Constructing Musical Scales Espen Slettnes

Transcription

1 Berkele Mah Circle Sepember, 08 Inervals Consrucing Musical Scales Espen Slenes Definiion. A pure wave is a sound wave of he form = A sin (f ( + c)), where A is he ampliude and f is he pich of he wave. Definiion. An inerval is a posiive raio r beween wo piches. For example, he inerval from A 3 (= 0 Hz) o E 4 ( 330 Hz) is approximael 3. Definiion 3. An ocave is he inerval. Definiion 4. A perfec fifh is he inerval 3. Figure : A pure sine wave, a pure sine wave plaed wih he ocave above i, and a pure sine wave plaed wih he perfec fifh above i. Figure : A perfec fifh in he -noe scale. Alhough he waveform changes subl over ime, he change is oo suble for mos of us o noice. Exercise. Inuiivel, when does an inerval beween wo piches sound good? Beas Consider he wave sin(f x) + sin ((f + ɛ) x) wih frequencies ɛ f. B he sum-o-produc ideniies his becomes sin (( ) ( f + ) ɛ x cos ɛ x). Since ɛ is small, he period of cos ( ɛ ) is much larger han he period of sin (( ) f + ) ɛ x ; hus our ears hear a noe wih frequenc f + ɛ fading in and ou ever ime cos ( ɛ x) = 0, which occurs ever seconds. ɛ Figure 3: Two close pure ones plaed ogeher. Page of 5

2 Berkele Mah Circle Sepember, 08 Overones Definiion 5. The n h harmonic and he (n ) h overone is he inerval n. When ou hear a noe from a musical insrumen, ou do no hear a pure one. You hear a series of overones. Figure 4 shows how much of each overone is produced in he aack (beginning) of he A 3 0 Hz noe, plaed on various insrumens. Guiar Piano Flue Oboe Clarine Horn Figure 4: Overone series of various insrumens [] When he inerval m wih posiive inegers m, n small is plaed on an insrumen, no onl is he period of he n resuling waveform small, bu some of he harmonics line up. In addiion, when plaing an inerval approximaing bu no exacl equal o m, some harmonics will inerfere wih each oher. n Exercise. The ones D 4 and A 4 oscillae a approximael Hz and 440 Hz, respecivel. Wha frequenc of beas produced b he 3rd harmonic of D 4 and he second harmonic of A 4? Scales An iniial approach Definiion 6. A se S is a F-scale, where S, F are ses of posiive real frequencies ha saisfies he following properies:. F S,. s S, s, s S f f 3. f, f, f 3 S, S, f 3 4. S is no dense a an posiive real number. Exercise 3. Find he smalles {0 Hz}-scale. Exercise 4. Find he smalles {0 Hz, 330 Hz}-scale. Page of 5

3 Berkele Mah Circle Sepember, 08 Take Definiion 7. A se S is a F-scale wih error δ > 0, where S, F are ses of posiive real frequencies ha saisfies he following properies:. F S,. s S, s, s S ( 3. f, f, f 3 S, + δ ff, ( + δ) ff ) S f 3 f 3 4. S is no dense a an posiive real number. Exercise 5. Find a {0 Hz, 330 Hz}-scale wih error 0% b finding a power of 3 a power of. wihin a facor of. from Exercise 6. Repea exercise 5 wih error %. Definiion 8. A perfec major hird is he inerval 5 4. Exercise 7. Complee one ocave of inervals in he incomplee {, 3, 5 4} -scale shown in figure 5, using denominaors 0 for all bu wo noes D E F G A B C D Figure 5: One ocave saring from D, wih some inervals noed Modern Approach: Equal Temperamen Definiion 9. A se S is an equal empered F-scale wih error δ > 0 if. S is of he form k Z/n, where n N and Z/n = { } m/n m Z (S is said o have n noes per ocave and base frequenc k), ( ). f F, f, ( + δ)f S. + δ Exercise 8. Find he infinum (roughl speaking, minimum) error of he equal-empered {, 3 } -scale wih base frequenc and noes per ocave. Exercise 9. Find he infinum error of he equal-empered {, 3, 5 4} -scale wih base frequenc and noes per ocave. Exercise 0. Find he infinum error of he equal-empered F-scale, where F is he se of inervals ou found in Excercise 7 / Figure 5. Page 3 of 5

4 Berkele Mah Circle Sepember, 08 Finding good equal-empered microonal scales for he perfec fifh Definiion 0. Raional approximaion is he process of finding raional numbers ha deviae from an irraional number b a small disance for some meric. Exercise { }. Find an irraional number such ha is raional approximaions will ield good equal-empered, 3 -scales wih base frequenc. () Definiion. The coninued fracion for an irraional number a is he unique fracion of he form a = a 0 + for ineger a 0 and posiive inegers a, a, a 3,.... a + a + a 3 + Exercise. Work ou he firs few erms of he coninued fracion for our answer o exercise. The coninued fracion is ofen used for raional approximaion b sopping he fracion afer a couple of laers. Exercise 3. Use he coninued fracion ou found in exercise o find a leas wo equal-empered scales wih more han noes.... Finding good equal-empered microonal scales for more inervals [] Exercise 4. Think abou wh approximaing he firs few erms of N in an equal-empered scale will give good -sounding inervals (see exercise ). Exercise 5. Find a rigonomeric funcion in erms of n and N ha peaks when N is well-approximaed b a noe in he equal-empered n-noe scale wih base frequenc. () Exercise 6. Wrie ou he firs few erms of N= N a f N(n), where a > and f N (n) is he funcion ou found in exercise 4. Exercise 7. Inerpre wha i means musicall for he sum in exercise 6 o be high. Exercise 8. Wha is he role of he parameer a in exercise 6? Wha do high and low a mean? Exercise 9. Simplif he expression in exercise 6 using he subsiuion cos(x) = R (e ix ). Definiion. For R(s) >, he Riemann zea funcion is ζ(s) = Exercise 0. Wrie he expression from exercise 9 in he form R (ζ(s)) for some s in erms of n and a. Exercise. Find some scales which our approach wih coninued fracions in exercise 3 failed o discover. Hins: N= N s. () () For wha values of x does cos (πx) peak? m/n 3 Page 4 of 5

5 Berkele Mah Circle Sepember, 08 References [] Maria Bell. Fourier Analsis in Music. url: hps : / / www. projecrhea. org / rhea / index. php / Fourier_analsis_in_Music. [] genewardsmih. The Riemann Zea Funcion and Tuning. url: hps://xenharmonic.wikispaces. com/the+riemann+zea+funcion+and+tuning. Page 5 of 5