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Tempering Notation

The equal-tempered (ET) twelve-note scale has now dominated music for more than a century. During this period many important changes have occurred, not least the entire development of electronic musical instruments and very detailed theoretical ideas underlying them. Most of these assume that the ET scale is the only one in use, especially in such practical manifestations as the MIDI interface and music programming languages. Whilst it is true that both of these make provision for other scales using the convention of microtones, this is treated as a specialized modification of the "normal" (i.e. ET) scale and the resulting solutions are clumsy to work with.

Familiarity with the Natural Harmonic Sequence (NHS) produces a very different understanding of the theory of musical scales, and suggests a way of defining them based on it. This new definition takes the notes of the NHS as reference points and defines notes of the scale as deviations from them. The advantage of this for musicians is that scales from other musical traditions, or based on different tempering schemes, can then be defined in a simple and consistent manner.

However, there is another advantage of such a definition that extends beyond music. Modern physics is beginning to recognize and investigate the harmonic relations within energy structures, but uses traditional mathematical descriptions of them that often obscure the simple but important relationships underlying musical theory and performance. Both physics and music would benefit greatly from a cooperative approach to defining these relationships. The ear is an amazingly sensitive organ, and can often perceive patterns and relationships in sound that are difficult or impossible to recogize in a mathematical representation. By giving expression to physical structures and events as sound patterns, new and valuable scientific insights may well appear. Musicians for their part can make use of these correlations between sound and physical structure to produce new musical compositions and forms.

Practical applications of these ideas may result in extensions or modifications to the MIDI interface and such popular computer music languages as Csound and Cmix, and a project to create these is in process of definition as discussed below.

Table 1   OCTAVES OF THE NHS
Octave# notesNotes
111st
221st, 5th
341st, 3rd, 5th, 7th
48Diatonic
516Chromatic
632Microchromatic

The NHS can be considered as a sequence of Octaves, each containing twice as many notes as the previous one. Table 1 shows the first six Octaves, the number of notes in each, and the degrees of the notes in their approximate correspondence to the diatonic ET scale. The musical possibilities of the first three Octaves are too limited to be of practical use, and the intervals in Octaves above the sixth too close to be of general interest. Octaves four, five, and six are therefore the "musical octaves" for most practical purposes.

When writing out the notation for a piece of music, standard practice is first to specify the clef upon which the notes are to be written, then the key signature, the time signature, and possibly an indication of the speed of performance in beats per minute. Tempering adds a further optional requirement, that being the relative positions of the notes used in relation to the NHS. There are obviously several ways of doing this. Eventually a preferred standard will emerge, but until it does, a simple, workable system with which to begin is given here.

A useful comparison might be made with the tab stops on a typewriter. Standard tab stops are at every eighth character, but when they vary it is usual to specify their absolute positions (e.g. 4, 8, 14, 22 etc) rather than the distance between them (4, 4, 6, 8 in the previous example). When specifying notes in a scale, the first indication should be the Octave of the NHS being used as a reference. Each Scale Note should correspond to a Natural Note in that Octave, but the choice then arises as to whether a strict one-to-one mapping is maintained, or whether multiple mappings are allowed. Some practical examples will clarify this and other issues.

Table 2   NOTATION OF THE ET SCALE
HarmonicNote¢ET note¢Spec
16A'1200C'12004A+0
15H1088B11004H+12
14G969--4G-0
13F841A9004F+59
12E702G7004E-2
11D471F5004D+29
10C386E4004C+14
9B204D2004B-4
8A0C04A+0

Let us begin by mapping the ET scale to the Fourth Octave of the NHS. Because the use of musical cents is so well established (see here for an explanation of cents) they can be retained for the sake of consistency and simplicity. The specification of each note can begin with a number indicating the Octave used as a reference. This can be followed by a letter naming the note of the Natural Scale, and beginning with A. Finally, a signed number indicates the difference in cents between the Natural Note and the Scale Note.

Table 2 shows how this works out in practice, and should be read from bottom to top. The ET scale has seven notes, and if we take C as the root, its specification is 4A+0. The next note D is slightly flatter than its equivalent in the Natural Scale, and is specified as 4B-4. The scheme can be followed to the note A, when two problems occur. The A note is about halfway between the thirteenth and fourteenth harmonics, so the closer of the two, the thirteenth, is used as a reference. There is no ET note corresponding to the fourteenth harmonic, so the convention is adopted of using "minus zero" to indicate this, and the scheme then covers all requirements whilst maintaining consistency.

A further problem arises if we wish to use a scale with an inconvenient number of notes, say nine. This will require one of two alternatives; either we can move to the next Octave having sixteen notes, or we can use multiple mapping by assigning two Scale Notes to one Natural Note. Table 3 shows that the latter possibility yields a simple and workable solution, with the fourteenth harmonic doubly mapped.

Table 3   MAPPING A NINE-NOTE SCALE
HarmonicNote¢ET note¢SpecNote #
16A'1200C'12004A+01
15H1088B11004H+129
14G969Bb10004G+318
14G969A9004G-697
13F841Ab8004F+596
12E702G7004E-25
11D471F5004D+294
10C386E4004C+143
9B204D2004B-42
8A0C04A+01

If we now consider applying this scheme to the MIDI interface, a final requirement is to give each note a unique name or number. This requirement has two parts, the former being the musical octave in which the note occurs, and the latter the note itself. Table 4 shows the designation schemes commonly used today, with the octave above Middle C shown shaded. As to be expected, these are based on the diatonic ET scale and keyboard naming conventions. The last line gives a suggested scheme for adapting them to tempered scales of the Fourth Octave. For Fifth Octave scales with notes from A to P the designations would be A.0 - P.0, A.1 - P.1 etc. The Sixth Octave has thirty-two notes, six more than the number of letters in the English alphabet. It will be used so rarely that it is not worth abandoning the scheme to accomodate it, and likely solutions include the use of Greek letters, the numbers one to six, and similar extensions to the alphabet. Those needing to use it will undoubtedly evolve suitable symbols.

Table 4   COMMON & SUGGESTED NOTE DESIGNATIONS
HelmholtzC'' - B''C' - B'C - Bc - bc' - b'c'' - b''
ModernC0 - B0C1 - B1C2 - B2C3 - B3C4 - B4C5 - B5
Nominal C frequency (Hz)163264128256512
Concert C frequency (Hz)16.432.765.4130.8261.6523.3
Suggested schemeA.0 - G.0A.1 - G.1A.2 - G.2A.3 - G.3A.4 - G.4A.5 - G.5

The complete notation for a note can combine this with the above: 4A+0.4 then becomes the tempered notation for Middle C in the Fourth Octave, 5A+0.4 in the Fifth Octave, and 6A+0.4 in the Sixth.

The suggested scheme is simple and consistent, and should be suitable for use in specifying appropriate extensions to musical interfaces and programming languages. It can also be used as a basis for the design of new integrated circuit sound generation chips.