Jou nal o Ma hema ics and Music,2025
h ps://doi.o g/10.1080/17459737.2025.2560922
Dissonance, sound spec um and musical scale o ancien
idiophones and ae ophones
Vic o E xeba ia Ecena o ∗
Depa men o Elec ici y and Elec onics, Facul y o Science and Technology, Uni e si y o he Basque
Coun y UPV/EHU, Leioa, Spain
(Recei ed 22 May 2025; accep ed 10 Sep embe 2025)
We do no know how he ea lies musical ins umen s—such as idiophones and ae ophones—we e played,
bu hei acous ic p ope ies can p o ide aluable clues. As a i s s ep, we p esen he e he concep
o dissonance cu es o a sound o a gi en spec um. These cu es show he ela i e dissonance ha
esul s o all in e als o a gi en musical ins umen . This idea leads o he associa ion o spec a and
scales, which a e ela ed because he dissonance cu e has minima in he in e als ha de ine he scale.
A compu a ional me hod o calcula ing dissonance cu es is p esen ed and se e al examples o i s use in
p ac ical cases, bo h o Wes e n and Eas e n musical ins umen s, a e gi en and in e p e ed. These esul s
allow us o explain om a physical poin o iew he eason o exis ence o well-known mode n 12-no e
scales, as well as some uncommon bu documen ed scales o a ious ins umen s in ea ly musical his o y.
Keywo ds: Musical ins umen s; e hnomusicology; dissonance; musical scale; spec um
1. In oduc ion
Idiophones o hy hm and ae ophones o melody a e p obably wha ou ances o s played o e
50,000 yea s ago (Mo ley 2013). The e olu ion o ancien musicali y is unknown, bu we can
y o deduce how ea ly musical ins umen s sounded by analyzing hei acous ic p ope ies and
combining his wi h he impo ance o hei uning and empe amen (Ba bou 1951).
A chaeological si es ha e e ealed many musical ins umen s made o s one, bone, wood o
me al. All ci iliza ions ha e de eloped such ins umen s simul aneously, and al hough Wes e n
and Eas e n music pa ly de eloped independen ly, we can ind common g ound in bo h musical
a eas by compa ing ancien ins umen s de eloped in bo h geog aphical ends.
Documen ed s yles and scales o ea ly idiophones such as he Ja anese gambang o he Thai
ena (Mo on 1976)in heEas ,and heGambianbala o he Basque xalapa a in he Wes
(Jones 1971;Bel an 2013), seem o be absolu ely di e en . Al hough hei espec i e cul u es
ha e e ol ed sepa a ely, we may ind simila ins umen s whose acous ic p inciples we can
analyze.
Thai cul u e has been in con ac wi h o he ci iliza ions o cen u ies, and Thai music and
musical ins umen s ha e been in luenced by China, Indonesia and India, among o he s. Some o
he musical ins umen s used in Thai classical music a e a ype o xylophone ( he ena ek and i s
∗Email: ic o [email p o ec ed]
© 2025 In o ma UK Limi ed, ading as Taylo & F ancis G oup
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2V. E xeba ia Ecena o
low-pi ched e sion, he ena hum), and melodic ae ophones such as he pi and o he melodic
ins umen s such as he jakeh (a ype o zi he ).
The xalapa a is an ancien idiophone, o iginally om he Basque Coun y (Eu opean egion
in he Wes e n Py enees) whose peculia i ies as a pe cussion ins umen and i s mys e ious
his o y ha e a oused g ea in e es among musicologis s and his o ians. The i s his o ical e -
e ence o he xalapa a appea s in 1882 in a book on cide p oduc ion in he Basque Coun y
(Agui e-Mi amon 1882, 129), al hough he e a e ea lie men ions o obe as (a me al a ian
o he xalapa a). The i s o hese is in a legal documen om 1688 (Lekuona 1920,52–53).
Melodic ins umen s such as he alboka (an ancien ype o cla ine ) we e men ioned in 1443 in
he Basque Coun y (Donos ia 1952). Ve y li le is known abou he p ac ice o he xalapa a
and alboka be o e he wen ie h cen u y, bu an h opologis s, his o ians, musicians and o he
schola s ha e placed he ins umen on a new pa h o g ow h, use and cul u al enewal o g ea
in e na ional in e es .
Music is usually conside ed o be a pu ely social science, belonging only o he humani-
ies, since i deals wi h knowledge conside ed o ha e been in en ed by and o mankind. The
ac ha ce ain musical scales exis is s ill an impo an s udy o musicologis s, an h opolo-
gis s o psychologis s (Gill and Pu es 2009). O he social scien is s also s udy ela ed musical
p ope ies, such as consonance o dissonance o sound, also app oaching he s udy o human
musicali y (Bowling and Pu es 2015)and he uningo musicalins umen sasanaspec mainly
ela ed o human p e e ence (F iedman e al. 2021). The s udy o cul u al p e e ence o musi-
cal psychoacous ics (Ee ola and Lahdelma 2021)isana ea ha s ill equi esmuch esea ch.
Impo an ad ances in he pe cep ion o amilia o un amilia music (Smi and Milne 2021)and
human p e e ences o musical ha mony (S olzenbu g 2015)ha ebeenmadeby esea che sin
mul idisciplina y ields.
La ge scale analysis o he pe cep ion o consonance and dissonance, such as he impo -
an Ha ison and Pea ce psychoacous ic model (Ha ison and Pea ce 2020)hasiden i ied h ee
main ca ego ies ( oughness, ha monici y, amilia i y), o e alua e o p edic dissonance. O he
esea che s (Ee ola and Lahdelma 2021)ha eincludednewelemen ssuchasspec alen elope
as an addi ional ca ego y. The o de o impo ance o hese ea u es on he pe cep ion o conso-
nance and dissonance emains an impo an ield no comple ely known which equi es u he
esea ch, especially in he con ex o c oss-cul u al s udies o uning.
The main objec i e o his s udy is o p o ide physical easons o explain he exis ence o
speci ic musical scales, ins ead o using he classical ma hema ical empe amen s. Based on he
calcula ion o ce ain dissonance cu es, he o dina y Wes e n 12 equal empe amen and he spe-
cial Eas e n se en equal empe amen scales appea na u ally. In his con ex , he non-ha monic
spec um o ancien idiophones is a key componen o he p oposed acous ic applica ion. The
exis ence o ce ain musical scales and he associa ed uning o musical ins umen s can be
ea ed om he poin o iew o acous ics as a b anch o applied physics. Ou app oach he e
is based on he basic acous ic p ope ies o bo h mode n idiophones as well as some ancien
musical ins umen s, which can gi e us some insigh in o how hey migh ha e been played.
The s udy is s uc u ed as ollows: sec ion 2 in oduces ma e ials and me hods, including
ab ie e iewo heclassicaliden i ica iono dissonanceapplied omusicalsounds.Then,
sec ion 2.1 p esen s a basic me hod o calcula ing dissonance in sounds o any spec um, and
we p opose a compu a ional echnique o easy calcula ion o dissonance cu es. In sec ion 3,
esul s o he me hod applied o bo h ha monic and non-ha monic ins umen s a e gi en. In
pa icula , i is shown how he well-known mode n 12 semi one scale, app oxima ely equal
empe ed, is ob ained di ec ly. Also, by applying he dissonance cu es o mode n xylophones,
aluable in e p e a ions o hei musical cha ac e is ics a e de i ed. In sec ion 3.3 he p oposed
echnique is applied o he non-ha monic ena and xalapa a and hei ela ed adi ional Thai
and Basque ins umen s, espec i ely. In e es ing ea u es o hese ancien ins umen s as well as
Jou nal o Ma hema ics and Music 3
Figu e 1. A eas o oughness calcula ed and plo ed by H. Helmhol z in 1870 when wo iolin no es a e played
simul aneously (Helmhol z 1954).
app op ia e bu uncommon scales o hei use a e ound. In he inal sec ion, he ob ained esul s
a e discussed, and he main conclusions o he s udy a e summa ized.
2. Ma e ials and me hods
One o he mos amous s udies o he pe cep ion o consonance and dissonance in music was
by Helmhol z (1954), who p oposed a model o dissonance based on he phenomenon o bea s.
When wo pu e ones o close equencies sound simul aneously, he in e e ence o he wo ones
p oduces bea s. The bea s become slowe as he equencies o he ones become mo e simi-
la , and disappea when he equencies coincide. Slow bea s a e ypically pe cei ed as smoo h
wa es, bu as bea s end o be ough and unpleasan , wi h maximum oughness obse ed when
he bea s occu a ound 32 imes pe second. Conside ing ha e e y sound can be b oken down
in o sinusoidal pa ials, Helmhol z concluded ha he dissonance pe cei ed when lis ening o wo
ones simul aneously is caused by he apid bea ing o he pa ials. Thus, acco ding o Helmhol z,
consonance is he absence o such dissonan bea s.
Assuming ha he oughness o all in e ac ing pa ials o wo ones add up, he dissonance o
any in e al can be calcula ed simply by conside ing all possible combina ions o pai s o pa ials
and summing hei con ibu ions. Helmhol z pe o med his ype o calcula ion o ha monic
sounds such as hose p oduced by iolins (E xeba ia and Rie a 2002), and p esen ed he esul s
g aphically in diag ams such as he one shown in Figu e 1, aken di ec ly om Helmhol z (1954).
The ho izon al axis ep esen s he in e al be ween he wo ones. One emains a a cons an
equency (labelled c’), and he o he one mo es om c’ o he uppe oc a e (labelled c’’). The
heigh o he cu es ( e ical axis) is p opo ional o he oughness p oduced by he pa ials
whose equency a ios a e labelled on he plo , using he maximum oughness c i e ion o 32
Hz bea s. The esul is a plo ha has minima (in e als whe e minimum oughness occu s) nea
many o he in e als o he majo scale, sugges ing a ela ionship be ween he phenomenon o
bea s and he musical no ions o consonance and dissonance.
Helmhol z’s me hod o calcula ing oughness is na a ed in his book (Helmhol z 1954)on
pages 192–194, whe e he admi s he was “ o ced o assume a somewha a bi a y law” and
chose “ he simples ma hema ical o mula,” which he does no speci y analy ically bu on his
g aph. Ins ead o ma hema ically modelling dissonance, he coun ed bea s when musical in e -
als appea (c’–g i h; c– ou h; c–c oc a e, e c.) and judged he maximum o oughness wi h
his 32 Hz bea s c i e ion. This ema kable idea is one o he i s psychoacous ic expe imen s in
4V. E xeba ia Ecena o
his o y. In e ms o audi o ium acous ics, when wo simul aneous sine wa es a e e y close in
equency, a single pleasan one wi h slow a ia ion in loudness (bea s) is hea d. Fu he apa
in equency, bea s become as e (and dissonan ). Fu he , he wo ones a e pe cei ed indi id-
ually and dissonance dec eases. These h ee s eps—single sound pleasan , single sound ough,
wo independen , non-in e e ing sounds—a e pe cei ed when one o he wo simul aneous ones
changes equency and he bo de be ween consonance and dissonance is de ined.
Helmhol z’s wo k was o eno mous impo ance and opened up many a enues in psychoa-
cous ic esea ch, many o which a e s ill being de eloped. One o he mos amous e inemen s
o Helmhol z’s wo k on consonance and dissonance is due o Plomp and Le el (1965), who
ca ied ou a se ies o expe imen s on he pe cep ion o consonance and dissonance sensa ions
on olun ee s wi h no musical knowledge, using wo pu e ones whose ela i e dissonance was
judged by he lis ene s. These expe imen s wi h pu e ones made i possible o e ine Helmhol z’s
32 Hz c i e ion and o use mo e closely he concep o c i ical bandwid h, which is no indepen-
den o he equency. Based on hei esul s, Plomp and Le el we e able o calcula e dissonance
cu es o non-pu e ha monic ones.
Acco ding o he heo e ical p ocedu e desc ibed in Se ha es (1993), which in u n builds on
he wo k o Plomp and Le el (1965), we p esen a compu a ional me hod o calcula ing gene al
dissonance cu es. The me hod allows he calcula ion o dissonance cu es o bo h ha monic
and non-ha monic sounds. This makes i possible o ela e a sound spec um (o an ins umen )
o a musical scale (de ined by in e als ha ha e dissonance minima).
2.1. Calcula ion o dissonance cu es
Nex , we show he calcula ion me hod o ob aining dissonance cu es. The i s s ep in ob aining
aclosed o m o calcula ingdissonanceasa unc iono in e alis oencapsula ePlompand
Le el ’s pu e one cu e in a ma hema ical o mula.
The dissonance cu e o wo simul aneous pu e ones ob ained expe imen ally by Plomp and
Le el (Figu e 2)canbecon enien lypa ame e izedbyamodelo he o m:
d(x)=e−b1x−e−b2x(1)
whe e x ep esen s he equency di e ence be ween he wo sinusoids, and b1and b2de e mine
how quickly he cu es ise and all. Using a leas squa es i , he espec i e alues o b1=3.5
and b2=5.75 a e ob ained.
On he o he hand, he Plomp–Le el cu es depend on he absolu e equency (and no only
on he di e ence be ween he equencies o he wo ones), as shown in Figu e 3.This amilyo
cu es can be exp essed in a single unc ional as desc ibed in (Se ha es 1993):
d( 1, 2,a1,a2)=a1a2[e−b1s( 2− 1)−e−b2s( 2− 1)](2)
whe e 1and 2a e he equencies ( 1≤ 2) o he sinusoidal ones and a1and a2a e he
espec i e ampli udes. The pa ame e shas he o m:
s=x∗
s1 1+s2
(3)
whe e x∗is he maximum o (1). Fo he abo e alues o b1and b2 his esul s in x∗=0.24. The
pa ame e s sin (3) allow he unc ional o in e pola e be ween he di e en cu es in Figu e 3,
by mo ing he dissonance cu e along he equency axis so ha i s a s a 1and ha he max-
imum dissonance occu s a he co esponding equency. Using he Plomp-Le el cu es, he
pa ame e s can be adjus ed o he alues o s1=0.0207 and s2=18.96.
Jou nal o Ma hema ics and Music 5
Figu e 2. Pa ame iza ion model o he dissonances obse ed by Plomp and Le el (1965), in a se ies o expe imen s
o wo simul aneous pu e ones sounding a close equencies, o e ine he phenomenon o bea s obse ed by Helmhol z.
Figu e 3. Plomp-Le el dissonance cu es in expe imen s o di e en base equencies (Plomp and Le el 1965). I
is obse ed ha he dissonance depends no only on he di e ence be ween he equencies o he wo pu e ones, bu on
he absolu e equency as well.
In gene al, a sound Fo undamen al equency 1is a collec ion o nsine wa es o equencies
1< 2<... < nand ampli udes aj, so ha he in insic dissonance o F can be calcula ed as
he sum o he dissonances o all he pai s o pa ials:
DF=
n
!
i=1
n
!
j=1
d( i, j,ai,aj)(4)
Finally, i wo ones o F sound simul aneously in an in e al o a io α(i.e. Fand αFsound,
whe e αFcon ains he equencies α 1,α 2,... ,α n,wi hampli udesaj), hen he dissonance o
Fin he in e al αwill be he sum o he wo in insic dissonances o he wo ones, plus he sum
o he dissonances o he pai s o pa ials aken one a each one:
DF(α)=DF+DαF+
n
!
i=1
n
!
j=1
d( i,α j,ai,aj)(5)
6V. E xeba ia Ecena o
Thus, he dissonance cu e gene a ed by Fis de ined as he unc ion DF(α) o allin e also
in e es α.
3. Resul s
In he Appendices A and B we lis he p og ammes p o ided by Se ha es on his websi e: h ps://
se ha es.eng .wisc.edu/comp og.h ml.Theseha ebeen ansla ed oPy honandwillbeused o
gene a e he dissonance cu es in his Resul s sec ion. These appendices a e included in his
a icle so ha i emains as a comple e and sel -con ained ex , bu no e ha he ma hema ical
compu a ion is based on (Plomp and Le el 1965)and(Se ha es 1993,2004). The main cal-
cula ion is ob ained h ough he code shown in A. This code akes a ec o o equencies and
ampli udes as a gumen s and calcula es he dissonance acco ding o he me hod desc ibed abo e.
The main p og amme – shown in B – calls epea edly he p e ious unc ion o he pa ame e s
de ining he co esponding sound spec um we wan o conside .
Equi alen compu e adap a ion o Se ha es’ dissonance measu emen unc ions can also be
ound as an elec onic supplemen o his a icle o he eade . A good sou ce is in he Gi hub
si e: h ps://gis .gi hub.com/endoli h/3066664, whe e he e is a p og amme called se ha es.py
ha implemen s he same unc ionali y ha can be used di ec ly in any s anda d compu e .
These compac codes allow e y as and e icien calcula ion o dissonance cu es o a bi-
a y sounds, and will be used in he ollowing o ob ain he esul s. The dissonance cu es can be
calcula ed o desc ibe he expec ed cha ac e is ics o gene al kind o musical ins umen s aking
in o accoun hei co esponding spec a. No e ha phase di e ences be ween pa ials may also
occu in a musical ins umen depending on he mode o sound p oduc ion. Playing s yle may
modi y he cohe ence be ween componen equencies and he eby educe he luc ua ions in dis-
sonance alues. In ou compu a ions we will assume a single playing s yle and phase di e ence
o each ins umen .
3.1. Dissonance cu es o ha monic sounds
Using he p oposed compu a ional model lis ed in he Appendix, Figu e 4is calcula ed, which is
he dissonance cu e co esponding o a ha monic sound wi h a undamen al equency o 500
Hz and six addi ional ha monics, each o hem o equal ampli ude. The esul is e y illus a i e.
As i can be seen, he cu e has minima whe e he equencies a e ela ed by simple a ios. In
addi ion, he consonan in e als o he oc a e (2/1), he pe ec i h (3/2), he majo six h (5/3),
he pe ec ou h (4/3), and he majo and mino hi ds (5/4and6/5, espec i ely) s and ou .
O he in e als ha can be app oxima ely iden i ied a e he augmen ed six h (abou 7/4), he
augmen ed ou h (abou 7/5) and he majo second (abou 7/6). Finally, he dissonan in e als
o he mino second (app oxima ely 8/7), mino six h (8/5) and se en h (app oxima ely 9/5) a e
also iden i ied.
O e all, he dissonance cu e o ha monic sounds allows he de ini ion o he 12-semi one
scale in common use, simply by iden i ying he in e als o he scale o minimum dissonance. In
his sense, i can be said ha he ha monic spec um and he 12-semi one scale ( ypical o mos
mode n Wes e n musical ins umen s) a e closely ela ed.
3.2. Dissonance cu e o xylophone ba s
Ha ing applied he me hod o dissonance cu es o ha monic sounds, we can now ask wha
consonances and dissonances a e o be expec ed when non-ha monic spec um ins umen s a e
Jou nal o Ma hema ics and Music 7
Figu e 4. Dissonance cu e calcula ed using he p oposed compu a ional model o a ha monic sound wi h a unda-
men al equency and six addi ional ha monics. F om his esul , i is e y impo an o no e ha he dissonance minima
coincide e y closely wi h he usual 12 musical in e als. This e eals he clea associa ion be ween app oxima e musical
scales and dissonance.
used. In his sense, i is qui e possible ha in e als ha a e commonly conside ed o be consonan
o dissonan will change hei cha ac e is ics when played on non-ha monic ins umen s. I is
also possible ha , gi en he dissonance cu e o a non-ha monic ins umen being s udied, a
mo e app op ia e associa ed scale can be de ined, be e han he usual one o 12 app oxima ely
equal empe ed ones.
The expe imen al se up consis s o a pe sonal compu e wi h Py hon 3.11 o la e ins alled
in he ope a ing sys em, which can be ei he Windows, macOS o Linux. The p og amme
code o he compu a ion o dissonance cu es shown in Appendix A should be ins alled in
he co esponding Py hon en i onmen . The expe imen s a e ca ied ou using he calling code
(Appendix B) ocompu eandplo eachdissonancecu e,de ining o eachexample heco -
esponding spec um ec o eq and he ampli ude ec o o each pa ial amp shown in
Appendix B.
The ollowing applica ion esul s udies he expe imen al esponse spec um o xylophone ba s
p esen ed in B e os, San ama ia, and Alonso Mo al (1997). Acco ding o hese expe imen al da a,
he esponse o a “Royal Pe cussion” S udio-49 (Ge many) xylophone ba uned o an A4 is no
ha monic and has a spec um:
F=[ ,4.0 ,9.1 ,14.8 ,19.9 ,25.5 ]; =437.1 Hz (6)
Using hese da a, and assuming ha all pa ials ha e he same ampli ude o simplici y, he
dissonance cu e shown in Figu e 5is calcula ed. The dissonance p edic ed by he cu e is
gene ally lowe han ha o he ha monic ones (compa e Figu es 5and 4). Pe haps he mos
no able di e ences be ween he wo cu es a e he ela i e dissonance expec ed o he majo
six h in e al (in e al 1.67) and he p edic ed consonance o he se en h in e al (in e al 1.89)
in he xylophone ba s, bo h o which a e he opposi e o he ha monic sounds.
Using he da a om B e os, San ama ia, and Alonso Mo al (1997)again oweigh he ela i e
con ibu ion o he pa ials, hei ela i e ampli udes in he spec um o he measu ed ba s a e
app oxima ely as ollows:
A=[1, 0.631, 0.3162, 0.2818, 0.1585, 0.1] (7)
8V. E xeba ia Ecena o
Figu e 5. Dissonance cu e calcula ed using he p oposed compu a ional model o he non-ha monic spec um sound
o he mode n Royal Pe cussion S udio-49 (Ge many) xylophone.
Figu e 6. Dissonance cu e calcula ed using he p oposed compu a ional model o he non-ha monic spec um sound
o he mode n Royal Pe cussion S udio-49 (Ge many) xylophone, using he measu ed eal dec easing ampli udes.
Taking (6) and (7), a new dissonance cu e can be compu ed, as shown in Figu e 6.Asi
can be seen, he dissonances calcula ed o his las cu e a e much lowe han hose gi en in
Figu e 5, which is no su p ising since he high equency pa ials a e now much less impo an
in con ibu ing o dissonance.
3.3. Applica ion o Eas and Wes adi ional ins umen s
The o dina y Basque xalapa a consis s o wooden planks laid ho izon ally on wo suppo s and
s uck wi h wooden s icks by ou hands. Due o he u al o igins o he ins umen , i is e y
common o use planks om local ees such as oak, ches nu o alde , and ash s icks, simila o
he sho handles o u al ools. The me al ba used in he old smi hies o he Basque Coun y
was a b onze ube weighing se e al kilos, wi h a sligh ly la ened conical shape, and as such i
Jou nal o Ma hema ics and Music 9
Figu e 7. Dissonance cu e calcula ed using he p oposed compu a ional model o he non-ha monic spec um sound
o he Basque xalapa a and alboka o he Thai ena and jakeh. F om his esul , i is ound ha hese us ic ancien
ins umen s i wi h he a he uncommon se en one equal empe amen .
had all he p e equisi es o good sono i y o he obe as.Acco ding oall hee idence, hismus
ha e been he ins umen ha was o iginally played a weddings. This pe cussion ins umen has
been used oge he wi h ae ophones such as he alboka capable o p o iding pen a-, o hep a onic
melodies.
Asu p isingaspec o adi ionalThaimusicis ha i isplayedinascale ha is e yclose o
an equal empe ed hep a onic scale, which means ha i s in e als ne e coincide (excep in he
oc a e) wi h hose o he 12-semi one equal empe ed scale. This was in es iga ed in chap e 15
o Se ha es’ book (Se ha es 2004)whe eheexplo es he7- oneequal empe amen .The ena ,
o example, is a xylophone uned app oxima ely o a single equal empe ed hep a onic scale.
The modes o ib a ion o he ena ba s and he xalapa a planks a e simila o hose o an
ideal ba , whose spec um con ains he ollowing i s ou pa ials (Fle che and Rossing 1998,
628–629):
F xalapa a, ena =[ ,2.76 ,5.4 ,8.9 ](8)
Combining his wi h a ha monic sound such as he pi, hejakeh o he alboka and aking six
pa ials, he mixed dissonance cu e p oduced by playing melodic and pe cussion ins umen s
oge he can be calcula ed (Figu e 7).
As i can be seen in he plo , he cu e has dissonance minima in he ollowing in e als:
Dmin =[1, 1.22, 1.35, 1.49, 1.64, 1.80, 2] (9)
which la gely coincides wi h he di ision o he oc a e in o he a he uncommon se en no e
equal empe ed scale based on in e als 7
√2:
7eq emp =[1, 1.10, 1.22, 1.35, 1.49, 1.64, 1.81, 2] (10)
so ha he p oposed acous ical model based on dissonance cu es p o ides an explana ion o he
documen ed use o he unusual se en no e equal empe ed musical scale in ancien music.
