Sub-Harmonic Beat Calculator
by Graham Oxley
Contents
The sub-harmonic beat calculator is a program designed to find an appropriate tempo for a piece of music according to the key it is played in, or alternatively, to find an appropriate key for a given tempo. The relationship between key and tempo can be subtle, and can be responsible for the difference between an excellent performance and an okay one.
With the exception of pure sine waves, all repetetive sounds have harmonics whose frequencies are integer multiples of the repetition frequency. Thus a simple 4:4 kick drum at 120 bpm (2Hz) has harmonics in the audio range, particularly noticable in the bass. At the simplest level, you could add some other percussive sounds to the kick drum, say some semiquaver hihats (8Hz), or demisemiquavers (16Hz).
These frequencies, whilst they are still below the threshold of audible perception, can be relevant. You can hear on some house/techno tracks where at the end of a section the snare drum gets faster each bar until just before the break you can hear the snare as a bass tone.
Even though most times the percussion will not be playing so fast that you can hear it as a bass tone, nevertheless the harmonics of the rythmic parts do exist in the same frequency range as the bass part, which generally sets the key of the piece.
The bottom note of a bass guitar is approximately 41.203 Hz. You only have to go down another octave (20.6Hz) and you are in the realm of tempo (20.6Hz is demisemiquavers at 154.5 bpm).
This relationship between key and tempo is not always this direct. A more obvious example is the beat frequency between fifths. Consider a piece of music that starts with an opening A chord, followed by the drummer beating time on the hi-hats. Because of the difference in frequency between a 'perfect' fifth and an equally tempered fifth, there will be a beat frequency formed in the chord. So, at what tempo should the drummer come in? The guitarists A note will have a frequency of 110Hz, and the E note in the chord (that's E below middle C) a frequency of 164.814 Hz. The third harmonic of the A note (330Hz) will clash with the second harmonic of the E note (329.63Hz) and thus there will be a perceptable beat of 0.372Hz, about once every 3 seconds, or one bar's worth at 89.28 bpm.
The effect is more noticeable if distortion is applied to the guitar sound (try it). It is also used by piano tuners to get the tuning right. If a piano were tuned so that there were no beat between fifths, by the time the tuner got back to the note s/he started with, s/he would be a semitone out.
Things are not always this straightforward. Take for example the guitar chord mentioned above. In addition to the A's and E's in the chord, let us assume it is A major, so there will be an additional C# to consider. There will be a beat between the C# and the 5th harmonic of A. There will be the third harmonic of C# (that's close to a G# but not quite) beating with the 5th harmonic of E (that's also close to G# but not quite).
The richness of sound of many instruments is related to
presence of harmonics and their inter-relationships. The purpose of bpm-calc is to
calculate the tempo of the main beat frequencies. With this information,
you can understand why perahps a particular piece always seems to drag,
or why the key-change for the middle eight sounds so wonderful.
When you start the program, you are presented with the calculator window. The current version can only calculate the BPM from keys, not the other way round. (i.e., it cannot tell you what key a succession of double-hemi-demi-semi-quavers at a particular tempo will sound like.
Choose the calculation mode from the group on the left, and click on the desired key. The tempo will be displayed on the right automatically. (the calculate BPM button is unnecessary).
That's more or less it.
When fundamental mode is selected, the frequency of the key is divided by two until it is at foot tappable speed, when it is multiplied by 60 to give beats per minute.
Fifth mode: The frequency of the third harmonic is calculated, and also the frequency of a perfect fifth above the key. The difference between these frequencies is used.
Major Third: The difference betweeh the frequencies of the fifth harmonic and a major third above the key.
Minor Seventh: The difference between the frequencies of the seventh harmonic and a minor seventh above the key.
The dialog should be made to work both ways, i.e. Calculate the appropriate key for a tempo.
Since the range of the visible spectrum is approximately an octave (i.e. The edge of UV is twice the frequency of the top if infra red), then perhaps the appropriate colour for a key should be displayed!
Other beats should be made available, to allow for complex chords.
The direct beat frequency should be displayed, not just the foot tappable adjusted bpm.
The frequency edit control should accept input, or at least have a increment/decrement button.
Perhaps also a reverb-calc, so bands can know what to expect when they go into a venue and the back wall (opposite the stage) is x metres from the stage. What is the tempo of the slap-back?
This help file could be converted to HTML so it can be used as help.
A=440Hz, -> 103.125 bpm
The frequency ratio between
a semitone is 1.0594630943593
1.4983070768767
Fifth ratio: 1.4983070768767 (2 to the power of 7/12)
E=659.255Hz -> 154.513 bpm or 77.256 bpm