As mentioned in the previous section, frequency is defined as the number of cycles or periods per second. Frequency is measured in Hertz. If a tone has a frequency of 440Hz it completes 440 cycles every second. Given a tone's frequency, one can easily calculate the period of any sound. Mathematically, the period is the reciprocal of the frequency and vice versa. In equation form, this is expressed as follows.

Frequency = 1/Period Period = 1/Frequency

Therefore the frequency is the inverse of the period, so a wave of 100Hz frequency has a period of 1/100 or 0.01 seconds, likewise a frequency of 256Hz has a period of 1/256, or 0.004 seconds. To calculate the wavelength of a sound in any given medium we can use the following equation:

λ = Velocity/Frequency

For instance, a wave of 1000 Hz in air (velocity of diffusion about 340 m/s) has a length of approximately 340/1000 m = 34 cm.

It is generally stated that the human ear can hear sounds in the range 20Hz to 20,000Hz (20kHz). This upper limit tends to decrease with age due to a condition known as presbyacusis, or age related hearing loss. Most adults can hear to about 16 kHz while most children can hear beyond this. At the lower end of the spectrum the human ear does not respond to frequencies below 20 Hz, with 40 of 50 Hz being the lowest most people can perceive.

So, in the following example, you will not hear the first (10 Hz) tone, and probably not the last (20 kHz) one, but hopefully the other ones (100 Hz, 1000 Hz, 10000 Hz):

**EXAMPLE 01B01_LimitsOfHearing.csd**

<CsoundSynthesizer> <CsOptions> -odac -m0 </CsOptions> <CsInstruments> ;example by joachim heintz sr = 44100 ksmps = 32 nchnls = 2 0dbfs = 1 instr 1 prints "Playing %d Hertz!\n", p4 asig oscils .2, p4, 0 outs asig, asig endin </CsInstruments> <CsScore> i 1 0 2 10 i . + . 100 i . + . 1000 i . + . 10000 i . + . 20000 </CsScore> </CsoundSynthesizer>

A lot of basic maths is about simplification of complex equations. Shortcuts are taken all the time to make things easier to read and equate. Multiplication can be seen as a shorthand for repeated additions, for example, 5x10 = 5+5+5+5+5+5+5+5+5+5. Exponents are shorthand for repeated multiplications, 3^{5} = 3x3x3x3x3. Logarithms are shorthand for exponents and are used in many areas of science and engineering in which quantities vary over a large range. Examples of logarithmic scales include the decibel scale, the Richter scale for measuring earthquake magnitudes and the astronomical scale of stellar brightnesses. Musical frequencies also work on a logarithmic scale; more on this later.

Intervals in music describe the distance between two notes. When dealing with standard musical notation it is easy to determine an interval between two adjacent notes. For example a perfect 5th is always made up of 7 semitones. When dealing with Hz values things are different. A difference of say 100Hz does not always equate to the same musical interval. This is because musical intervals as we hear them are represented in Hz as frequency ratios. An octave for example is always 2:1. That is to say every time you double a Hz value you will jump up by a musical interval of an octave.

Consider the following. A flute can play the note A at 440Hz. If the player plays another A an octave above it at 880 Hz the difference in Hz is 440. Now consider the piccolo, the highest pitched instrument of the orchestra. It can play a frequency of 2000Hz but it can also play an octave above this at 4000Hz (2 x 2000Hz). While the difference in Hertz between the two notes on the flute is only 440Hz, the difference between the two high pitched notes on a piccolo is 1000Hz yet they are both only playing notes one octave apart.

What all this demonstrates is that the higher two pitches become, the greater the difference in Hertz required for us to recognize the spacing as the same musical interval. We can use simple ratios to represent a number of familiar intervals; for example the unison: (1:1), the octave: (2:1), the perfect fifth (3:2), the perfect fourth (4:3), the major third (5:4) and the minor third (6:5); but it should be noted that most of these intervals are only represented with absolute precision when using just intonation. In equal temperament, the dominant method used in the tuning of many instruments, only unison and the octave are represented with these precise ratios.

The following example shows the difference between adding a certain frequency and applying a ratio. First, the frequencies of 100, 400 and 800 Hz all get an addition of 100 Hz. This sounds very different, though the added frequency is the same. Second, the ratio 3/2 (perfect fifth) is applied to the same frequencies. This spacing sounds constant, although the frequency displacement is different each time.

**EXAMPLE 01B02_Adding_vs_ratio.csd**

<CsoundSynthesizer>

<CsOptions>

--env:SSDIR+=../SourceMaterials -odac -m0

</CsOptions>

<CsInstruments>

;example by joachim heintz

sr = 44100

ksmps = 32

nchnls = 2

0dbfs = 1

instr 1

prints "Playing %d Hertz!\n", p4

asig oscils .2, p4, 0

outs asig, asig

endin

instr 2

prints "Adding %d Hertz to %d Hertz!\n", p5, p4

asig oscils .2, p4+p5, 0

outs asig, asig

endin

instr 3

prints "Applying the ratio of %f (adding %d Hertz) to %d Hertz!\n", p5, p4*p5, p4

asig oscils .2, p4*p5, 0

outs asig, asig

endin

</CsInstruments>

<CsScore>

;adding a certain frequency (instr 2)

i 1 0 1 100

i 2 1 1 100 100

i 1 3 1 400

i 2 4 1 400 100

i 1 6 1 800

i 2 7 1 800 100

;applying a certain ratio (instr 3)

i 1 10 1 100

i 3 11 1 100 [3/2]

i 1 13 1 400

i 3 14 1 400 [3/2]

i 1 16 1 800

i 3 17 1 800 [3/2]

</CsScore>

</CsoundSynthesizer>

So what of the algorithms mentioned above. As some readers will know the current preferred method of tuning western instruments is based on equal temperament. Essentially this means that all octaves are split into 12 equal intervals. Therefore a semitone has a ratio of 2^{(1/12)}, which is approximately 1.059463.

So what about the reference to logarithms in the heading above? As stated previously, logarithms are shorthand for exponents. 2^{(1/12)}= 1.059463 can also be written as log2(1.059463)= 1/12. Therefore musical frequency works on a logarithmic scale.

Csound can easily deal with MIDI notes and comes with functions that will convert MIDI notes to Hertz values and back again. In MIDI speak A440 is equal to A4 and is MIDI note 69. You can think of A4 as being the fourth A from the lowest A we can hear; well, almost hear.

*Caution: like many 'standards' there is occasional disagreement about the mapping between frequency and octave number. You may occasionally encounter A440 being described as A3.*

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