Musical Mathematics

on the art and
science of acoustic instruments

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CHAPTER 12: *ORIGINAL INSTRUMENTS*

Section 12.12

The Diamond Marimba in Plate 7 is based on a 13-limit tonality diamond.
Max F. Meyer (1873–1967) first described the concept of a two-dimensional
tonality diamond in his book *The Musician’s Arithmetic*, published in 1929.
On p. 22, Meyer shows the diagram of a 7-limit Tonality Diamond that
includes 16 just intoned frequency ratios.
(See Chapter 10, Figure 57.) In
1946, Harry Partch (1901–1974) transformed and expanded Meyer’s original design
and built an 11-limit Diamond Marimba with 36 just intoned bars.
(See Chapter 10, Figure 59.) With
respect to my Diamond Marimba, Figure 12.5 shows the 49 bars required by a
13-limit tonality diamond. Here the
frequency ratios of the diagonals that ascend from left to right include odd
numbers 1, 5, 3, 7, 9, 11, 13 — or “octave-multiples” of these numbers — in the *numerators**;*
conversely, the frequency ratios of the diagonals that descend from left
to right include odd numbers 1, 5, 3, 7, 9, 11, 13 — or “octave-multiples” of
these numbers — in the *denominators*. A careful
examination of Meyer’s 7-limit, Partch’s 11-limit, and my 13-limit diamond
reveals that the row that runs through the center of these designs represents a
sequence of unisons. For this reason, I
refer to the center row as the *neutral
axis*. On the 11-limit and 13-limit
Diamond Marimbas, the neutral axis sounds the tone of the tonic, ratio 1/1,
below all the bars in the *upper *
halves of the diamonds.
Furthermore, on the 13-limit Diamond Marimba, the neutral axis produces
the tone of the “octave,” ratio 2/1, above the following 15 bars in the
*lower *half of the diamond:
14/13, 12/11, 10/9, 8/7, 14/11, 4/3 (12/9), 18/13, 10/7, 14/9, 8/5, 18/11, 5/3,
22/13, 12/7. And it produces the tone of
the “double-octave,” ratio 4/1, above the following 6 bars in the
*lower *
half of the diamond:
16/13, 16/11, 20/13, 16/9, 20/11, 24/13.

Now, a bar that sounds the fundamental frequency, ratio 1/1, below the lowest
bar, or below the “sharp minor third,” ratio 16/13, is
*
not* a part of the diamond.
Also, a bar that sounds the “octave,” ratio 2/1, between the “sharp minor
seventh,” ratio 24/13, and the “sharp minor second,” ratio 14/13, is not
included. Consequently, I decided to
append the basic structure of the diamond design.
In the lower part of the instrument, Figure
12.5 illustrates that I added a
bar for the fundamental G3 at 196.0 cps, and a bar for the “octave” G4 at 392.0
cps. The neutral axis now produces the
interval of the “double-octave” G5 at 784.0 cps above the fundamental.
Figure
12.5 shows that I also included
three more bars that produce the intervals of the “fourth,” ratio 4/3, the
“fifth,” ratio 3/2, and the “sharp minor sixth,” ratio 13/8, above the
fundamental. Therefore, the Diamond
Marimba in Plate 7 has a total number of 49 bars + 5 bars = 54 bars.

**
****
**

The Diamond Marimba stand consists of six parts:
a lower base, four poles, and an upper platform.
The Honduras rosewood bars are mounted on a terraced platform that
consists of fourteen rows of bars.
Beginning with the second row, each succeeding row rises a half inch above the
previous row, so that the difference in height between the first row and the
last row equals 13 ×
1/2 in. = 6 1/2 in.
Underneath the platform, I mounted a quarter-wavelength resonator for
each bar. (See Chapter 7, Sections 10–11.) The resonators are made from cast acrylic tubes, and the stand
poles, from cast acrylic rods. In Plate
7, note that the profile of the fundamental bar in the first row shows the
triple-arch design used to tune the first *
three* modes of vibration. (See
Chapter 6, Sections 10–14.) For the
others, I tuned the first *two* modes of
the bars in the 16/13–7/5 range, and only the
*first* or fundamental mode of the bars
in the 13/9–13/8 range. (See Chapter 6,
Section 15.)