For input in days (for example to model orbital periods of planets, etc.), use the following:
This page is an exposition of the technique of aural modelling of periodic objects based on the aesthetic relatedness of octaves.
Another way to arbitrarily demonstrate the aesthetic relatedness of octaves is to note the example of western classical music notation: pitches are assigned classes, for example F# (F sharp): if you look at the piano keyboard, F sharp is the left-most of the groups of three black keys. Every left-most black key from a group of three is called F sharp. When you play them, they sound aesthetically related, and this is because they are related to each other in the ratio of the octave. Taking any F sharp on the piano keyboard and then moving to the next F sharp to the right, the frequency ratio will be 1:2 and if two people were to sing those two notes they could be said to be singing in "octave unison".
The human ear can interpret a wide range of periodic objects as sound, from roughly 20hz (or safely 50hz) up to somewhere in the region of 15000hz (or much higher in people with very good hearing - the higher end of the range decreases typically with age).
Frequencies of sound are measured in hertz. This simply means cycles per second. The pitch modern orchestras tune to (the 'A natural' delivered by the oboe or the piano) is a period object which cycles 440 times in one second, 440hz. This means that each iteration of this wave is 1/440 seconds long.
The 'A natural' above that 'A natural' will always be at the ratio 1:2 so is bound to be 880hz. (Please note that whilst the frequency doubles, this means logically that the period of one iteration of the wave halves, since the frequency is the reciprocal of the period in seconds, ie. the number of times it 'fits' into one second.)
Logically the series of octaves can be extended indefinitely to 440 x 2 x 2 x 2 x 2.... or to 440 / 2 / 2 / 2 / 2... and will still, from a theoretical viewpoint, retain its aesthetic relatedness to 440hz.
One person may be unable to hear the frequency 20000hz and another may be able to hear it. One would therefore report aesthetic relatedness to 10000hz and the other would report silence. nb. The perception of aesthetic relatedness is prone to serious error in very high and very low frequencies - a better example would be someone with severely impaired hearing being unable to perceive 1000hz.
The set of doubles and halves, ie. the set of things in the iterated ratio 1:2, is infinite. It follows then that any periodic object has an intrinsic aesthetic relatedness to a pitch-class within the audible spectrum. Another way to put it is that all periodic objects belong to a set of doubles/halves containing around eight members which fall within the human audible spectrum. The human ear can distinguish frequencies of at most around 1hz in general so the set of sets of doubles/halves will range from the set containing 50hz through to the set containing 99hz, such that the set containing 100hz will be the same set containing 50hz due to their 1:2 (2:1) ratio.
If we were to model our solar system visually we would choose a scale which would allow us to see all of the objects we needed to see (for example the planets) within the bounds of our diagram (for example an A4 sheet of paper or a 1024x768 pixel array on a monitor screen).
With visual depictions, the scale factor can be arbitrarily assigned and the relations between the objects will retain their correct meaning.
With aural depictions, the scale factor must be a power of 2. This can be demonstrated by imagining an aural model of the 'A natural' to which orchestras tune. Obviously this is already an audible pitch, but imagining a listener with severely impaired hearing could only hear frequencies up to 200hz, we would have to 'scale' the pitch. In order for the person to recognise it as an 'A natural' we would have to double or halve its frequency. Any other arbitrary scale factor would destroy its aesthetic relatedness to 'A natural'. In principle this need not be a problem in some cases: a melody can be transposed so that a note which was an 'A natural' becomes a 'G natural' and the melody will retain much of its character. In a strict sense this is limited by the range of audible frequencies, such that a transposition of this kind might render the lowest or highest notes of the melody literally silent. An added complication is that people with 'absolute pitch' may also perceive the melody as different. The purest scale factors will always be powers of 2 to preserve the strict aesthetic relatedness of the starting and finishing pitches.
In its simplest form then, the act of modelling periodic objects is simply to divide (or multiply) them by 2 until such a time as they fall into the audible spectrum of the intended audience.
Taking the solar system as an example, each planet orbits the sun in a repeating period. The Earth for example orbits every 365.25 days.
To determine which set of doubles/halves contains the Earth's orbit, and which audible frequencies are members of that set and are therefore aesthetically isomorphic to the Earth's orbit, we can apply either a formula or a procedure.
Generally speaking the audible spectrum is around 20hz to 15000hz (depending on individual hearing) and this, in seconds, is 0.05 down to 0.000066666666667.
So to summarise the procedure:
Take the orbital period in days and convert to seconds.
Divide this number by two until such a time as it falls between 0.05 and 000067 seconds.
Convert seconds to hertz by dividing one by your number, eg. 1/0.05 = 20hz.
For input in seconds (for example to model the period of a pendulum), use the following:
For input in days (for example to model orbital periods of planets, etc.), use the following:
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