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fifth and fourth equal 702 and 498 cents (as against 700 and 500 in the tempered scale). The 9:8 tone by which the fifth exceeds the fourth is (702 498 =) 204. It is easy to work out that when two such tones are marked off within a fourth, the residue is an interval of 90 cents, a little less than half a tone. |
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Intervals are a matter of relative pitch. The absolute pitch of a piece of ancient music cannot be established exactly, and anyway it may have differed in different performances. But we can make a good guess at it by comparing the compass of the widest-ranging vocal compositions with the natural range of the male voice. This is explained more fully in Chapter 9. Many pieces lie in the tenor range, and in transcriptions of them it is most convenient to use the treble (G) clef with an '8' under the clef-sign to show that it is transposed an octave lower:  = middle C. Transcription exactly at the best-guess pitch would sometimes involve key-signatures with several sharps or flats. In the interests of easier reading I have avoided this by shifting to a nearby pitch (never more than a tone away from the best-guess pitch) so that no more than one sharp or flat is required, adding the annotation 'original a tone (semitone) lower (higher)'.16 Similarly, in discussing abstract scales or the scales of particular compositions I treat them so far as possible as being in the 'natural' key, that is, the one in which their notes appear as white notes. |
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Notes of a specific octave are designated by italic letters and differentiated according to the following convention: |
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C to B is the octave below the one below middle C.
c to b is the octave below middle C.
c' to b' is the octave from middle C upwards.
c'' to b" is the next above that. |
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Besides the standard inflection-signs,  sharp,  flat, and  natural,17 it will sometimes be necessary to employ symbols to specify that a note is raised or lowered by something less than a semitone. For this I use the self-explanatory signs  and  . Where the interval is to be defined more precisely, a cent number is appended to them. Thus  will signify 'e sharpened by about 40 cents'. |
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I think we can now proceed. |
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16 In this 1 follow the recommendation of Sachs, WM 31. |
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17 Double sharps and double fiats will not be required. |
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