Pitch and tuning of the bells
The Federation Bells have a pitch range of four octaves, from D2 (two octaves below middle C) to D6 (two octaves above middle C). This corresponds to a frequency range from 73 Hertz (i.e. 73 sound wave cycles per second) up to 1172 Hertz.
Unlike most Western instruments in use today, the Federation Bells are just tuned. This is a tuning system that follows the harmonic series that exists in nature, and has been in use for 5 000 years. Rather than dividing every octave into 12 equal divisions (as is done with the equal-tempered tuning system we are used to hearing), just tuning intervals are derived from frequency ratios of a fundamental pitch.
In the case of the Federation Bells, the fundamental pitch is a D (or 73 Hertz in the lowest octave of the Bells). A ratio of 2/1 represents one octave above (i.e. D3 or 146 Hertz). Other ratios used in the Federation Bells are 3/2, 4/3, 5/3, 5/4, 7/4, 6/5, 7/5, 8/5, 9/5, 9/8 and 15/8. The absolute frequencies and the “notes” that they correspond to are listed in the information table below.
Note that in some cases the just tuning ratio and the equal temperament tuning is almost identical. For example, the ratio 3/2 (used in Bells 2, 6, 21 and 33) is almost identical to an interval of a fifth above the fundamental (‘A’ in the case of the Federation Bells).
However, for other intervals, the ratio varies significantly from its equal tempered equivalent. For example, the ratio 7/4 (used in Bells 7, 10, 24 and 36) roughly corresponds to a minor/dominant 7th above the fundamental (i.e. ‘C’ for the Federation Bells), but it is 31.2 cents below an equal tempered ‘C’. In other words, it is almost one-third of a semitone flat compared to an equal tempered instrument.
For the uninitiated this all simply means that the Federation Bells have a unique sound and that music created for the Federation Bells must be composed with this tuning system in mind. Many modern-day Western songs or tunes would not sound particularly effective performed on the Federation Bells; however, conversely, the tuning system of the Federation Bells opens up possibilities that are not achievable on other instruments.
Most of the Federation Bells are harmonic bells. Seven bells are polytonal bells, meaning that they are tuned to play more that one pitch with each strike. The table shows that bells 9 to 14 are two-tone bells and bell 8 has three distinct frequencies. These bells give the impression of a musical chord.
The harmonic bells are very pure in the tone compared to other bells. Each bell is tuned to seven harmonic partials, meaning that the fundamental pitch of the bell can be clearly perceived.