THE SCIENCE AND APPLICATIONS OF ACOUSTICS - H. H. Arnold ...

520 18. Music and Musical Instrumentsa musician depends on his or her own sense of subjectivity to obtain the properintensity or range of intensities. The common notations and abbreviations forloudness are as follows: Pianissimo (ppp): softly as possible Pianissimo (pp): very soft Piano (p): soft Mezzo piano (mp): half soft Mezzo forte (mf ): half loud Forte ( f ): loud Fortissimo (ff ): very loud Fortisissimo (fff ): extremely loudLoudness can vary in musical passages. An increase in loudness can be indicatedby the term crescendo or the abbreviation cres or the sign . A crescendoconnotes a gradual increase in the intensity of the music. A decrescendo is adecrease in loudness and is thus the converse of a crescendo. It is denoted by theword decrescendo (also diminuendo) or the abbreviation decresc or the symbol.18.7 Harmony and DiscordConsider two tuning forks being sounded together. Let us keep the pitch of one forkfixed at 261 Hz, while the pitch of the other begins at 262 and is gradually raised.As the pitch raises, beats can be heard, due to the difference in the frequenciesfor a time and then can longer be discerned. The sound of the combined tonesstarts out by sounding pleasant to the ear and then it becomes gradually moreunpleasant. The unpleasantness reaches a maximum at about 23 beats per second,and then begins to abate. This unpleasantness or discord declines only slightly,and the discord remains at a fairly uniform level until the octave-marking value of522 Hz is reached, at which point the unpleasantness disappears.If this experiment is repeated with violin strings, radically different results willbe obtained. The discord does not stay at a uniform level but fluctuates erratically.It almost vanishes at the interval of a major third, and again at the intervals of thefifth and octave. At the precise points at which the minimums of the unpleasantnessoccur, the frequency ratios of the variable to the fixed tone are found to have thevalues: 5/4, 4/3, 3/2, and 2/1.It has been observed that tone tones sound well together when the ratio of theirfrequencies can be expressed in terms of small numbers. The smaller the numbers,the better is the consonance. Table 18.1 lists the intervals in order of increasingdissonance.The further away from small numbers, the more we encroach into the realm ofdiscord. Pythagoras knew this fact more than 2500 years ago when he associatedconsonance with the ratios of small numbers. The premise of the Pythagoreandoctrine “all nature consists of harmony arising out of number,” may be somewhat