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Article A Geometrical Method for Sound-Hole Size and Location Enhancement in Family Musical Instruments: The Golden Method

Soheil Jafari 1,* and Mohammad Mahdi Karbasbaf 2

1 Department of Engineering and Informatics, University of Sussex, Falmer, Brighton BN1 9QT, UK 2 Edinburgh College of Art, The University of Edinburgh, Edinburgh, EH3 9DF, UK; [email protected] * Correspondence: [email protected]; Tel.: +44-7459-16-14-12

Academic Editor: Ellen Fallowfield Received: 5 May 2017; Accepted: 20 November 2017; Published: 28 November 2017

Abstract: This paper presents a new analytical approach, the Golden Method, to enhance sound-hole size and location in musical instruments of the lute family in order to obtain better sound damping characteristics based on the concept of the golden ratio and the instrument geometry. The main objective of the paper is to increase the capability of lute family musical instruments in keeping a note for a certain time at a certain level to enhance the instruments’ orchestral characteristics. For this purpose, a geometry-based analytical method, the Golden Method is first described in detail in an itemized feature. A new is then developed and tested to confirm the ability of the Golden Method in optimizing the acoustical characteristics of musical instruments from a damping point of view by designing the modified sound-hole. Finally, the new-developed instrument is tested, and the obtained results are compared with those of two well-known instruments to confirm the effectiveness of the proposed method. The experimental results show that the suggested method is able to increase the sound damping time by at least 2.4% without affecting the frequency response function and other acoustic characteristics of the instrument. This methodology could be used as the first step in future studies on design, optimization and evaluation of musical instruments of the lute family (e.g., lute, , , , , and etc.).

Keywords: sound-holes; lute family; musical instruments; golden ratio; golden spiral; optimization; damping characteristics; orchestral characteristics; stringed-musical instrument

1. Introduction The stringed musical instrument family is the largest family in an orchestra. Therefore, the quality of these instruments plays a vital role in the performance and quality of the whole orchestra. The main orchestral requirements for these instruments are continuous playing range, fast melodic and run playing, ability to play several musical notes at the same time, and playing numerous different articulations. These requirements should be translated into acoustic characteristics to enable instrument makers to provide more practical instruments. One of these characteristics is the instrument capability in keeping a note for a certain time at a certain level (damping characteristics). In other words, the higher the damping time of a stringed musical instrument, the wider the playing range of the instrument. In stringed musical instruments, the soundboard plays a vital role in the instrument performance. In fact, this element performs the most important effect in the frequency response of the resonator to the acoustical waves because it transfers sound waves between strings and the resonator. In 2015,

Arts 2017, 6, 20; doi:10.3390/arts6040020 www.mdpi.com/journal/arts Arts 2017, 6, 20 2 of 15

resonator. In 2015, Kamalian and Mobasseri (2015) (and especially setar1 in their research) showed that one third of the instrument sound is generated by the sound box and the rest (2/3) is from the sundboard plate. soundboards usually contain openings called sound‐holes to help acoustic instruments project their sound more efficiently (Figure 1). Sound‐holes have different shapes and sizes in different instruments depending on the area of the soundboard, range of frequencies derived by resources (strings), volume of the air inside the sound box, and stiffness of the plate. In 2015,

Artsresearchers2017, 6, 20 at MIT published an analysis charting the evolution and improvements in effectiveness2 of 15 of F‐hole design over time (MIT News 2015; Nia et al. 2015). This paper will focus on sound‐ holes in lute family musical instruments. KamalianBarbat and can Mobasseri be considered(2015 as) (and one of especially the first engineered setar 1 in their prototypes research) of showedlute and thatone of one the third oldest of theinstruments instrument in the sound world, is which generated probably by the originated sound box in central and the Asia rest (Marcel (2/3)‐Dubois is from 1941). the sundboard The oldest plate.pictorial soundboards representations usually of this contain instrument openings are called found sound-holes in the first‐century to help BC acoustic (Vyzgo instruments 1980) of Ḵa projectḷčayān theirin North sound more (present efficiently‐day (Figure South 1Uzbekistan),). Sound-holes while have a more different “clear shapes cut” depiction and sizes of in the different barbat instrumentsfrom depending sculpture on dates the area to the of the2nd–4th soundboard, centuries range AD. ofAccording frequencies to derivedEncyclopedia by resources Iranica (strings),(1988), this volume type of of instrument the air inside could the have sound been box, introduced and stiffness by the of Kushans the plate. and In was 2015, later researchers adopted by at MITthe Persians. published By an the analysis 7th century, charting the the barbat evolution was developed and improvements by the in effectiveness into its current of violin form, F-holecalled designthe oud. over The time modern(MIT Persian News 2015 barbat; Nia is et almost al. 2015 the). This same paper as the will oud, focus although on sound-holes differences in lute include family a musicalsmaller body, instruments. longer neck and a slightly raised .

Figure 1. Elements of a typical lute. Figure 1. Elements of a typical lute. Even in initial formations of these instruments, sound‐holes as effective elements in acoustic behaviourBarbat of can instruments be considered were as considered one of the firstby designers engineered and prototypes manufacturers. of lute andHistorically, one of the it oldestis not instrumentspossible to exactly in the world,discover which the emergence probably originated of the sound in central‐holes Asiain ancient (Marcel-Dubois musical instruments, 1941). The oldest but a pictorialcomprehensive representations historical ofsurvey this instrument on sound‐hole are design found inshows the first-century that in the middle BC (Tamara ages they Semenovna could be Vyzgo 1980) of Kalˇcayan¯ in North Bactria (present-day South Uzbekistan), while a more “clear cut” observed in some¯ . classical paintings (Figure 2) showing stringed instruments like the , depictionDulcimer and of the Psaltery barbat in from Europe Gandhara and barbat sculpture in Persia dates (Rault to the2014) 2nd–4th. There centuriesis other evidence AD. According that shows to Encyclopaediacircular sound‐ Iranicaholes in(1988 early), this violin type ancestors of instrument in the could 10th havecentury been (Engel introduced 1883), bysemi the‐circular Kushans in and the was12th latercentury adopted (Baines by 1992), the Persians. crescent in By the the 13th 7th century century, (Sandys the barbat and was Forster developed 1884), semi by the‐circular Arabs strips into itsin the current 13th form,century called (Van the der oud. Straeten The modern 1968), and Persian C‐holes barbat in the is 13th almost and the 14th same centuries as the oud,(Coates although 1985). differencesBetween the include 13th and a smaller 17th centuries, body, longer the necksound and‐hole a slightly shape and raised size fingerboard. were approximated by simple Even in initial formations of these instruments, sound-holes as effective elements in acoustic behaviour of instruments were considered by designers and manufacturers. Historically, it is not possible1 Setar is to a Persian exactly member discover of lute the family emergence musical of instruments. the sound-holes in ancient musical instruments, but a comprehensive historical survey on sound-hole design shows that in the they could be observed in some classical paintings (Figure2) showing stringed instruments like the Rebec, and Psaltery in Europe and barbat in Persia (Rault 2014). There is other evidence that shows circular sound-holes in early violin ancestors in the 10th century (Engel 1883), semi-circular in the 12th

1 Setar is a Persian member of lute family musical instruments. Arts 2017, 6, 20 3 of 15 century (Baines 1992), crescent in the 13th century (Sandys and Forster 1884), semi-circular strips in the 13th century (Van der Straeten 1968), and C-holes in the 13th and 14th centuries (Coates 1985). Between theArts 13th 2017 and, 6, 20 17th centuries, the sound-hole shape and size were approximated by simple3 circular of 15 and elliptical geometries in lute family musical instruments (Hutchins 1990; Cremer 1984), and by circular and elliptical geometries in lute family musical instruments (Hutchins 1990; Cremer 1984), f-shape geometry in instruments (Hideo and Kumagai 1952; Shaw 1990). The instrument and by f‐shape geometry in violin family instruments (Itakawa and Kumagai 1952; Shaw 1990). The makers usually modified the errors in sound-holes using empirical fitting factors (Bissinger 1992). instrument makers usually modified the errors in sound‐holes using empirical fitting factors The last basic changes occurred in the 16th and 17th centuries. The lute rosset size and shape were (Bissinger 1992). The last basic changes occurred in the 16th and 17th centuries. The lute rosset size modified by Warwick Hans in 1540, Wendelin Tieffenbrucker in 1590, Wendelio Venere in 1592, Padov and shape were modified by Warwick Hans in 1540, Wendelin Tieffenbrucker in 1590, Wendelio inVenere 1595, and in 1592, Sebastian Padov Schelle in 1595, in and 1744 Sebastian (Barber Schelle and Harris in 1744 2016 (Barber). Subsequently, and Harris 2016). the main Subsequently, shape of the sound-holesthe main shape in violin of the and sound lute‐holes families in violin remained and lute almost families unchanged. remained almost unchanged.

Figure 2. Initial paintings show the emergence of sound‐holes in musical instruments Roman fiddle Figure 2. Initial paintings show the emergence of sound-holes in musical instruments Roman fiddle adapted from the manuscript Br. Libr., Arundel. 91 (fol°218v°), eginning of the 12th Century adapted from the manuscript London Br. Libr., Arundel. 91 (fol◦218v◦), eginning of the 12th Century (left), Vièle recreated for the Museu de la música in Barcelona, 2006 (center), Gothic fiddle, Cantigas (left), Vièle recreated for the Museu de la música in Barcelona, 2006 (center), Gothic fiddle, Cantigas de de Santa Maria, Spain, 13th Century (right) (Rault 2014). Santa Maria, Spain, 13th Century (right) (Rault 2014). In recent centuries, studies on air resonance fundamentals with application to musical instrumentsIn recent centuries,were initiated studies by on Savart air resonance (1976), Von fundamentals Helmholtz with (1860; application 1954), Rayleigh to musical (1945), instruments Lamb were(1932), initiated and Mcpherson by Savart (2008).(1976), However,Von Helmholtz with respect(1860; to1954 Tavakoli), Rayleigh Nia (2010,(1945 ),p. Lamb3), “the(1932 design), andknowledge Mcpherson in such(2002 sound). However,‐holes accumulated with respect by totime Tavakoli is still Niauncovered(2010,” p.. In 3), 2011, “ the Tavakoli design knowledge Nia started in such sound-holesanalysing accumulated sound‐holes by and time the is still concept uncovered behind”. In their 2011, geometrical Tavakoli Nia design. started He analysing showed sound-holes that the andemergence the concept of behindcircular their sound geometrical‐holes in the design. lute family He showed was due that theto passing emergence the ofmajority circular of sound-holes air flow inthrough the lute the family near‐ wasthe‐edge due toarea passing of the opening the majority (Tavakoli of air Nia flow 2010). through He comprehensively the near-the-edge investigated area of the openingthis area (Tavakoli and finally Nia concluded 2010). He that comprehensively his approach could investigated be a good thisstarting area point and finallyfor the concludedmodification that hisof approach the instruments could be for a musicians good starting and instrument point for the makers. modification of the instruments for musicians and instrumentThere makers. are several parameters that have effects on acoustic characteristics of a musical instrument (e.g., material, age of the instrument and used wood, weather, manufacturing procedure and There are several parameters that have effects on acoustic characteristics of a musical instrument mechanism, geometry, etc.). All of the above‐mentioned parameters can affect the instrument (e.g., material, age of the instrument and used wood, weather, manufacturing procedure and behaviour, quality, and capabilities. Some of them, like material, age, wood, and weather, are mechanism, geometry, etc.). All of the above-mentioned parameters can affect the instrument behaviour, dynamic indices and subject to change and/or deteriorate over time and therefore require quality, and capabilities. Some of them, like material, age, wood, and weather, are dynamic indices comprehensive research and facilities for examination purposes. However, other parameters like andgeometrical subject to design change and and/or manufacturing deteriorate procedure over time and and mechanism therefore are require not time comprehensive dependent and research can andbe facilitiesstudied more for examination easily. purposes. However, other parameters like geometrical design and manufacturingAlthough procedure the whole andbehaviour mechanism of a musical are not instrument time dependent is the result and can of the be studiedsuperposition more easily.of the effectsAlthough of all parameters, the whole this behaviour paper, as of the a musical first step, instrument will focus on is the the resulteffect of of sound the superposition‐hole design and of the effectslocation of all on parameters, the capability this of paper, the musical as the first instrument step, will in focus keeping on the a effectnote which of sound-hole can enhance design the and locationorchestral on the characteristics capability of of the stringed musical musical instrument instruments. in keeping Since a note the which circular can sound enhance‐hole the has orchestral been characteristicsproved as the of optimized stringed musical shape for instruments. the lute family Since based the circular on the sound-holeliterature and has especially been proved an MIT as the researchers’ report (Tavakoli Nia 2010), the assumption here is that the circle is the optimized shape and only the size and location will be discussed. For this purpose, an analytical approach based on

Arts 2017, 6, 20 4 of 15 optimized shape for the lute family based on the literature and especially an MIT researchers’ report (Tavakoli Nia 2010), the assumption here is that the circle is the optimized shape and only the size and location will be discussed. For this purpose, an analytical approach based on the geometry of instrument, the concept of the golden ratio, and their magical effects is proposed using the several years of experience and academic studies of the authors in the fields of optimization and musical instrument making. This methodology is called the Golden Method by the authors.

1. The method is described in an itemized flowchart form in chapter two. 2. Two well-known instruments will be checked with this approach as case studies in order to show the effectiveness of the Golden Method. 3. A new musical instrument will be developed using the proposed method and its damping capability would be compared with that of two well-known instruments to confirm the effectiveness of the golden method in optimizing the sound-hole size and location in lute family musical instruments from a damping point of view.

2. Methodology The golden ratio concept is used for the first time here for sound-hole size and location design in musical instruments of the lute family in order to enhance the damping capability. The golden ratio has been claimed to have held a special fascination for at least 2400 years, though without reliable evidence (Markowsky 1992). According to Mario Livio: “Some of the greatest mathematical minds of all ages, from Pythagoras and Euclid in ancient , through the medieval Italian mathematician Leonardo of Pisa and the Renaissance astronomer Johannes Kepler, to present-day scientific figures such as Oxford physicist Roger Penrose, have spent endless hours over this simple ratio and its properties. But the fascination with the Golden Ratio is not confined just to mathematicians. Biologists, artists, musicians, historians, architects, psychologists, and even mystics have pondered and debated the basis of its ubiquity and appeal. In fact, it is probably fair to say that the Golden Ratio has inspired thinkers of all disciplines like no other number in the history of mathematics (Livio 2002).” This ratio has also inspired the researchers in the field of acoustics characteristics of instruments. Takegawa(2008) invented a with optimized air vent holes using the golden ratio in 2008, and King(2004; 2005) presented a comprehensive research study on locating the Stradivari’s violin hole locations using the golden ratio. The authors spent several hours analyzing, modelling and simulating different assumptions and the manufacturing process to find the optimized sound-hole for specific musical instruments to determine the maximum sound radiation capability. The results of all analytical and trial and error procedures are summarized here entitled: Golden Method. The assumptions used for this method are: a. The circular sound-hole has been assumed as the optimized shape for the lute family (based on the literature). b. The material and mechanical properties of the instrument are not discussed in this paper because of their dynamic-index nature. These parameters can be studied in a separate standalone research work. c. The main objective of the proposed method is to enhance the musical note keeping capability of the instrument for orchestral consideration.

Moreover, without loss of generality, the lute is basically used to describe the method because it is a world-wide well-reputed instrument. However, the procedure is exactly the same for all lute family musical instruments. The Golden Method is presented for a typical professional lute: In a musical instrument of the lute family, to optimize the sound-hole location and size (or to check that a sound-hole is optimized) from a damping capability point of view, perform the following steps: Arts 2017, 6, 20 5 of 15

1. Plot the rectangle ABCD covering the instrument soundboard and tangent to it as shown Artsin 2017 Figure, 6, 203a. 5 of 15 2. ArtsPlot 2017 the, 6, 20 golden rectangle and its logarithmic spiral from one side of the ABCD as shown5 of 15 2.in FigurePlot the3b. Fullgolden details rectangle on how and to plotits logarithmic such a golden spiral rectangle from andone spiralside of are the very ABCD straightforward as shown in 2. Plot the golden rectangle and its logarithmic spiral from one side of the ABCD as shown in andFigure can be 3b. found Full details in ( on how 2002 to, Hemenway plot such a golden 2005). rectangle and spiral are very straightforward Figure 3b. Full details on how to plot such a golden rectangle and spiral are very straightforward 3. Repeatand stepcan be 2 fromfound another in (Chang side 2002, of the Hemenway ABCD as 2005). shown in Figure3c and draw another golden 3. Repeatand can step be found 2 from in another (Chang side2002, of Hemenway the ABCD 2005).as shown in Figure 3c and draw another golden rectangle and logarithmic spiral that mirror those of step 2. 3. rectangleRepeat step and 2 logarithmicfrom another spiral side thatof the mirror ABCD those as shown of step in2. Figure 3c and draw another golden 4. Plotrectangle a circle C and1 tangent logarithmic over the spiral 3 lines that AB,mirror BD, those and CDof step as shown 2. in Figure4a. 4. Plot a circle C1 tangent over the 3 lines AB, BD, and CD as shown in Figure 4a. 5. 4.Plot Plot the a first circle golden C1 tangent circle over C2 so the that 3 lines AB, BD, and CD as shown in Figure 4a. 5. Plot the first golden circle C2 so that 5. Plot the first golden circle C2 so that •  TheThe vertical vertical diameters diameters and and upper upper vertex vertex of of the the two two circles circles are are common. common.  The vertical diametersd(c1) and upper vertex of the two circles are common. •  TheThe diameter diameter ratio: ratio: (diameter ratio) = 1.618 . = Φ  The diameter ratiod(:c 2) . 6. 6.Plot Plot the the second second golden golden circle circle CC33;; this this circle circle is is drawn drawn like like C2 C(but2 (but C3 should C3 should follow follow the above the‐ 6. Plot the second golden circle C3; this circle is drawn like C2 (but C3 should follow the above‐ above-mentionedmentioned conditions conditions for C for2) (Figure C2) (Figure 4c). 4c). mentioned conditions for C2) (Figure 4c). 7. Now, if C3 touches the two golden spirals (Figure 5a), this can locate the precise location and 7. Now, if C3 touches the two golden spirals (Figure5a), this can locate the precise location and size 7. Now, if C3 touches the two golden spirals (Figure 5a), this can locate the precise location and of thesize optimized of the optimized sound-hole. sound‐hole. size of the optimized sound‐hole.

(a) (b) (c) (a) (b) (c) FigureFigure 3. (a )3. Step (a) one:Step tangentialone: tangential rectangle, rectangle, (b) step (b two:) step the two: first the golden first spiral,golden ( cspiral,) step three:(c) step the three: second the Figure 3. (a) Step one: tangential rectangle, (b) step two: the first golden spiral, (c) step three: the goldensecond spiral. golden spiral. second golden spiral.

(a) (b) (c) (a) (b) (c) Figure 4. (a) Step four: Circle C1, (b) step five: the first golden circle, (c) step six: the second golden Figure 4. (a) Step four: Circle C1, (b) step five: the first golden circle, (c) step six: the second golden Figurecircle. 4. ( a) Step four: Circle C1,(b) step five: the first golden circle, (c) step six: the second golden circle. circle. Note: It rarely happens that in an oval soundboard (with an almost correct geometry and proportion) the secondNote: Itgolden rarely circle happens does that not in touch an oval the soundboard golden spirals (with and an does almost not correctfollow geometrythis golden and method. proportion) the second golden circle does not touch the golden spirals and does not follow this golden method.

Arts 2017, 6, 20 6 of 15

Note that C3 usually touches the spirals in the 4th golden rectangle (Figure 5b); it can be seen in Figure 5 that C3 precisely tangents on the of the manufactured lute. In this way, one can Artsoptimize2017, 6 ,the 20 sound hole of any lute family instrument by relation between geometry and the golden6 of 15 ratio (Φ) or appraise the manufacturing process from the sound‐hole design and location point of view.

Arts 2017, 6, 20 6 of 15

Note that C3 usually touches the spirals in the 4th golden rectangle (Figure 5b); it can be seen in Figure 5 that C3 precisely tangents on the sound hole of the manufactured lute. In this way, one can optimize the sound hole of any lute family instrument by relation between geometry and the golden ratio (Φ) or appraise the manufacturing process from the sound‐hole design and location point of view.

(a) (b)

FigureFigure 5.5. ((a)) Step seven:seven: check the situation, ((b)) thethe optimized sizesize andand locationlocation ofof thethe sound-hole.sound‐hole.

3. Experimental Evaluation Note: It rarely happens that in an oval soundboard (with an almost correct geometry and proportion)In order the to confirm second the golden effectiveness circle does of the not proposed touch the Golden golden Method, spirals an and experimental does not follow case study this goldenis presented method. in this section. The setar is selected as the case study. A setar (from seh, meaning “three” and tNoteār, meaning that C3 usually“string”) touches is a Persian the spirals musical in theinstrument. 4th golden It rectangleis a member (Figure of the5b); lute it can family, be seen which in is played with the index finger of the right hand. Two and a half centuries ago, a fourth string was Figure5 that C 3 precisely tangents on(a) the sound hole of the manufactured lute.(b) In this way, one can optimizeadded to thethe sound setar (Khaleghi hole of any 1963). lute family It has instrument25–27 moveable by relation which between are geometry usually made and the of goldenanimal ratiointestines (ΦFigure) or or appraise 5.silk. (a) StepIt theoriginated seven: manufacturing check in thePersia situation, process before ( fromb the) the thespread optimized sound-hole of Islam.size and design Figure location and 6 of locationshows the sound a point typical‐hole. of view.setar (dimensions are in millimetres). The sound box structure of this instrument is similar to that of the 3.lute3. Experimental Experimental but only with Evaluation Evaluation respect to the size of soundboard and the type of sound‐holes. InIn order order to to confirm confirm the the effectiveness effectiveness of of the the proposed proposed Golden Golden Method, Method, an an experimental experimental case case study study isis presented presented in in this this section. section. TheThe setarsetar isis selectedselected asas thethe casecase study.study. AA setarsetar (from(from seh,seh, meaningmeaning “three”“three” andand t ar,t¯ār, meaning meaning “string”) “string”) is is a a Persian Persian musical musical instrument. instrument. It It is is a membera member of of the the lute lute family, family, which which is playedis played with with the indexthe index finger finger of the of right the right hand. hand. Two andTwo a and half a centuries half centuries ago, a fourthago, a stringfourth was string added was toadded the setar to the (Khaleghi setar (Khaleghi 1963). It has1963). 25–27 It has moveable 25–27 moveable frets which frets are which usually are made usually of animal made intestinesof animal orintestines silk. It originated or silk. It inoriginated Persia before in Persia the spread before of the Islam. spread Figure of Islam.6 shows Figure a typical 6 shows setar a (dimensions typical setar are(dimensions in millimetres). are in Themillimetres). sound box The structure sound box of this structure instrument of this is similarinstrument to that is similar of the luteto that but of only the withlute respectbut only to with the sizerespect of soundboard to the size of and soundboard the type of and sound-holes. the type of sound‐holes. Figure 6. A typical setar

Setar makers believe that the holes adjust the sound of the instrument. Usually after final shaving of the soundboard, they punch the holes. A review of several setar patterns shows that there is no fixed pattern for making the sound‐holes; the setar makers design their pattern based on their manner and experience. One of the main differences between setar patterns is the templates of sound‐holes that are bored in the soundboard and guttoral are of the setar. The manufacturers usually find the location of the main hole by using a tuning fork at note of A = 440 Hz. In this way, they vibrate the tuning fork, put it on the area and move it to the neck. The point where the sound reaches its FigureFigure 6. 6.A A typical typical setar. setar peak is marked, and the procedure is repeated several times to specify the main sound‐hole location precisely.Setar The makers manufacturers believe that punch the holes a 5 adjust‐mm hole the soundand the of other the instrument. holes should Usually be along after the final perimeter shaving Setar makers believe that the holes adjust the sound of the instrument. Usually after final shaving ofof thisthe holesoundboard, (Figure 7) they (Shirazi punch 1990). the holes. A review of several setar patterns shows that there is no of the soundboard, they punch the holes. A review of several setar patterns shows that there is no fixedSome pattern researchers for making claim the sound that the‐holes; main the air setar resonance makers frequencydesign their might pattern be basedapproximated on their manner by the fixed pattern for making the sound-holes; the setar makers design their pattern based on their manner linearand experience. superposition One of of the the effect main of differences each hole (Shawbetween 1990; setar Takegawa patterns 2008).is the Sometemplates conventional of sound ‐typesholes and experience. One of the main differences between setar patterns is the templates of sound-holes ofthat setar are sound bored‐ holein the patterns soundboard are shown and guttoralin Figure are 8. of the setar. The manufacturers usually find the that are bored in the soundboard and guttoral are of the setar. The manufacturers usually find the location of the main hole by using a tuning fork at note of A = 440 Hz. In this way, they vibrate the location of the main hole by using a tuning fork at note of A = 440 Hz. In this way, they vibrate the tuning fork, put it on the bridge area and move it to the neck. The point where the sound reaches its tuning fork, put it on the bridge area and move it to the neck. The point where the sound reaches its peak is marked, and the procedure is repeated several times to specify the main sound‐hole location peak is marked, and the procedure is repeated several times to specify the main sound-hole location precisely. The manufacturers punch a 5‐mm hole and the other holes should be along the perimeter of this hole (Figure 7) (Shirazi 1990). Some researchers claim that the main air resonance frequency might be approximated by the linear superposition of the effect of each hole (Shaw 1990; Takegawa 2008). Some conventional types of setar sound‐hole patterns are shown in Figure 8.

Arts 2017, 6, 20 7 of 15 precisely. The manufacturers punch a 5-mm hole and the other holes should be along the perimeter of thisArts 2017 hole, 6 (Figure, 20 7)(Shirazi 1990). 7 of 15

Arts 2017, 6, 20 7 of 15

Figure 7. Empirical sound‐hole design and development for . Figure 7. Empirical sound-hole design and development for setars.

Some researchers claim that the main air resonance frequency might be approximated by the linear superposition of the effect of each hole (Shaw 1990; Takegawa 2008). Some conventional types of setar sound-hole patternsFigure 7. areEmpirical shown sound in Figure‐hole8 design. and development for setars.

Figure 8. Conventional types of setar sound‐hole patterns.

For comparison, two well‐reputed setar patterns are selected in this study:

3.1. Case Study I: Delroba2 FigureFigure 8.8. ConventionalConventional typestypes ofof setarsetar sound-holesound‐hole patterns.patterns. The first one is called the Delroba, shown in Figure 9a, designed by the legendary classical MehdiFor Kamalian comparison, (1918–1997 two well) and‐reputed is a well setar‐preserved patterns exampleare selected of his in thiswork. study: Here, we analyze the air For comparison, two well-reputed setar patterns are selected in this study: mode schema of this pattern for optimizing the sound‐holes using the Golden Method. 2 3.1.3.1. CaseCaseUsing StudyStudy the I:I:Golden DelrobaDelroba Method2 described in Section 2, the following pattern shown in Figure 9b is achievedThe firstfor theone Delroba.is called theAs Delroba,shown in shown this figure, in Figure the 9a, middle designed holes by asthe well legendary as the classicalmain hole luthier are locatedThe inside first one the is golden called circle. the Delroba, shown in Figure9a, designed by the legendary classical luthier Mehdi Kamalian (1918–1997) and is a well‐preserved example of his work. Here, we analyze the air Mehdi Kamalian (1918–1997) and is a well-preserved example of his work. Here, we analyze the air mode schema of this pattern for optimizing the sound‐holes using the Golden Method. mode schema of this pattern for optimizing the sound-holes using the Golden Method. Using the Golden Method described in Section 2, the following pattern shown in Figure 9b is Using the Golden Method described in Section2, the following pattern shown in Figure9b is achieved for the Delroba. As shown in this figure, the middle holes as well as the main hole are achieved for the Delroba. As shown in this figure, the middle holes as well as the main hole are located located inside the golden circle. inside the golden circle.

2 Mean enchanting; one of the most famous Setar designs.

(a) (b)

2 Mean enchanting; one of the most(a famous) Setar designs. (b)

2 Mean enchanting; one of the most famous Setar designs.

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Figure 7. Empirical sound‐hole design and development for setars.

Figure 8. Conventional types of setar sound‐hole patterns.

For comparison, two well‐reputed setar patterns are selected in this study:

3.1. Case Study I: Delroba2 The first one is called the Delroba, shown in Figure 9a, designed by the legendary classical luthier Mehdi Kamalian (1918–1997) and is a well‐preserved example of his work. Here, we analyze the air mode schema of this pattern for optimizing the sound‐holes using the Golden Method. Using the Golden Method described in Section 2, the following pattern shown in Figure 9b is Arts 2017, 6, 20 8 of 15 achieved for the Delroba. As shown in this figure, the middle holes as well as the main hole are located inside the golden circle.

Arts 2017, 6, 20 (a) (b) 8 of 15

FigureFigure 9. 9. (a()a )Delroba Delroba pattern, pattern, (b (b) )Application Application of of the the Golden Golden Method Method to to the the Delroba. Delroba.

3.2.3.2. Case Case2 StudyMean Study enchanting; II: II: Eshghi Eshghi 3one 3 of the most famous Setar designs.

TheThe second second case case study study is is the the Eshghi Eshghi pattern pattern by by Mohammad Mohammad Navaei Navaei (1903–1987). (1903–1987). Masters Masters of of setar setar makingmaking believe believe that that this this pattern pattern is is the the best best scale scale pattern pattern ever ever designed designed and and implemented. implemented. This This model model hashas fewer fewer holes holes than than the the Delroba. Delroba. Note Note that that all all Eshghi Eshghi holes holes are are located located in in the the golden golden circle circle as as shown shown inin Figure Figure 10. 10 .

(a) (b)

FigureFigure 10. 10. (a()a )Eshghi Eshghi pattern, pattern, (b (b) )Application Application of of the the Golden Golden Method Method to to the the Eshghi. Eshghi.

3.3. Case Study III: Mahava4 3.3. Case Study III: Mahava4 Finally, in order to prove the effectiveness of the Golden method in enhancing the sound Finally, in order to prove the effectiveness of the Golden method in enhancing the sound radiation radiation ability, a new musical instrument with an optimized sound‐hole was developed to compare ability, a new musical instrument with an optimized sound-hole was developed to compare with with the above‐mentioned instruments. We called this the Mahava. It is worthwhile to mention that the above-mentioned instruments. We called this the Mahava. It is worthwhile to mention that the the Delroba, Eshghi, and Mahava are exactly the same in sound‐hole size, location and pattern to Delroba, Eshghi, and Mahava are exactly the same in sound-hole size, location and pattern to ensure ensure fair comparison. Figure 11 shows the construction procedure of the Mahava. fair comparison. Figure 11 shows the construction procedure of the Mahava.

3 Means “related to love”; one of the most famous Setar designs. 4 Means “like the moos sound”.

(a)

3 Means “related to love”; one of the most famous Setar designs. 4 Means “like the moos sound”.

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Figure 9. (a) Delroba pattern, (b) Application of the Golden Method to the Delroba.

3.2. Case Study II: Eshghi3 The second case study is the Eshghi pattern by Mohammad Navaei (1903–1987). Masters of setar making believe that this pattern is the best scale pattern ever designed and implemented. This model has fewer holes than the Delroba. Note that all Eshghi holes are located in the golden circle as shown in Figure 10.

(a) (b)

Figure 10. (a) Eshghi pattern, (b) Application of the Golden Method to the Eshghi.

3.3. Case Study III: Mahava4 Finally, in order to prove the effectiveness of the Golden method in enhancing the sound radiation ability, a new musical instrument with an optimized sound‐hole was developed to compare with the above‐mentioned instruments. We called this the Mahava. It is worthwhile to mention that theArts Delroba,2017, 6, 20 Eshghi, and Mahava are exactly the same in sound‐hole size, location and pattern9 of to 15 ensure fair comparison. Figure 11 shows the construction procedure of the Mahava.

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3 Means “related to love”; one of the most famous Setar designs. 4 Means “like the moos sound”.

(b) (c)

(d)

Figure 11. New optimized musical instrument‐manufacturing procedure. (a) Raw application of the Figure 11. New optimized musical instrument-manufacturing procedure. (a) Raw application of theGolden Golden Method; Method; (b) (Manufacturingb) Manufacturing procedure procedure based based on on the the Golden Golden Method; Method; ( (cc)) Final musical instrument, the Mahava; (d) Comparison between thethe Mahava andand Delroba.Delroba.

4. Experimental Apparatus and Procedure 4. Experimental Apparatus and Procedure After developing the Mahava, experimental tests were carried out on all discussed instruments After developing the Mahava, experimental tests were carried out on all discussed instruments to confirm the ability of the Golden Method to optimize the sound‐hole placement and size. The to confirm the ability of the Golden Method to optimize the sound-hole placement and size. experimental procedure was to elicit the same note in all instruments and to record the response until it decayed to zero to analyze the sound damping capability of the instruments.

4.1. Data Acquisition System Figure 12 shows a photograph of the microphone array. Identical RØDE NT1000 Cardioid polar pattern microphones (RØDE Microphones, Silverwater, Australia) were positioned at the vertices and front side of the samples, pointing directly inward. The microphone position was adjusted so that there was exactly 18” (45.72 cm) between the microphone and sample; the axis was aimed toward the opposite vertex in the array. The array was suspended in the center (near the floor) of an 8 × 6 × 3 m studio. The microphones were routed through preamps with a flat frequency response to a Focusrite Scarlett 6i6 sound card (Focusrite Audio Engineering Ltd, High Wycombe, UK). All microphone signal paths were normalized to within 1 dB, using a test tone generator.

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The experimental procedure was to elicit the same note in all instruments and to record the response until it decayed to zero to analyze the sound damping capability of the instruments.

4.1. Data Acquisition System Figure 12 shows a photograph of the microphone array. Identical RØDE NT1000 Cardioid polar pattern microphones (RØDE Microphones, Silverwater, Australia) were positioned at the vertices and front side of the samples, pointing directly inward. The microphone position was adjusted so that there was exactly 18” (45.72 cm) between the microphone and sample; the axis was aimed toward the opposite vertex in the array. The array was suspended in the center (near the floor) of an 8 × 6 × 3 m studio. The microphones were routed through preamps with a flat frequency response to a Focusrite Scarlett 6i6 sound card (Focusrite Audio Engineering Ltd., High Wycombe, UK). All microphone signal paths were normalized to within 1 dB, using a test tone generator. Arts 2017, 6, 20 10 of 15

Figure 12. Experimental Apparatus. Figure 12. Experimental Apparatus. 4.2. Test Procedure 4.2. Test Procedure The main issue in the test procedure was that there was no control over the force of the strike on The main issue in the test procedure was that there was no control over the force of the strike on the string. In fact, one could easily imagine that the harder the pluck, the louder the sound, and the the string. In fact, one could easily imagine that the harder the pluck, the louder the sound, and the longer the sound would take to decay; this problem would affect the analysis results. To avoid this longer the sound would take to decay; this problem would affect the analysis results. To avoid issue, three musicians who were not aware of the target of the research were selected, and the this issue, three musicians who were not aware of the target of the research were selected, and the experiments were performed by them. experiments were performed by them. Then, sets of “perfect” impulse responses were selected based on the excitation signal. This was doneThen, by eliminating sets of “perfect” any double impulse hammer responses hits, were eliminating selected based any recordings on the excitation where signal. the researcher This was donecommented by eliminating during the any experiment double hammer that it was hits, a eliminatingbad hit, then any finally recordings choosing where from thethe researcherremaining commentedpossibilities the during force the hammer experiment impulse that with it was the a narrowest bad hit, then and finally highest choosing peak. Selected from the signals remaining were possibilitiestransferred digitally the force to hammer a computer impulse for analysis. with the narrowest and highest peak. Selected signals were transferredThe experiment digitally to was a computer performed for on analysis. the third string of samples that was tuned to 440 Hz. The thirdThe string experiment was composed was performed of phosphor on the bronze third string or brass. of samples It has a that minimum was tuned tensile to 440 strength Hz. The of third 900 string was composed of phosphor bronze or brass. It has a minimum tensile strength of 900 N/mm2. N/mm2. The diameter of the third string of a Persian setar is normally 0.02 mm thicker than that of Thesteel diameter strings. (The of the diameter third string of steel of a strings Persian is setar normally is normally 0.2 mm, 0.02 and mm the thicker thickness than of that bronze of steel strings strings. is (The0.22 mm). diameter of steel strings is normally 0.2 mm, and the thickness of bronze strings is 0.22 mm). 5. Results and Analysis 5. Results and Analysis Figure 13 shows one time record graph for the Mahava. It is worth mentioning that experimental Figure 13 shows one time record graph for the Mahava. It is worth mentioning that experimental tests were conducted several times as discussed in the previous section, and the worst-case scenario tests were conducted several times as discussed in the previous section, and the worst‐case scenario results were selected to gain confidence about the validity of the analysis. Figure 14 compares results were selected to gain confidence about the validity of the analysis. Figure 14 compares the the time response of the Delroba (Kamalian), Eshghi, and Mahava. The results obtained from the time response of the Delroba (Kamalian), Eshghi, and Mahava. The results obtained from the experiments are summarized in Table1. As shown in this table, the Golden Method improved the experiments are summarized in Table 1. As shown in this table, the Golden Method improved the radiation time of the Mahava by 2.4% and 3.2% compared with the Delroba and Eshghi, respectively. radiation time of the Mahava by 2.4% and 3.2% compared with the Delroba and Eshghi, respectively. Moreover, the maximum recorded amplitude of the Mahava was slightly higher than that of the two other instruments.

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Moreover, the maximum recorded amplitude of the Mahava was slightly higher than that of the Artstwo 2017 other,, 6,, 20 instruments. 11 of 15

FigureFigure 13. 13. MahavaMahava time time record record history. history.

Figure 14. Cont.

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Figure 14. Time record history of the Delroba (top), Eshghi (middle), and Mahava (bottom). Figure 14. Time record history of the Delroba (top), Eshghi (middle), and Mahava (bottom).

Table 1. Time response results. Table 1. Time response results. Maximum Amplitude (mV) Maximum Amplitude Time (ms) Damping Time (ms) § Delroba Maximum Amplitude962 (mV) Maximum Amplitude42.9 Time (ms) Damping3004 Time (ms) § Eshghi 855 40.6 2955 Delroba 962 42.9 3004 EshghiMahava 1000 855 44.8 40.6 3155 2955 Mahava§ The time in which (and 1000 after which) the response amplitude 44.8 would remain in the band 3155 of ±5% of the§ The final time amplitude. in which (and after which) the response amplitude would remain in the band of ±5% of the final amplitude. In other words, Table 1 confirms that the Golden Method is able to enhance the capability of the setar Inin note other keeping, words, Table even 1in confirms comparison that with the Golden well‐reputed Method setars. is able Consequently, to enhance the the capability results confirm of the thesetar effectiveness in note keeping, of the even proposed in comparison method with in well-reputedsound‐hole size setars. and Consequently, location design the resultsin lute confirmfamily musicalthe effectiveness instruments of the to proposedpromote their method orchestral in sound-hole capabilities. size and location design in lute family musical instrumentsMoreover, to promotethe frequency their orchestral response capabilities. function (FRF) of the three instruments is also shown in FigureMoreover, 15. As shown the frequencyin this figure, response the FRF function trends are (FRF) the same of the for three all instruments instruments and is alsoconsequently, shown in theFigure golden 15. Asmethod shown does in this not figure,affect the the frequency FRF trends characteristics are the same of for the all instruments. instruments andIn other consequently, words, it is not a trade‐off approach.

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As shown in the above figures and tables, the Mahava is superior to the other well‐known instrumentsArts 2017, 6, 20 from the sound radiation point of view, which is an important orchestral characteristic13 of 15 of a musical instrument. The other characteristics of the Mahava are the same as those of the Delbora and Eshghi. Consequently, the Golden Method could be introduced as a new successful approach to the golden method does not affect the frequency characteristics of the instruments. In other words, design and locate optimized sound‐holes in musical instruments of the lute family. it is not a trade-off approach.

(a) (b)

(c)

Figure 15. Frequency response of Delroba (a), Eshghi (b), and Mahava (c). Figure 15. Frequency response of Delroba (a), Eshghi (b), and Mahava (c).

6. Conclusions As shown in the above figures and tables, the Mahava is superior to the other well-known instrumentsThe paper from focused the sound on enhancing radiation pointthe damping of view, characteristic which is an important of lute family orchestral musical characteristic instruments of whicha musical is one instrument. of the most The important other characteristics orchestral characteristics of the Mahava of are stringed the same instruments. as those of The the DelboraGolden Method,and Eshghi. a geometry Consequently,‐based theanalytical Golden approach Method could using be the introduced golden ratio as a concept, new successful is introduced approach as toa noveldesign procedure and locate to optimized optimize sound-holesthe size and inlocation musical of instruments the sound‐holes of the in lute musical family. instruments of the lute family (e.g., lute, oud, barbat, mandolin, setar, and etc.). The proposed method claims that if the golden6. Conclusions circle generated by the golden method is tangent to the golden spirals of the soundboard, it locates the optimized location and size of the sound‐hole. To prove this claim, three case studies were The paper focused on enhancing the damping characteristic of lute family musical instruments considered including two well‐reputed musical instruments and a new setar, Mahava, developed to which is one of the most important orchestral characteristics of stringed instruments. The Golden show the effectiveness of the Golden Method in increasing the sound damping capability of musical Method, a geometry-based analytical approach using the golden ratio concept, is introduced as a novel instruments. The results from experiments confirm that the instrument constructed based on the procedure to optimize the size and location of the sound-holes in musical instruments of the lute family Golden Method is superior to other well‐known instruments from a sound radiation point of view (e.g., lute, oud, barbat, mandolin, setar, and etc.). The proposed method claims that if the golden circle without affecting other acoustic characteristics. Consequently, the Golden Method could be used in generated by the golden method is tangent to the golden spirals of the soundboard, it locates the future studies to optimize the sound‐hole design procedure in lute family musical instruments. optimized location and size of the sound-hole. To prove this claim, three case studies were considered including two well-reputed musical instruments and a new setar, Mahava, developed to show the Author Contributions: S.J. and M.M.K. conceived and designed the experiment; M.M.K. collected the data; S.J. analyzedeffectiveness the data; of the S.J. Golden and M.M.K. Method wrote in and increasing revised the paper. sound damping capability of musical instruments. The results from experiments confirm that the instrument constructed based on the Golden Method Conflictsis superior of Interest: to other The well-known authors declare instruments no conflict from of interest. a sound radiation point of view without affecting other acoustic characteristics. Consequently, the Golden Method could be used in future studies to optimize the sound-hole design procedure in lute family musical instruments.

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Author Contributions: S.J. and M.M.K. conceived and designed the experiment; M.M.K. collected the data; S.J. analyzed the data; S.J. and M.M.K. wrote and revised the paper. Conflicts of Interest: The authors declare no conflict of interest.

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