Tuning by Ratios

• Chord Progression Example
• Traditionally Notated Score
• Csound Orchestra
• Csound Score

Chord Progression Example

The following example is a familiar chord progession, namely, I-vi-ii-V-I. To obtain the frequencies in Hertz for each note, simply multiply the fraction beneath the note by 55, which is an octave equivalent of A 440. Both the traditionally notated and Csound scores are shown below.

Notice the low B natural in the bass on measure two, beat one, is tuned as 20/9. In the same measure but in the treble and on beat three, there is another B natural which is tuned as 9/1, which is not an octave equivalent of 20/9. Such redefined notes are known as "mutable tones".

It may seem unnatural to have octave equivalents of two B naturals in adjacent chords, however, it is absolutely necessary to have two different B's in this particular passage when the following conditions are assumed:

1. The note A natural is a common tone between the first two triads (A major and F# minor).
2. The note F# is a common tone between the second two triads (F# minor and B minor).
3. The E natural in the fourth triad (E major triad) is related to tonic A as an octave equivalent of 3:1.
4. The relative frequencies for a major triad are always octave equivalents of 1, 3 and 5.
5. The relative frequencies for a minor chord are always octave equivalents of 3, 5 and 15.

Both the traditionally notated and Csound scores are shown below.

Traditionally Notated Score

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example1.MP3 (283K)

Csound Orchestra

sr=44100
kr=441
ksmps=100
instr 1
a1 pluck 9000, p5/p6*55, p5/p6*55, 0, 1, 0, 0
; display a1,.005
out a1
endin

instr 2
a1 pluck 9000, p5/p6*55, p5/p6*55, 0, 1, 0, 0
; display a1,.005
out a1
endin

instr 3
a1 pluck 9000, p5/p6*55, p5/p6*55, 0, 1, 0, 0
; display a1,.005
out a1
endin

instr 4
a1 pluck 9000, p5/p6*55, p5/p6*55, 0, 1, 0, 0
; display a1,.005
out a1
endin

Csound Score

f11 0 1024 10 1
f51 0 513 5 .00195 1024 1 ; exponential increase over 1024 points
t 0 40

i1 0 2 0 10 1
i1 2 2 0 8 1
i1 4 2 0 32 3
i1 6 2 0 9 1
i1 8 4 0 10 1

i2 0 2 0 6 1
i2 2 2 0 20 3
i2 4 2 0 20 3
i2 6 2 0 15 2
i2 8 4 0 8 1

i3 0 2 0 6 1
i3 2 2 0 5 1
i3 4 2 0 20 3
i3 6 2 0 6 1
i3 8 4 0 6 1

i4 0 2 0 2 1
i4 2 2 0 10 3
i4 4 2 0 20 9
i4 6 2 0 3 1
i4 8 4 0 2 1