# Harmonograph

53 52 The ratios and intervals in this book concern frequencies, normally expressed as cycles per second, or Hertz. Classical tuning sets C at 256 Hz. Modern tuning is higher, fixing A at 440 Hz. The period T of a wave is the reciprocal of its frequency f T 1f. The speed of sound in dry air is roughly 331.4 0.6Tc ms, where Tc is the temperature in degrees celsius. Its value at room temperature, 20oC, is 343.4 ms. Gravitational acceleration on earth, g, is 9.807 ms2. 1 gravitational acceleration 2 pendulum length Frequency of a Pendulum. consTanTs anD equaTions The greeK moDes The white notes on a piano give the seven notes of the seven modes of ancient Greece. Medieval transcription errors have left us with modern names which dont fit the ancient ones. Each mode, or scale, has its own pattern of wholetones and halftones, only two surviving as our major and natural minor scales. Other scales include modal pentatonics which forbid semitones, the harmonic minor with its minor 3rd and 6th, 1 2 37 4 5 67 7 8, and many others. speed of sound area of opening 2 volume of cavity x length of Resonant frequency of a cavity with an opening. 1 string tension 2 x string length string mass string length Fundamental frequency of a tensioned string. speed of sound 2 x length of cylinder Fundamental frequency of an open pipe or cylinder. The beat frequency between f1 and f2 is the difference between them, fb f2 f1. The ratio ab converts to cents where a b logalogb x 1200 log2. To covert cents into degrees multiply by 0.3. Clapping in front of a rise of steps produces a series of echoes with a perceived frequency equal to v2d, where v is the speed of sound, and d is the depth of each step. Clapping in a small corridor width w produces a frequency vw. The arithmetic and harmonic means are central to Pythagorean number theory. The arithmetic mean of two frequencies separated by an octave produces the fifth between them 32, the harmonic mean producing the fourth 43.