practice

  1. According to Galileo …
    Especially harsh is the dissonance between notes whose frequencies are incommensurable; such a case occurs when one has two strings in unison and sounds one of them open, together with a part of the other which bears the same ratio as the side of a square bears to the diagonal ….
    Identify this interval on the equal tempered scale.
  2. Determine the beat frequency between C4 and G4 (a perfect fifth) when played on an equal tempered scale where A4 = 440 Hz.
  3. Write something different.
  4. Write something completely different.

numerical

  1. The often quoted ideal range of human hearing is 20 Hz to 20 kHz. How many octaves is this?
  2. Determine the frequencies of the 13 labeled notes on the fragment of a keyboard shown below for an equal tempered scaled based on A = 435 Hz (International Pitch of 1891).
                    C♯
     
         		   E♭
     
         		     F♯
     
         		   G♯
     
         		   B♭
     
         		                    
            C
     
         		 D
     
         		 E
     
         		 F
     
         		 G
     
         		 A
    435
    Hz    		 B
     
         		 C
     
         		        
  3. Read the following passage on pigeons (Columba livia) …

    Pigeons have extraordinarily sensitive vision and hearing, scientists say, They can see beyond the spectrum of light visible to humans, into the ultraviolet range. They can hear in extremely low frequencies far beneath what humans can detect, down to something like 13 octaves below middle C on the piano. That ability is probably critical to a bird that is meant to migrate long distances and is at the mercy of weather patterns. (Source: Angier, Natalie. New York's Tough Pigeons Fight Predators for Survival. New York Times. 8 July 1991.)

    According to the information this reporter found, how low can a pigeon hear?

  4. The lowest note in the universe has been playing for the last 2.5 billion years. The instrument is a black hole 250 million light years away from earth in the center of a galaxy called Perseus A. We think of space as being empty and it almost is, but in actuality space is filled everywhere with a gas or plasma that ranges in density from thin (one atom per cubic centimeter in the space between the planets) to very, very, very, very, very, very thin (one atom per cubic meter in the space between galaxies). The black hole at the center of Perseus A is consuming matter and belching out energy in a manner that has imprinted a pattern on the intergalactic medium that looks just like a sound wave as the images below show.

    Perseus Cluster
    Source: NASA/CXC/IoA/Fabian [magnify]

    In the words of the original discoverers.

    If the ripples are pressure (sound) waves moving at constant speed (about 1170 km s−1 for a temperature of 5 keV) then their separation (wavelength) of about 11 kpc means a period of about 9.6 × 106 yr. (Source: Fabian, A.C., et al. A deep Chandra observation of the Perseus cluster: shocks and ripples. Monthly Notices of the Royal Astronomical Society. Vol. 344 (2003): L43 arXiv:astro-ph/0306036v2.)

    Apply a bit of physics to these numbers and you will find a bit of a problem. The most reliable measurement has got to be the wavelength. This is a measurement determined by the geometry of the image. Everything else has to be calculated or inferred.

    1. Use the wavelength and period to calculate the wave speed. How does your answer compare to the reported wave speed?
    2. Use the wavelength and wave speed to determine the period. How does your answer compare to the reported period?
    3. What's going on with these numbers? (I don't know the answer to this question, by the way.)

    With such an incredibly long period, this sound wave (assuming it is a sound wave) has to be "the lowest note in the universe". Somewhere in the report of this discovery to the mass media, that's how it was phrased. Since I live in New York City, let's assume that the New York Times was the first to report this to the general public.

    Astronomers say they have heard the sound of a black hole singing. And what it is singing, and perhaps has been singing for more than two billion years, they say, is B flat — a B flat 57 octaves lower than middle C. (Overbye, Dennis. Music of the Heavens Turns Out to Sound a Lot Like a B Flat. New York Times. 16 September 2003: page unknown.)

    Assigning particular frequencies to notes is not that easy to do. Music isn't about frequencies, it's really about intervals. Notes are defined in terms of the ratios they make with some standard frequency, not by what frequency they have. You change that standard frequency and you change the note. Does Perseus A emit a note equivalent to an extremely low B♭ or is it closer to an extremely low A or B? Multiply the frequency of Perseus A by 57 octaves (257) and see where it lands on each of the following scales. What "note" is the black hole at the center of this galaxy emitting if we use …

    1. the American Standards Association (ASA) of 1936
    2. the International Pitch value of 1891
    3. Hermann von Helmholtz's "Scientific" or "Philosopher's" scale
     
    Apply Three Musical Scales to This Event
    standard year A4 B♭4 B4
    d. American Standards Association 1936 440.00 Hz 466.16 Hz 493.88 Hz
    e. International Pitch 1891 435.00 Hz 460.87 Hz 488.27 Hz
    f. Helmholtz's Scale   426.67 Hz somewhere
    in between    		 480.00 Hz
     

statistical

  1. Complete the following table for the twelve intervals of the just intonation scale …
    interval consonance
    name ratio per cent rank
    unison 1:1 100% 1
    minor second 16:15  
    major second 9:8 11%  
    minor third 6:5  
    major third 5:4 20%  
    perfect fourth 4:3 25% 4
    tritone 7:5  
    perfect fifth 3:2 33% 3
    minor sixth 8:5  
    major sixth 5:3 20%  
    minor seventh 16:9  
    major seventh 15:8 7%  
    octave 2:1 50% 2