboka or the Thai renat and jakeh. From this result, it is found that these rustic ancient instruments fit with the rather uncommon seven tone equal temperament. had all the prerequisites for good sonority of the toberas.Accordingtoalltheevidence,thismust have been the instrument that was originally played at weddings. This percussion instrument has been used together with aerophones such as the alboka capable of providing penta-, or heptatonic melodies. AsurprisingaspectoftraditionalThaimusicisthatitisplayedinascalethatisverycloseto an equal tempered heptatonic scale, which means that its intervals never coincide (except in the octave) with those of the 12-semitone equal tempered scale. This was investigated in chapter 15 of Sethares’ book (Sethares 2004)whereheexploresthe7-toneequaltemperament.Therenat, for example, is a xylophone tuned approximately to a single equal tempered heptatonic scale. The modes of vibration of the renat bars and the txalaparta planks are similar to those of an ideal bar, whose spectrum contains the following first four partials (Fletcher and Rossing 1998, 628–629): Ftxalaparta,renat =[f,2.76f,5.4f,8.9f](8) Combining this with a harmonic sound such as the pi,thejakeh or the alboka and taking six partials, the mixed dissonance curve produced by playing melodic and percussion instruments together can be calculated (Figure 7). As it can be seen in the plot, the curve has dissonance minima in the following intervals: Dmin =[1, 1.22, 1.35, 1.49, 1.64, 1.80, 2] (9) which largely coincides with the division of the octave into the rather uncommon seven note equal tempered scale based on intervals 7 √2: 7eq temp =[1, 1.10, 1.22, 1.35, 1.49, 1.64, 1.81, 2] (10) so that the proposed acoustical model based on dissonance curves provides an explanation of the documented use of the unusual seven note equal tempered musical scale in ancient music.