Musical Mathematics
on the art and science of acoustic instruments
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© 2000–2018 Cristiano M.L. Forster
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CHAPTER 12: ORIGINAL INSTRUMENTS
Section 12.12
The Diamond Marimba in Plate 7 is based on a 13limit tonality diamond. Max F. Meyer (1873–1967) first described the concept of a twodimensional tonality diamond in his book The Musician’s Arithmetic, published in 1929. On p. 22, Meyer shows the diagram of a 7limit 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 11limit 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 13limit 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 “octavemultiples” 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 “octavemultiples” of these numbers — in the denominators. A careful examination of Meyer’s 7limit, Partch’s 11limit, and my 13limit 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 11limit and 13limit 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 13limit 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 “doubleoctave,” 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 “doubleoctave” 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 quarterwavelength 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 triplearch 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.)
