Modeling and measuring of the oriental musical scale

WARS Aa, A. Rabbaa, A. Modeling and Measuring of the Oriental Musical Scale Abstract: After a brief survey of the western system, we present a study of the oriental musical scale, based on the (diatonic) models built by European mathematicians. The main feature being the presence of the three-quarter-tone interval or medium third, we explain why a modeling by an equaltempered scale of 24 quarter-tones of exactly 50 cents is not suitable. Our model starts with a combination of equal and pythagorean scales, displayed as a series of whole-tone and %-tone intervals. Practical music performance on lute and violin, when measured, show current pitches higher than equal-tempered theory would suggest. Some empirical adjustments of about 15 cents were needed to reach these practical pitches. Checking of some of the most common oriental keyboards proves that their %-tones are not in good accordance with real performance.

H ACU Vol. 92 (2006) 807~811 STICA Modeling and Measuring of the Oriental Musical Scale Ahmed Rabbaa . Professor of Physics at Faculté des Sciences amd Fiir ef Mesic at the Conservatoire, both in Casablanca. rabbaa_a@ yahoo.fr | Summary After brief survey of the western sx of the oriental musical scale, based on the (diatonic) models built by European matiematiiciame. The maim feature being the presence of the three-quarter-tone interval equal -tempered scale of 24 quarter-tones of exactly 50 cents By am ling or medium third, we explain why a mode is not suitable. Our model starts wah a cambian of equal and pythagorean scales, displayed as a series of whole-tone and %-tone itervads È 7xPrsac performance on lute and violin, when measured, show current I saezest. Some empirical adjustments of about 15 cents were Dex pitches higher than© ime of some of the most common oriental keyboards proves that practical gua needed to reach these 1. their 34-tones are not B a = al performance. PACS no. 43.75.Be cali direction several In mm “Oriental” refers to music and ta turks, persians, kurds and other = =|, Middle-East peoples use something similar, butin and Central-Asia. They are based on Me Sesa ental keyboards” refer to keybex Sam ter tone intervals. OsÈ| ra 4 The oriental musical scale, tical or mathematical viewposst& Eetted, so it allowed pitches such as Eb located beeen Eb and E and divided this semitone into two smaller Im roughly equal to one quarter of a tone. This re- E y - más us of the blues tonalities used in American black theorists tried to build some € mms. These notes, foundin an empirical way, do not have did not have the necessary = the 19th and 20th centunes. === theorists and have neglected Ge m = as te me = percisely defined frequencies. ‘We have been inspired by the Western model to build an Al in;ea) scale based on a combination of equal-tempered ' adopted as a standard. Now on temperaments. The resulting minor tones |E well defined, their frequencies are accurately evalare concerned with this problem and the values are compared with some existing carry out deep research to addre in English language deal with Ame they perform some notes not existing in the theo| mal scale. Their lute, unlike the European version, was MR anil = [1, 2], has not received an extensas their scales lack rigor and scale, Kirnberger, Rameau, Werckmeister, Holder =i op physical studies have been carried out that gave 10 different models (Zarlino, mean-tone, equal-temp- = and with practical performance. Ù reference is a manuscript by CBs The purpose of this paper IS; = The Western/European musical scale { ey) ental musical scale, with ¢ parison between the theoretical am so-called Y tone interval (oftes & ern writings), whose frequent oretically determined with enough=| empirically, is slightly different acer styles. diatonic musical scale is a series of 7 notes or pitch ses, from low to high. The name of these 7 notes varies cording to countries, the most used are Do, Re, Mi, Fa, sii The Greeks defined the western musical se ER = Sol, La, Si in latinic, arabic and middle-eastern countries. For the anglo-saxon countries, they are designated as C, Since the Ancient Greeks, an equivalence has been set D.E.F.G, A,B. acousticalimathematical way and Since de cy links the interval batecen two notes and the string Received 15 September 2003, lengths sounding these two notes: if you divide a string in two equal parts you will obtain the octave, in three equal accepted 20 June 2006. parts the (perfect) fifth, in % a tone, etc.... Otherwise, if

Vol. 92 (2006) 2 string lengths /; and /; give 2 sounds of frequencief, s and f>, we get /; /l> = 72/7). Music theorists use the cent. a unit which defines the interval betwe these 2 notes en as: f= 1200 log, PAIS = 1200 log,,(f2/f1)/ logy, 2. So, an equal-tempered half tone has 100 cents, and an octave 1200 cents. To raise the pitch by one cent we must multiply its frequency by 21/1200 = 1.000578. Pythagorean scale 2.1. Starting from any note (C for instance), we can form the ascending fifth with another note whose frequency is 1.5 times higher (G). Another fifth gives us D (1.5-1.5 = 2.25), etc... Continuing, we obtain a set of 7 notes whose frequencies brought inside the span of an octave are shown ; Table I. If we continue this procedure beyond the 7th note by forming another circle of fifths, we will find a new set of. notes higher than the first one by about half a tone. > + notes are expressed by CA, G#, D#, A#, F#, BH, F#. designated as E# and B# are very slightly higher than F 3 amd C whereas they must be their enharmonics. The differs 1200 108>(1.5'2/2”) = 23.5 cents J is about one ninth of a tone and is known as a (Py goa a GE do tia A IS ysE tems. Except E-F and B—C, the interval between 2 cessive notes is 9/8 = 1.125 or 203.9 cents, and this de- i fines the (Pythagorean) “whole-tone”. E-F and BC nil have a ratio of 256/243 = 1.0535 (90.2 cents), ma. slightly smaller than half a tone and is called a "imma of This “anomaly” is the cause and origin of the existence several different tuning systems; the most common is the Equal Temperament used for keyboard and fretted instraments. 2.2. Equal temperament Very suitable for keyboard and fretted instruments, the equal tuning has diatonic and chromatic semi-tones that are all equal. One octave is then composed of 12 equal semitones, whose height is expressed as the twelvet h root of 2. V2 =2'/? x 10594631 or I = 120010g,(2'/!2) = 1200- > = 100 cents. 2.3. Holderean scale (also called Holder-Kaufmann scale [4, 5]) Oriental theorists. Designed in Europe in the 1. The first [8th cénnity."” This ¿qn ¡many POS

Rabbaa: The Oriental musical scale Vol. 92 (2006) ai Table I. Pythagorcan scale and its intervals. le D 1 E | F sisi A | B Cc 9/8=1.125 | 81/64+ 12656 | 4/3 13333 | 3/2=15 | 27/16= 16875 | 243/128 + 1.8984 0 203.9 407.8 | 298 C-D DE | EF 203.9 2039 | 2 me 59 FG | | 2 1109.8 GA A-B B-C 2039 203.9 90.2 1200 9/8= 1.125 | 9/8= 1.125 | 256/24 = 10535 Tape RS | 9/8=1.125 | 9/8= 1.125 | 256/24 = 1.0535 mms si sax Table II. Equal scale and its intervals. C D i F 0 200 400 .».. | co C-D D-E 200 200 i 22/12 ~ 4.1225 | 22/2 = 1.1225 D | Ein = =i C-D 1200 Ga A-B B-C | 200 200 100 G 203.8 A B Cc B= 14999 | 22/5 ~ 1.6873 | 2/5 x= 18981 | 257% G-A | +! - Cal 203.8 on ps E = ne mostly, of empirical charts based on the Gvision of = A-B B-C AZ = 1.1249 | 29% = 1.1249 | 24% = 1.0537 ss wa) 1200 1109.4 905.7 i DE 29/53 = 1.1249 | 29/3 = 11249 203.8 1100 i: 7S 203.8 900 e 912/12 1.8877 | 2/2 = 1.1225 | 2/2 2 1.1225 | 2/2 = 1.0595 20/3 | 298 À 11249 | 0 B | Table II. Holderean scale and its è c A 14983 | 2% = 1.6818 | 21/2 20/12 | 22/12 ~ 1.1225 | 24% = 12599 | DSS -. 203.8 90.6 ta æems ©.2. the need of an accurate method in order to de- LT zu mstrements and computer cards and the convenience 2. In the 19th century the Lebanese Michel Mesh af equal-tempered Ge scale for transposition and modulatried to build (1820) some new model. but he rather practician, and theory was beyond Bes scope the [9, 10, 11]. The turc Raouf Yekta Bay explomed Dom. led us, initially, to experiment with such a scale. Baz. unlike blues tonality, an underlying equal temperstring length. a Holderean scale with its 53 commas or turksents 10 evaluate some non-equal intervals amet cannot work; oriental musicians always refer to them: favourite (unfretted string instrument) lute and prefer chromatic semi-tones higher than diatonic ones. However, it may be used as a basis which needs further little adjust- C—D— Eb— Feine 9 9 9 8 5 but as we will see, dividing the 3/2 tone D-F and A-C in 8 and 5 does not comply with practice. ments, and for the same reasons as in European music, we may not ignore the existence of the Pythagorean scale. We have finally chosen a scale that combines features of both equal and Pythagorean ones by merely taking the average of both corresponding frequencies of Table I and Table II. The gap between this new tuning and each of the pre- 4. vious ones is <0.17% (except for E and B), avoiding any Modeling of an oriental scale further debate about the choice among the two main scales. The %-tone is rather scarce; what really exists is the %- This debate will not affect our work because the main contone (minor tone). We can find the most common %-tone cern in oriental scale is to determine the length of the minor tone interval. So, because of the variation in practice, intervals in the widely used rast tuning [1, 3, 6] (something intermediate between minor and major modes, very similar to the blues tonality [12]). With T denoting a wholewe do not need to achieve such an accuracy. Considering E semi-flat and B semi-flat, we proceed by taking the geometric mean of the frequency ratio of (D-F) tone, rast tuning can be represented as follows: and (A-C). C— D— Eb— F—G—A—Bb— C. T Y % «T ‘i % % The idea of dividing an octave into equal parts (24 quarters of tone) may seem adequate and has been exploited in designing low price musical keyboards. Several au We then obtain the following intervals which form the rast tuning: | tone: 1.1237 (201.9 cents), C-D, F-G, G-A, % tone: 1.0896 (148.6 cents), D-Eb, Eb-F A-Bé, Bb-C,

Vol. 92 (2006) Table IV. Theoretical Rast tuning. Cc D Eb 1 0 1.1237 201.9 1.2244 350.5 GeADmSAe, Table V. Roland MIDI sequencer Arabian temperament. E D Eb 1 0 1.1251 204 1.2276 355 — Table VI. Practical/experimental Rast tuning. c D Eb 1 1.124 1.234 0 202 364 5:0 Y 14 tone: 1.0600 (101.0 cents), 4 tone: 1.0296 (50.5 cents). C-E® (often quoted by theors thirdum Our medi of 1.224 or 350.5 cents (1 26 for the maga 1 a value for the minor one). 5. Oriental keyboards [1] Almost all keyboard makers have their “oriental” models intended for Middle-East and North Africa markets. Their and main features is that they provide % tone intervals and ts instrumen typical some also but tonalities, related oriental (equal) pre-set a have models e Low-pric rhythms. tuning, i.e. with a quarter tone of exactly 50 cents, whichis not suitable for performers. Some advanced models allow the musician to set the tuning himself and find the % tone that suits his music, and dare not give a fixed value that, maybe, not everybody agrees with [1]. Very few attempts have been made to offer non equal pre-set tuning to account for real pitches. Let us see the example of Roland MIDI sequencer. Its user’s manual [13] gives a so-called Arabian temperament compared to an equal tuning. We have listed this scale, expressed in terms of (relative) cents and frequency ratios in Table V: I have chosen this example because it raises the real problem [1]. The medium thirds (C-Eb and G-Bb) of 355 cents are higher than the equal ones of 350 cents, but not enough. The % tone intervals (D-F and A-C) are composed of non equal 3% tones of 151 and 143 cents but their size must be rectified. 6. Experiment Although the interval of %-tone varies very slightly acsitionof our o matedesoma Lg {ead a enge EEE de especia 49 Gem a and 32.4 cm . So, applying the relation f/f, = Iı/I.the frequency ratio of the two medium thirds are 60/48.6 and 40/32.4 and are both roughly equal to 1.234: fof fir = fl = 1.234. After empirical adjustment, our work led us to divide each minor diatonic third D-F and A-C into 2 non equal % tone intervals (Table VI). The first ones (D-Eb and ABb) have a size of 163 cents, and are higher than the second ones whose size is 134 cents. The rest of the notes, i.e. the diatonic scale, are quite in accordance with Table IV. The value of the medium thirds (C-Eb or G-Bb) is 365 cents implying a gap of about +15 cents, compared with equal temperament, which can be easily detected by beginner musicians. As for the accuracy, the measured values have 3 significant figures: 486mm (Eb) and 324 mm (Bb). Both of them are means of several other values, which vary very cording to regions and styles, we can consider that a stanslightly according to players and melody styles, and are dard does exist: it was imposed by the strongly invading already rounded to the nearest whole millimeter. The ap- Egyptian style which has been adopted by many other parent accuracy shown in Table VI results only from arithcountries. metic operations.

Vol. 92 (2006) 7. Conclusion and tuning. Michigan State 18 Da51 Capo Press, . New-York, 1972. i This work presents some theoretical and Y pracucal desi a search on oriental musical scale. We first quasi-standard diatonic scale from a combination tempered and Pythagor ones. We ean then = _- Le E be t 1 pri i ofequal dem vals from this theoretical standard scale pS al sea ma | cents) plus two larger intervals of 3/2 tome (297 1977 Fast n - The theory based on the 53 N ag Etude à 3 ia) iti Arabe. : uchet/Chastel, Paris, (originally written in German). M. Mushagah: Lettre sur l'art musical. Early 19th century. > 2 medium thirds of 385 cents. (translated in English by R.E. Smith in 1949, and in French L.S.J.hes Ronzevalle: “Una traité de musique musi Ares arabe modcurrent practice and, in addition. 0 } 3. €. Chabrier: Un mouvement de réhabilitation longer convenient for any te scale tice very well, and is better a | 4 223 pages (in French). 794 2 i a lit Àge | scale (section 3 and [4]) Donval is a pscudonym of A Musique sme naaedi ootde 198 e DR d’ Erlanger: La Musique Arabe. Librairie Orienta a apre ae: liste (134 = about 365 cents (Table VI) are 15 € [1] S. Donval: History of of middle-eastern music. 5d 1992. nei J ee) > =~LÉ Also, the measured medium thirds (E ce References ri pessistion D-F and A-C are both composed of a lar Bel: Iderean ee rere bandsinan.co.uk/downlosd/castern pdf 1 D-F and AC. The conclusion is that our re AA krgboncl, pp ES and E17. N Home: scale (Table VI) contains three intervals of 1 om : medium third of 350 cents. 11:57. = raSth Table comparison with measured values given by cents or 55%) and a small %-tone andatıo n.org/musical_mathematic | s gene ues of the whole-tone, half-tone and « formance we came to the conclusion that the u il de la Musique Arabe et du Luth Oriental. Doctoral thesis. Paris. | Sorbonne, 1976. M2) E. Kriss: Begi piano ning blues, p.8. - Acom Music Press 1977, Amsco Publications 1984, New-Y ork. 3! Users Manual of ROLAND Personal Music _ PMA-5, p.135. TJ. Mathiesen: An annotated translation Assistant of Euclid’s divi- 4 sion of a monochord. Journal of Music Theory 19 (1975