Fuzzy tuning systems: the mathematics of musicians

Liern, V Fuzzy tuning systems: the mathematics of musicians Fuzzy sets and systems. 2005, 150, 1, p 35-52 We present some mathematical properties which determine tuning methods. We introduce the concept of fuzzy tuning systems and we analyze four of the systems coexisting within the current orchestras: Pythagorean, Just Intonation, Holder's and Equal Temperament systems. We show that the theoretical and practical! tuning methods are the same. We introduce the idea of compatibility between tuning systems and we give some sufficient conditions to determine an appropriate number of notes into which the octave must be divided. Losas

www.elsevier.com/locate/fss Fuzzy tuning systems: the mathematics of musicians Vicente Liern∗ Departamento de Matematicas Economico-Empresariales, Universitat de Valencia, Avda. de los Naranjos s=n, 46071 Valencia, Spain Received 3 October 2003; received in revised form 4 March 2004; accepted 5 April 2004 Abstract We present some mathematical properties which determine tuning methods. We introduce the concept of fuzzy tuning systems and we analyze four of the systems coexisting within the current orchestras: Pythagorean, Just Intonation, H3older’s and Equal Temperament systems. We show that the theoretical and practical tuning methods are the same. We introduce the idea of compatibility between tuning systems and we give some su9cient conditions to determine an appropriate number of notes into which the octave must be divided. c 2004 Elsevier B.V. All rights reserved.
Keywords: Tuning systems; Fuzzy sets; Fuzzy numbers; Continued fractions 1. Introduction A tuning system is the set of sounds that music uses. By this, we mean that from the set of the frequencies of all the possible sounds, R+ , a subset containing the appropriate frequencies is selected. Di>erent criteria have been used to make this selection but, at least since the 4th century B.C., most tuning systems have been obtained by means of mathematical arguments [10–12]. It is undeniable that the numerical nature of these systems made instrument manufacturing easier and also facilitated their transmission [7]. However, the crispness of the mathematical arguments relegated these tuning systems to theoretical studies, while in practice musicians tuned in a more Cexible way. In fact, if we represent graphically the frequencies at an instant t produced by each one of the instruments in an orchestra, the great di>erences observed between sounds considered as well tuned would be surprising. Nevertheless, the ensemble sensation is very pleasant [15]. This phenomenon  This work has been partially supported by the Ministerio de Ciencia y TecnologEFa of Spain, TIC2002-04242-C03-03. Tel.: +34-96-3828369; fax: +34-6-3828370. E-mail address: Vicente.Liern@uv.es (V. Liern). ∗ c 2004 Elsevier B.V. All rights reserved. 0165-0114/$ - see front matter  doi:10.1016/j.fss.2004.04.002

allows us to introduce the concept of compatibility between notes by means of an index of consistency, introduced by Zadeh [17], which measures how possible it is for the fuzzy numbers associated to the notes to be equal. In this paper, we show that, as in many other human activities, what musicians do is to apply fuzzy decision rules for their selection criteria [8,18]. Actually, we will see that the theoretical tuning systems and the well-tuned sounds (as tested by a chromatic tuner) that musicians use in practice are the same. On the other hand, the compatibility between two notes becomes insu9cient when the tuning of more than one instrument is analyzed. In this case, it is necessary to study when two tuning systems can coexist and the concept of -compatibility between tuning systems appears naturally, where  represents the level of similarity between these systems. 2. Previous concepts In this paper, we will identify each musical note with the frequency of its fundamental harmonic (the frequency that tuners measure) because we will work with tuning systems. The usual way to relate two frequencies is through their ratio and this number is called the interval. It is well known that, in the middle zone of the audible Neld, the “pitch sensation” changes approximately according to the logarithm of the frequency, so the distance between two notes sounds whose frequencies are f1 and f2 can be estimated by means of the expression


f1


d(f1 ; f2 ) := 1200
log2 ; (1) f2
where the logarithm in base 2 and the factor 1200 have been used in order to express d in cents [10]. Undoubtedly, the octave is the interval which is more generally used and it can be deNned as follows: Denition 1. Given two sounds with frequencies f1 and f2 , we say that f2 is an octave higher than f1 if f2 is double f1 . Two notes an octave apart from each other have the same letter-names. This naming corresponds to the fact that notes an octave apart sound like the same note produced at di>erent pitches and not like entirely di>erent notes. Based on this idea, we can deNne in R+ (the subset of all the frequencies of all the sounds) a binary equivalence relation, denoted by R, as follows [13]: f1 Rf2 if and only if ∃n ∈ Z such that f1 = 2n f2 : (2) Therefore, instead of dealing with R+ , we can analyze the quotient set R+ =R, which for a given Nxed note f0 (diapason) can be identiNed with the interval [f0 ; 2f0 [. In 1955 the International Organization for Standardization Nxed as diapason or concert pitch the frequency of A4 , the A above middle C, at 440 Hz (see [2]). However, for the sake of simplicity, we will assume that f0 = 1 and work in the interval [1; 2[.

Let us introduce the deNnition of a tuning system Denition 2. Let f1 =f2 be an interval and by the set = | log2 (f1 =f2 )|. We call the tuning system generated S := {2cn | cn = n −  n ; n ∈ Z} ⊂ [1; 2[; (3) where x is the integer part of x. Some systems are generated by more than one interval and in such cases, it is necessary to specify when and how many times each interval appears. Denition 3. Let  = { i }ki=1 ⊂ [0; 1[ and a family of functions fi : Z → Z, i = 1; 2; : : : ; k. We call the tuning system generated by the intervals {2 i }ki=1 (or simply by { i }ki=1 ) and F = {fi }ki=1 the set  F S := cn 2 | cn = k  i fi (n) − i=1  k   i fi (n)  ; n∈Z ⊂ [1; 2[: (4) i=1 If every element in the tuning system is a rational number, we say that it is a tuned system whereas if some element is an irrational number then the system is a temperament [16]. Remark 1. The advantage of expressing the tuned notes as 2cn is that if our reference note is 20 , by (1) the exponent cn provides the pitch sensation. Once we Nx a tuning system we are able to establish if a sound is tuned or not. Denition 4. A sound with frequency f is a well-tuned note in SF if n ∈ Z exists such that 2n ·f ∈ SF . Usually, musicologists feel more comfortable with ordering all or part of the notes in a tuning system by Nfths and then representing them as points in a circumference called a cycle or circle of 3fths. This cycle is not necessarily closed, i.e. the notes are not necessarily repeated, and actually to force the cycle to be closed one or even more Nfths must be modiNed [2,6]. However, in the Equal Temperament, for instance, the circle closes naturally with 12 equal Nfths. In order to deal with a circumference instead of an interval [f0 ; 2f0 [ we consider the function ’ : Z × S 1 → S 1 given by ’(n; ) =  + 2cn ; n ∈ Z; where S 1 = {:  ∈ [0; 2]}, and {cn }n∈Z is the sequence of exponents in (2) and (4), the sequence {’(n; 0)}n∈Z is equivalent to SF .

3. Mathematical translation of some known tuning systems Among the di>erent tuning systems used in the western music since the 6th century B.C., four of them are specially interesting because they still remain in our classic orchestras [4–6]: the Pythagorean, Zarlinean, Holderean and Equal Tempered systems. In fact, performers consider that they “sound” in the Equal Tempered system. However, some experiments [8] show that these four systems coexist and their simultaneity does not imply any loss of beauty in the ensemble. In this section, we express these four tuning systems in terms of DeNnition 4 and we brieCy show some of their advantages and disadvantages. 3.1. Tuned systems (a) Pythagorean system: This system is obtained by “transferring” the powers of 3 to the interval [1; 2[. Each time we multiply (resp. divide) by 32 a frequency f, we say that it goes up (resp. down) by a Nfth.  It is easy to prove that the Pythagorean system, generated by the Nfth interval, or by = log2 32 , is the set of notes given by S := {2cn | cn = n log2 (3=2) − n log2 (3=2) ; n ∈ Z}: (6) (b) Just Intonation: The Just Intonation can be viewed as a generalization of the Pythagorean system because it not only works with powers of 3, but also with powers of 5. Every time we multiply (resp. divide) a frequency f by 54 it is said that f goes up (resp. down) by one-third. In practice, the Just Intonation can be obtained by replacing some Nfths of the Pythagorean system 3 , by syntonic Nfths 40 (see [15]). Such a Nfth is called syntonic because it di>ers by a syntonic 2 27 = 81 . Depending on the number of Nfths substituted, comma [2] from the Pythagorean Nfth, i.e. 32 : 40 27 80 a di>erent variant is obtained [6]. In this paper, we use Zarlino’s approach which can be described as follows:   2   2   f1 ; f2 S 1 ; 2 := 2cn | cn = ; n∈Z ; (7) i fi (n) − i fi (n) where 1 = log2 3 2 i=1 , 2 = log2 f1 (n) = n − 4f2 (n); 5 4 i=1 and the functions f2 (n) = n+4 n+1 + : 7 7 Let us analyze the Pythagorean system and the Just Intonation in the circumference S 1 . As the generator intervals are irrational numbers, the sequence of exponents {cn }n∈Z veriNes that for all  ∈ S 1 , the sequence { + 2cn }n∈Z , is dense in S 1 (see [1]). Therefore, it is easy to show that Proposition 1. Let {cn }n∈Z be the sequence of exponents given in (6) or (7). (a) For n; m ∈ Z such that n = m then  + 2cn =  + 2cm . (b) Given 0 ; 1 ∈ S 1 such that 1 ∈= {0 + 2cn }n∈Z , then {0 + 2cn }n∈Z ∩ {1 + 2cn }n∈Z = ∅:

This result shows two disadvantages of the Pythagorean and Zarlinean systems. By (a) the circle of Nfth is not closed hence, to establish an appropriate number of notes in an octave, some additional criteria are necessary. According to (b), the point 0 ∈ S 1 determines the sequence {0 + 2cn }n∈Z , hence given the passage {f1 ; f2 ; : : : ; fk } obtained by multiplying by  (transpose down an interval ) a well-tuned passage {f1 ; f2 ; : : : ; fk } need not be tuned. 3.2. Tempered systems The temperaments appear as approximations of the tuned systems in order to avoid the problems described in Proposition 1. As in tempered systems some irrational numbers appear, so some tempered intervals do not correspond to the natural harmonics. However, the many advantages of temperaments have caused the words ‘tempered’ and ‘tuned’ to be considered synonymous in current musical practice. The most used temperaments are the cyclic temperaments that divide the octave into equal parts (in this way, the problems expressed in Proposition 1 are solved). Given a natural number q, the well-tuned notes are q −1 T q := {2k=q }k=0 : (8) In order to express T q in terms of DeNnition 2, it su9ces to take into account that given a natural number q, for each p ∈ N∗ such that (p; q) = 1; p¡q, T q = Sp=q holds. (c) Equal temperament (of 12 notes): This was utilized in at least 1482 by B. Ramos de Pareja in his book (Musica Practica [6,15]). However, it was not extended until the appearance of Das wohltemperierte Klavier I, (1721) of J.S. Bach. In this temperament the octave is divided into 12 equal parts, T 12 = {2k=12 ; 06k611}, hence we can express T 12 in terms of DeNnition 2 as  7 7 cn − n ; n∈Z : (9) S7=12 := 2 | cn = n 12 12 Nowadays, practically all musicians work with this tuning system and, in fact, it is called The Good Temperament [6]. (d) Temperament of H>older: Since the 17th century, hundreds of temperaments have arisen, but we will only work with the H3older’s temperament because it is still utilized in many theoretical studies. W. H3older (1614–1697) proposed a temperament that divides the octave into 53 equal parts, T 53 ={2k=53 }52 k=0 . In this way, a very good approximation of the Pythagorean system is obtained. Its notes can be expressed as  31 31 cn − n ; n∈Z : (10) S31=53 := 2 | cn = n 53 53 7 Notice that the choice of the values 12 and 31 in expressions (9) and (10), respectively, is not 53 unique. Theorem 1 justiNes this choice. In order to illustrate the di>erences between the tuning systems, let us consider the frequencies of the notes from three measures of the Third Movement of Music for Strings, Percussion and Celesta (1936) by BEela BartEok (see Fig. 1). For each note we compute the distance between its frequency in the Equal Tempered system Nxing A4 = 440 Hz and the remaining systems (see Fig. 2).

Fig. 1. Fragment of Music for Strings, Percussion and Celesta by BEela BartEok. 50 45 40 d(H,T) d(Z,T) d(P,T) Cents 35 30 25 20 15 10 Isotuned Band Garbuzov zone 5 0 0 1 2 3 4 5 6 7 8 9 10 11 12 Notes Fig. 2. Distances between the notes in the pentagram for the Holderean, H, Pythagorean, P, and Zarlinean, Z, systems. Notice that notes whose distance is greater than 5 cents can be distinguished by the human ear (“isotuned band”) [15]. Even if we accept as the same note two notes whose distance is less than or equal to 12 cents (Garbuzov zone for the unison [8]), several notes in the fragment analyzed would not be well tuned. As we analyze in Example 3, the distances from the Equal Tempered F ]] , E ] and A] to the same notes for the remaining tuning systems are too big, especially the distances to the Zarlinean system. 4. Some concepts of fuzzy musical notes A musical note must be understood as a band of frequencies around a “central frequency” f and, as we will show in this section, modelling by means of a fuzzy set f˜ becomes very suitable. The idea of modelling musical notes as fuzzy sets is not new (see [8]) and it can justiNed for several reasons: (a) Technical reasons: In fast passages, musicians choose comfortable although slightly out of tune positions, while, lip pressure, temperature, humidity, hall acoustics, etc. all modify the frequencies. (b) Psychological reasons: The perception of the intervals is not the same for all of us and even more signiNcantly, it depends on the mood of the performer (see [8,14]). To be more precise, we will consider a musical sound as a fuzzy number which should reCect the sensation that a frequency f produces, i.e. log2 (f) (see expression (1)), and whose membership function should model musicians usual practices. With this aim, we will use the information which

+31 CENT . -50 .0 AUTOMANL. 41 D+50 . CENT Fig. 3. Scheme of a standard chromatic tuner. an electronic chromatic tuner provides. These tuners, based on the 12 note tempered system (see Table 2), divide the octave into 12 equal parts. Each part is 100 cents wide, so if we represent it as a segment, the (crisp) tuned note would be in the middle, and the extremes would be obtained by adding and subtracting 50 cents from the central note. As we will see in Example 1, the deviation to the central note gives rise to a membership function. Example 1. Let us consider an electronic tuner in which we have set A4 = 440 Hz. If it detects a note N whose frequency is 299 Hz, then we would obtain (see Fig. 3): Note: D; Deviation: +31:1702 cents: Firstly, the tuner locates the tuned note closer to N which, in this case, is D and then it measures the deviation between N and D. This deviation is an indicator of the degree of truthfulness of the statement “N is the note D”, deviation 31:1702 =1 − = 0:3766: (11) 50 50 Other possibilities for deNning this degree of truthfulness could be valid, but the linear choice reCects musicians’ usual practices. In general, they consider that 50 cents represents 14 of a tone and, for a note whose deviation is 25 cents, they would say that “this note has deviated by 18 of a tone”. The tuner uses the Equal Temperament, hence in the interval [1; 2[ the well-tuned notes are tn = 2n=12 , 06n611, which correspond respectively to C, C ] , D, D] , E, F, F ] , G, G ] , A, A] , B. As our interest is in knowing the pitch sensation, we should work with the exponents, i.e. log2 (tn ); 06n611 (see Remark 1). When the diapason is Nxed at 440 Hz, note C4 is determined as C4 = 2−3=4 × 440 Hz, and the reference interval is 1− [f0 ; 2f0 [ := [2−3=4 · 440; 21=4 · 440[: (12) Once we have established f 0 , with the aim of translating each frequency f to the interval [1; 2[, and subsequently take the exponent corresponding to 2, we need to make use of the following transformation:     f f ∗ − log2 : (13) f := log2

1 0.5 0.3766 D*- 1 24 D* N* D*- 1 24 Fig. 4. Membership function of note N in Example 1. Taking this transformation into account, (11) can be generalized way considering  ∗ in an operative  1 1 1 that note D deNnes a symmetric triangular fuzzy number D̃ = D ; 24 = 6 ; 24 whose membership function is  1 ; 1 − 24|D∗ − x| if |D∗ − x| ¡ 24 "D̃ (x) = (14) 0 otherwise: Therefore, the membership degree of N is "D̃ (N ) = "D̃ (N ∗ ) = 0:3766 (see Fig. 4). Notice that the membership degree of any note whose distance to D is greater than 50 cents to D̃ is zero. Following this reasoning we can establish the following deNnition: Denition 5. Let t˜ = (t; #) be a symmetric triangular fuzzy number, where t; # ∈ [0; 1]. The triangular fuzzy number 2t˜ := (2t ; 2t −# ; 2t+# ) whose membership function is  2t − x   1 − ; 2t −# ¡ x 6 2t ;   2t − 2 t − #  x − 2t "2t˜(x) = (15)  ; 2t ¡ x 6 2t+# ; 1 −  t+# t  2 −2   0 otherwise; is a fuzzy musical note. Remark 2. The quantity $ := 1200# expresses, in cents, the tolerance that we admit. Hence, for a 1 1 1 chromatic tuner (based on 12 notes) we have # = 2q = 24 , therefore, the tolerance is $ = 1200 × 24 = 50 cents. The choice of a symmetric triangular membership function is justiNed by musicians’ usual practices as it was in (11). Once we have stated the concept of a fuzzy musical note, our interest is to determine when two notes sound well together. It is well known that a fuzzy number ã can be considered as a possibility

distribution and its membership function "ã (x) can be interpreted as the degree of possibility of the statement “x is in ã” [3,18]. Therefore the equality of two notes ã and b̃ restricted by "ã and "b̃ can be assessed by using the index of consistency Pos[ã = b̃] := sup min{"ã (x); "b̃ (x)} = sup "s˜∩t˜ (x) x ∈E x ∈E (16) introduced by Zadeh [17]. Although there is also a degree of intersection between "ã and "b̃ , it evaluates to what extent it is possible to Nnd a common value for ã and b̃. Denition 6. Let 2s˜ and 2t˜ be two musical notes, where s̃ = (s; #) and t˜ = (t; #). We deNne the degree of compatibility between 2s˜ and 2t˜ as Compat[2s˜; 2t˜] := Pos[s̃ = t˜] (17) and we say that 2s˜ and 2t˜ are -compatible,  ∈ [0; 1], if Compat[2s˜; 2t˜]¿. If we say that 2s˜ and 2t˜ are compatible, we mean to say that they are 12 -compatible. The next proposition allows us to ensure the -compatibility (see also [5]). Proposition 2. Two musical notes 2s˜, 2t˜, where t˜ = (t; #) and s̃ = (s; #), #¿0, are -compatible,  ∈ [0; 1], if and only if |t − s|62#(1 − ). Proof. We can assume that s¡t without any loss of generality. According to (16), when the intersection between s̃ and t˜ is non-empty, s̃ ∩ t˜ is the triangular non-normalized fuzzy number whose membership function is  0 if x6t − #;        1 − 1 (t − x) if t − # ¡ x6 s + t ; # 2 (18) "s˜∩t˜ (x) =  1 s + t   ¡ x6s + #; 1 − (x − s) if    # 2  0 if x ¿ s + #: Therefore, in general the compatibility between 2s˜ and 2t˜ is given by    |t − s| s+t s˜ t˜ = max 0; 1 − : Compat[2 ; 2 ] = sup "t˜∩s˜ = "t˜∩s˜ 2 2# (19) Then, Compat[2s˜; 2t˜]¿ if and only if |t − s|62#(1 − ). In particular, if # = 1=2q, q ∈ N, q = 0 (see Remark 2), for a given  ∈ [0; 1], s̃ and t˜ are compatible i> |t − s|¡(1 − )=q, hence they are compatible when |t − s| 6 1

Remark 3. It is usually more comfortable to calculate the compatibility between two notes in terms of their frequencies. Hence, given two notes with frequencies f1 and f2 , for which we admit a tolerance of $ cents, according to (19) the compatibility between f1 and f2 is given by  d(f1 ; f2 ) ; (21) Compat[f̃1 ; f̃2 ] := max 0; 1 − 2$ where d is their distance expressed in cents (see expression (1)). Our next purpose is to analyze the compatibility between tuning systems, so we introduce the deNnition of a fuzzy tuning system. Denition 7. Let #¿0,  = { i }ki=1 ⊂ R+ and a family of functions fi : Z → Z, i = 1; 2; : : : ; k. We call a fuzzy tuning system generated by the intervals { i }ki=1 and F = {fi }ki=1 to the set   k   k   S̃ F 2c̃n | c̃n = (22) i fi (n) − i fi (n) ; # ; n ∈ Z :  (#) := i=1 i=1 Denition 8. Let S˜q (#) = {2s˜i }qi=1 and T̃ q (#) = {2t˜i }qi=1 be two tuning systems with q notes. We say that S˜q (#) and T̃ q (#) are -compatible,  ∈]0; 1], if for each s̃i ∈ S˜q (#) there is a unique t˜j ∈ T̃ q (#) such that Compat[2s˜i ; 2t˜j ]¿: (23) The quantity  in (23) can be regarded as the degree of interchangeability between S˜q and T̃ q . The following result provides us a upper bound of the compatibility level: Proposition 3. Let S˜q (#) = {2s˜i }qi=1 and T̃ q (#) = {2t˜i }qi=1 be two tuning systems -compatible. Thus, the level of compatibility  veri3es  |ti − si | = min{Compat[2s˜i ; 2t˜i ]}; (24)  6 min 1 − i i 2# where for each s̃i = (si ; #), the number t˜i = (ti ; #) is the unique exponent such that 2s˜i and 2t˜i are -compatible. Proof. By DeNnition 8, given 2s˜i ∈ S˜q (#) there is a unique 2t˜i ∈ T̃ q (#) -compatible with it. By Proposition 2, 61 − (|ti − si |)=2#. Thus, this inequality holds for every note in these systems then, 6 mini {1 − (|ti − si |)=2#} and, according to expression (19), 1 − (|ti − si |)=2# = Compat[2s˜i ; 2t˜i ]. On the other hand, we can give some su9cient conditions for the non-compatibility as follows: Proposition 4. Let S˜q (#) = {2s˜i }qi=1 and T̃ q (#) = {2t˜i }qi=1 be two tuning systems. Then, (a) If there are 2s˜i ; 2s˜k ∈ S˜q (#), s̃i = s̃k , 2t˜k0 ∈ T̃ q (#) such that Compat[2t˜k0 ; 2s˜i ] ¿  Compat[2t˜k0 ; 2s˜k ] ¿ ; for a given  ∈ ]0; 1], then S˜q (#) and T̃ q (#) are not -compatible.

(b) If there are 2s˜i ; 2s˜k ∈ S˜q (#), s̃i = s̃k , 2t˜k0 ∈ T̃ q (#) verifying max Compat[2t˜j ; 2s˜i ] = Compat[2t˜k0 ; 2s˜i ] = 1 ; t˜j max Compat[2t˜j ; 2s˜k ] = Compat[2t˜k0 ; 2s˜k ] = 2 ; t˜j then S˜q (#) and T̃ q (#) are not -compatible for any  ∈ ]0; 1]. Proof. (a) Follows from the uniqueness required in DeNnition 8. (b) Let us suppose that, for instance, 1 62 . On one hand, if  ∈]0; 1 ] we have Compat[2t˜k0 ; 2s˜i ] ¿ and Compat[2t˜k0 ; 2s˜k ]¿, therefore, by (a) the systems are not -compatible. On the other hand, if  ∈]1 ; 1] as 1 = maxt˜j Compat[2t˜j ; 2s˜i ], there is no 2t˜j ∈ T̃ q (#) such that Compat[2t˜j ; 2s˜i ]¿. Then, by DeNnition 8, the systems cannot be -compatible. Notice that the concept of -compatibility between systems reCects not only the idea of proximity between the notes of two di>erent systems, but also that their conNguration is similar. In practice, musicians must know which note is close enough to which other one to be considered as interchangeable and, clearly, this criterion must be unique. As some tuning systems consist of a Nnite number of notes, it can happen that two systems were compatible or not depending on which terms are chosen (see (6) and (7)). We will see that in the following example: ˜ = {2s˜n }n∈Z and the Equal Temperament of Example 2. We consider the Pythagorean system S(#) 41 t˜n 40 41 notes T̃ (#) = {2 }n=0 , where  s̃n =  3 3 ;# ; n log2 − n log2 2 2  t˜n =  24 24 − n ;# : n 41 41 Thus,  1   1  1 s˜n 23 41 = {2t˜n }23 (a) S˜41 1  2·41  = {2 }n=−17 and T̃ n=−17 are -compatible, for ¿ 2 . 2·41 1 1 s˜n 40 41 (b) S˜41 = {2t˜n }40 2 2·41 = {2 }n=0 and T̃ n=0 are not compatible for any  ∈ ]0; 1]. 2·41 1 , By a direct calculus it is easy to prove that if n = m, n; m ∈ {−17; : : : ; 23}, then |tn − sn |6 82  1   1   1  1 1 41 41 41 |tn −sm |¿ 82 . And so, by applying (20), S˜1 2·41 and T̃ 2·41 are 2 -compatible. However, S˜2 2·41   and T̃ 41 2·141 are not compatible because 2t˜40 is the most similar note to both 2s˜28 and 2s˜40 (see Fig. 5), i.e. max Compat[2t˜j ; 2s˜28 ] = Compat[2t˜40 ; 2s˜28 ] = 0:661499; max Compat[2t˜j ; 2s˜40 ] = Compat[2t˜40 ; 2s˜40 ] = 0:338502:

1 p28 t28 p40 t40 0.5 0.78 0.79 0.8 0.81 0.82 0.83 0.84  1  41  1  and 2s˜28 ; 2s˜40 ∈ S˜2 2·41 . Fig. 5. Membership functions of the notes 2t˜28 ; 2t˜40 ∈ T̃ 41 2·41 5. Su+cient conditions of compatibility With the aim of obtaining su9cient conditions for the compatibility of tuning systems, let us recall some concepts of continued fractions: Denition 9. Given {ai }∞ i=0 , a sequence of natural numbers, where ai = 0; i¿0, we construct [a0 ] = a0 ; [a0 ; a1 ] = a0 + 1 ; a1 [a0 ; a1 ; a2 ] = a0 + 1 ··· a1 + 1=a2 (25) and we denote rn = [a0 ; : : : ; an ] = pn =qn (if a0 = 0, then p0 = 0; q0 = 1). The sequence {rn }∞ n=0 is said to be a continued fraction associated to {ai }∞ , and each rational number r is said to be a n i=0 convergent of the continued fraction. Each real number has a continued fraction {rn }∞ n=0 associated to it, and for a given convergent pn =qn , if a rational number p=q exists, (p; q) = 1, such that | − p=q|¡| − pn =qn |, in [1,9], for instance, it is proved that q ¿ qn : (26) This property allows us to prove the following lemma: Lemma 1. Let p=q be a convergent of the continued fraction of k p = k ; q −q + 1 6 k 6 q − 1: ∈ R+ . Thus, Proof. We consider = p=q (for = p=q the result is obvious). Let us assume that ¡p=q. If k ∈ {−q + 1; : : : ; q − 1} would exist such that [k ]¡[kp=q], we would get a contradiction. Let us

distinguish two cases: (a) If 0¡k6q − 1, there exists m ∈ N such that k ¡m6kp=q, thus ¡m=k6p=q. But p=q is a convergent and k¡q, and so by (26) we obtain a contradiction. (b) If −q+16k¡0, there exists m ∈ N such that k ¿−m ¿kp=q and consequently ¡−m =k6p=q. Taking into account (a) for −k, we obtain a contradiction. And following similar reasoning the case ¿p=q can be proved. q −1 Theorem 1. Let S˜q = {2s˜n }n=0 be a fuzzy tuning system generated by fuzzy temperament (generated by p=q) with q notes. If



p
−
¡ 1 ;
q
2q2 q −1 and T̃p=q = {2t˜n }n=0 the (28) then S˜q is compatible with T̃p=q . Proof. Given k ∈ {0; : : : ; q − 1}, we consider the notes 2s˜k , 2t˜k whose exponents are, respectively, the symmetric triangular fuzzy numbers       1 p p 1 ; t˜k = : (29) s̃k = k − k ; k− k ; 2q q q 2q By Lemma 1 and (28), we obtain











1 p p p p |sk − tk | =

k −  k − k + k

=

k − k

= k

−

¡ : q q q q 2q Applying Proposition 2 and (20), Compat[2t˜k ; 2s˜k ]¿ 12 . On the other hand, given k; k  ∈ {0; : : : ; q −1}, k = k  , let us see that |sk −tk
|¿1=2q. If we suppose that |sk − tk
|¡1=2q we would have |tk − tk
| = |tk − sk + sk − tk
| 6 |tk − sk | + |sk − tk
| ¡ 1 1 1 + = ; 2q 2q q and this is not true because, by construction, for each pair of notes t˜k ; t˜k  ∈ T̃p=q , |tk − tk
|¿1=q holds. Remark 4. The condition | − p=q|¡1=2q2 given in the above theorem holds easily because at least one of every pair of convergents of the continued fraction of veriNes this condition. Moreover, for p; q ∈ N, (p; q) = 1, verifying (28), p=q is a convergent of the continued fraction of (see, for instance [9]). Actually, Theorem 1 provides us with a constructive method for obtaining cyclic -compatible temperaments with a given tuning system for ¿ 34 .

Corollary 1. Let S˜ = {2s˜n }n∈Z be a tuning system generated by the positive irrational number . For a given p=q ∈ Q, such that | − p=q|¡1=2q2 , the systems S˜q = {2s˜n : −q=2 + 16n6q=2 } and T̃p=q = {2s˜n : −q=2 + 16n6q=2 }, where s̃n and t˜n are given in (29), are 34 -compatible. Proof. For a k ∈ {[−q=2] + 1; : : : ; [q=2]}, as | − p=q|¡1=2q2 , by Lemma 1 we obtain


q 1 p

1
= ; |sk − tk | = k
−
¡ 2 q 2 2q 4q and with the same arguments as Theorem 1 we obtain the result. However, these reasonings are not valid when the tuning system is generated by more than one interval. In this case, Nnding a cyclic temperament associated to the tuning system means appropriate divisions of the octave such that all the intervals can be approximated. In [2] a possible solution to this question is proposed: Theorem 2. If 1 ; 2 ; : : : ; k are real numbers, and at least one of them is irrational, then there exist an in3nite number of ways of choosing a denominator q and numerators p1 ; p2 ; : : : ; qk in such a way that the approximations p2 pk p1 ≈ 1; ≈ 2; : : : ; ≈ k; q q q have the property that the errors are all less than 1=q1+1=k . q −1 Therefore, the theoretical issue could be solved by using {2n=q }n=0 . However, the denominator q in Theorem 2 is not obtained by means of any constructive method. Hence, it is necessary to make use of other strategies which usually provide good approximations. Let us see what happens with the Just Intonation. In Section 3.1, we have seen that this tuning system is generated by     3 5 ; 1 = log2 2 = log2 2 4 and also that it can be obtained by modifying some of the terms of the Pythagorean system. Actually, 5 of each 7 terms are obtained by 32 and the other ones with 40 . The idea is to calculate an 27 3 40 intermediate interval in which the quantities 2 and 27 appear in the proportions above mentioned, i.e.      40 2 3 5 7 7 50 : (30) = 27 2 3  In this way a meantone temperament arises, S˜ , with = log2 7 50 , called temperament of 27 of 3 comma [6]. We can apply Corollary 1 for the system S˜ , i.e. we calculate some convergents of the continued fraction of , 0¡ 4 11 69 1 ¡ ¡ ¡ ¡ ··· ¡ 2 7 19 119 ¡ ··· ¡ 443 29 7 3 ¡ ¡ ¡ ¡ 1; 764 50

and the denominator values provide us with some possibilities for the cyclic temperament T̃ that approximates to S˜ , and also analyze the -compatibility between T̃ and the Zarlinean system. 7 The fraction 12 is a convergent of the continued fraction of the interval which generates the Pytagorean, Holderean and Equal Temperated  50  systems. In contrast, for the Zarlinean system, the 7 1 fraction 12 is a convergent of = 7 log2 3 (which is able to approximate to the Zarlinean system). This circumstance allows us to analyze the “goodness” of the su9cient conditions for -compatibility in the case of 12 notes. By using Proposition 3 and Table 3 in appendix, we see that for the Pythagorean, Holderean and Equal Temperated systems, the level of compatibility reaches the value 0.8827. However, the level of compatibility for the Zarlinean system is lower: 0.5699, 0.5733 and 0.6677 for the systems of Pythagoras, H3older and Equal Temperament, respectively. Finally, in Example 3 we analyze the -compatibility of the fuzzy notes in the pentagram in Fig. 1. We assume a tolerance of $ = 25 cents for all of them (it is approximately the double of the Garbuzov zone for the union [8]). Example 3. Let us consider that the notes in Fig. 1 are in the Equal Temperament T̃ 12 (#) with $ 1 # = 1200 = 48 . Table 1 shows the term of each tuning system corresponding to each note (column 2), i.e. di = 2 ∗ C ∗ ci ; i ∈ Z; where C is the frequency of C in each system (column 2), and ci is the ith term of the sequence that generates the tuning system (see expressions (6), (7), (9) and (10)). For each tuning system, the distances between the peaks of the fuzzy notes and the Equal Temperament appear in columns 3, 5 and 7. And, according to (21), we calculate the compatibility between the notes as Compat[f˜1 ; f˜2 ] = max{0; 1 − d(f1 ; f2 )=2$} (columns 4, 6 and 8). Table 1 Distances and compatibilities between the Equal Temperament and the Pythagorean, Zarlinean and Holderean systems for the notes in the pentagram described in Fig. 1 Note E A B A] G] F ]] F] A E] G] A] Term d4 d3 d5 d11 d8 d13 d6 d3 d11 d8 d10 Pythagorean Zarlinean Holderean d(p; t) Compat[p; t] d(z; t) Compat[z; t] d(h; t) Compat[h; t] 2.6251 0 3.9216 13.6868 9.7769 19.5491 5.8646 0 15.6413 9.7769 13.6868 0.8950 1 0.8431 0.4525 0.6089 0.2180 0.7655 1 0.3743 0.6089 0.4525 2.6251 0 3.9216 7.8184 11.7284 44.9689 5.8646 0 27.3714 11.7284 7.8185 0.8950 1 0.8431 0.6873 0.5309 0 0.7654 1 0 0.5309 0.6873 2.5464 0 3.7817 13.2151 9.4411 18.8634 5.6548 0 15.0886 9.4411 13.2151 0.8981 1 0.8487 0.4714 0.6224 0.2455 0.7738 1

By Proposition 3 the compatibility between two notes is an upper bound for their -compatibility, i.e. two notes with frequencies f1 and f2 are -compatible for 6Compat[f˜1 ; f˜2 ] (see (21)). Thus, for instance, the Zarlinean and Equal Tempered F ]] are not -compatible for any  ∈]0; 1]. Actually they do not sound together well. 6. Conclusions Most of the musicians who constitute a classic orchestra must adjust their instrument to obtain a good tuning. For example, wind instruments players modify the air pressure or the Nnger positions to adapt their notes to the ensemble. Because of this, many musicians feel that the mathematical arguments that justify the tuning systems are impractical. With the same arguments employed when a chromatic tuner is used, we make the concept of musical note Cexible. In this framework, fuzzy mathematical rules and practice are the same thing. In fact, the adjustments that the musicians make, constitute a method for increasing the compatibility level among systems. In this way, describing the tuning systems as fuzzy sets permits us to include in a mathematical structure the daily reality of musicians and their theoretical instruction. In my opinion, this constitutes a good model of reality. From the idea of -compatibility, the possibility of substituting a tuning system with another one arises. Therefore, when a tuning system presents many harmonic di9culties such as not allowing certain transpositions, we can use a compatible system to avoid these disadvantages. On the other hand, knowing the compatibility between notes allows musicians to improve their performances by choosing between di>erent tune positions, increasing lip pressures, etc. In fact, our current research is devoted to designing a user-friendly computer program which calculates the compatibility using records as its input. Finally, we would like to remark that our methods to ensure the -compatibility are constructive. Moreover, they allow us to determine an appropriate number of divisions of the octave for every tuning system. Acknowledgements I wish to thank my colleagues and friends JosEe MartEFnez Delicado, Carlos Ivorra and Teresa LeEon for their valuable help. In addition, I would like to thank the referees for their useful comments and suggestions. Appendix In Table 2, we show the frequencies in Hertz for the more usual (crisp) notes in the octave C4 , Nxing A4 = 440 Hz. Notice that in the Zarlinean system the order of sharps and Cats is opposite to that of the other systems. As the four systems have been generated by intervals of Nfths, it is practical to group them into groups of size seven and order them as F − C − G − D − A − E − B. Hence, from c−1 to c5 the natural notes appear, from c6 to c12 the notes with one sharp, from c13 to c19 the two sharp notes,

Table 2 Frequencies in Hertz for the more usual notes in four tuning systems Note Pythagorean system Equal Temperament H3olderean system Note Zarlinean system C D[ C] D E[ D] E F G[ F] G A[ G] A B[ A] B 260.7407 274.6898 278.4375 293.3333 309.0261 313.2422 330 347.6543 366.2531 371.2500 391.1111 412.0347 417.6562 440 463.5391 469.8633 495 261.6256 260.7716 274.7764 278.3936 293.3449 309.0991 313.1681 329.9870 347.7091 366.3830 371.2061 391.1419 412.1484 417.5739 440 463.6304 469.7337 494.9610 C C] D[ D D] E[ E F F] G[ G G] A[ A A] B[ B 264 275 285.1200 297 309.3750 316.8000 330 352 366.6667 380.1600 396 412.5000 422.4000 440 458.3000 475.2000 495 277.1826 293.6648 311.1270 329.6275 349.2282 369.9944 391.9954 415.3047 440 466.1638 493.8833 Table 3 Distances in cents between 12 notes Notes d(P; T) d(Z; T) d(H; T) d(H; P) d(Z; P) d(H; Z) C C] D E[ E F F] G G] A B[ B 5.86 7.82 1.95 11.73 1.95 7.82 5.86 3.91 9.77 0 9.77 3.91 15.64 13.69 1.95 31:28 1.95 13.69 15.64 17.60 11.73 0 33:23 3.91 5.66 7.55 1.89 11.32 1.89 7.55 5.66 3.77 9.43 0 9.44 3.77 0.20 0.27 0.7 0.41 0.07 0.27 0.20 0.14 0.34 0 0.34 0.14 21.51 21.51 0 43:01 0 21.51 21.51 21.51 21.51 0 43:01 0 21.30 21.23 0.07 42:60 0.07 21.23 21.30 21.37 21.16 0 42:67 0.14 Notice that we have underlined the distances greater than 25 cents. etc. If we consider the negative subscripts, from c−8 to c−2 we obtain the notes with one Cat, from c−15 to c−9 the two Cat notes, and so on. In western music it is usual to employ 12 notes, C, C ] , D, E [ , E, F, F ] , G, G ] , A, B[ , and B, which correspond to the terms for n = − 2 to n = 9 of Pythagorean (P), Zarlinean (Z), Holderean (H) and Equal Temperament (T) tuning systems. If in the four systems we Nx A4 = 440 Hz, the distances between the (crisp) notes of these systems, in cents, are in Table 3.

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