The Partch Hoax Doctrines
What’s in a new name? when the source from which it flows is old.
by Cris Forster
20 August 2015
© 2015–2019 Cristiano M.L. Forster
www.chrysalisfoundation.org
* * *
A print publication recently released inaccurate and misleading descriptions of my work as a builder of unique acoustic musical instruments. The writers of this dictionary do not lack historical and factual information about these instruments. At the website, www.chrysalisfoundation.org, launched in 2002, and in my book, Musical Mathematics: On the Art and Science of Acoustic Instruments, published by Chronicle Books in 2010, I have described the musical origins, overall construction, and special features of all my instruments. An obvious reason why my efforts have had no effect on these writers is that they have not read my texts, and therefore have opinions on subjects they know nothing about. Another and more insidious reason is economics. Large and small print and online publications tend to promote products that are most convenient for them to produce. To minimize costs and maximize gains — either in the form of monetary profits or political power and prestige — they not only analyze and respond to popular opinion, but have a vested interest in shaping it as well. Since predilections, favoritisms, and intentional omissions are never openly acknowledged, agendadriven writers, editors, and publishers always portray themselves as gallant visionaries, or as magnanimous providers of “educational” resources for future generations.
Over the past 40 years, I have spared no expenditures of time and energy in building, tuning, composing for, and writing about unique acoustic musical instruments. Also, as curator of the Harry Partch Foundation (1976–1980), I restored and tuned virtually all the instruments. Therefore, I am also qualified to speak about the origins, construction, and tunings of the instruments built by Harry Partch (1901–1974). With these skills and experiences, I will try to be objective, accurate, and truthful in my objections to the halftruths of this publication.
Before I begin, I would like to address a severe limitation that surrounds me on all sides. When someone ignores facts or suppresses evidence, I cannot prove that the perpetrators of such omissions seek to denigrate the truth by promulgating halftruths. All students of philosophy know that you can’t prove a negative. You can’t prove that something does not exist. Therefore, since I cannot prove a negative, I also cannot presume to know what exists in another human being’s soul.
I begin with The Grove Dictionary of Musical Instruments, the first edition (Grove1) published by Macmillan Publishers Limited in 1984, edited by Stanley Sadie, and the second edition (Grove2) published by Oxford University Press in 2014, edited by Laurence Libin. Grove2 appeared 30 years after Grove1 but used almost the exact same text to misrepresent my work. And again, despite my website and book, Grove2 perpetrated a largely incorrect description of my first concertsize instrument, Chrysalis I.
In his Preface, Laurence Libin draws attention to several distinctions between Grove1 and Grove2. In the following statement, he singles out discussions on tunings as a unique contribution:
“On the other hand, playing techniques (e.g. bowing, tonguing, fingering), pitch, and tuning, are given due attention as they are crucial to understanding how instruments work and sound.”
Similarly, he touts a greater inclusion of instrument makers by acknowledging their contribution to the development of music in this century:
“This edition pays substantially greater attention to electronic and
experimental instruments and to instrument design and manufacture, and discusses
more persons—acousticians, collectors, curators, dealers, as well as
makers—whose work has shaped our understanding of instruments and thus of music
in the 21st century.”
“Cris Forster made several instruments in 56note just tuning: two of them, the
Harmonic/Melodic Canon and Diamond Marimba, were inspired by Partch; a third,
Chrysalis, consists of a disc mounted vertically on a stand with 82 strings on
each face, which radiate out from an offcenter circular bridge.”
[ 1 ] — There is no such thing as a “California group.” Yes, I have lived and worked in California since 1961. If Grove wants to make up a group based on some arbitrary geographical location and then cast me into that fictitious group without acknowledging my activities over the past 30 years, the least they could do is find a more appropriate category than socalled “microtonal instruments.” The adjective ‘microtonal’ can apply to any and all scale and tuning theories known to man. ‘Micro’ is a relative term, as is ‘macro’, and therefore has no mathematical meaning. Experiencing new tunings is the principal reason why I build musical instruments. Scales and tunings cannot be intelligently discussed without numbers. So, without mathematics there will never be advancements on the subject of tuning, no matter how noble the aspirations to evoke change. Finally, just because a fiddle maker may have lived next door to Stradivarius does not justify throwing him into some nonexistent “Cremona group.”
[ 2 ] —
Although it is true that I have built “several instruments,” this inept
description gives the impression that my life’s work consists of only the three
instruments mentioned in the sentence. Here is a complete list of all my
instruments to date: [1] Little Canon (1975); [2] Chrysalis I
(1975–1976, restored 2015); [3] Harmonic/Melodic Canon (1976, rebuilt
1981, final version 1987); [4] Diamond Marimba I, with pernambuco bars,
after a design by Max F. Meyer, (1978, rebuilt 2019); [5]
Boo III, also known by the more descriptive name
New Boo I, was commissioned by the Harry Partch Foundation as a replacement
for Boo I, which had disintegrated and become unplayable, (1979); [6]
Glassdance (1982–1983); [7] Bass Marimba (1983, 1985–1986); [8]
Bass Canon (1989); [9] Diamond Marimba II, with Honduras rosewood bars,
(1989); [10] Just Keys, a restrung, retuned, and rebuilt console piano,
(1990); [11] Simple Flutes, based on the equations in Musical
Mathematics, Chapter 8, (1995); [12] Chrysalis II (2013–2015).
[ 4 ] —
The innovations in design and construction of my Harmonic/Melodic Canon
and Bass Canon enable these two instruments to function as true canons; in other
words, the ratios these instruments produce (up to a carefully described limit)
are absolutely predictable, and therefore have nothing to do with the
instruments Partch called “canons.” The veracity of modern science depends on
two unshakable principles: predictability and repeatability. The ancient Greeks
understood these principles perfectly, hence the word ‘canon’. From Musical
Mathematics, p. 65:
* * ^{ ↓} * *
The canon as described in the works of Ptolemy (see Section 10.19) and AlJurjānī (see Section 11.52) represents the mathematical embodiment of tuning theory. Although this instrument has a noteworthy history, it did not develop into a precise musical instrument because of a persistent mechanical problem: rattling bridges! When one places a triangularshaped bridge under a string, and then plucks the string, the applied force causes the bridge to rattle against the soundboard. To avoid this difficulty, it is possible to make a long bridge, so when one plucks a given string the other strings hold the bridge in place. Even so, after much playing such a bridge begins to creep due to the vibratory motion of the strings. To prevent the bridge from moving, it becomes necessary to increase the downbearing force (or downward force) that the strings exert on the bridge. An increase in the height of the bridge increases the strings’ deflection, which in turn increases this vertical force. However, because the downbearing force effectively increases the tension of the strings, all the stopped strings sound sharp.
* * ^{↑} * *
As a young instrument builder, Partch understood these problems all too well, and formally addressed these difficulties in the first edition of Genesis of a Music (Genesis1), published by the University of Wisconsin Press in 1949. Below, please find (1) a scanned photo from Genesis1, opposite p. 205, that shows Partch’s first Harmonic Canon (1945), and (2) a scanned line drawing of the bridge design for this instrument, p. 99.
Partch attempted to solve the problem of rattling and creeping bridges by gluing wood laths, or a system of rails, directly to the soundboard. The line drawing shows a notch in the lower right hand corner of the bridge that ran the length of the bridge and acted as a secondary restraint to stabilize or hold the bridge against the rails. Also, both the photo and drawing show a machine screw and wing nut assembly used to lock the string to the bridge, thereby eliminating the downbearing force needed to secure the string in a notch at the top of the bridge. For many reasons, all these design features were doomed to failure.
Finally, notice a ruler along the front edge of the soundboard, which proves Partch knew that all canons require accurately measured string lengths for the construction of length ratios.
Rulers do not appear on any of the five socalled “canons” built by Partch after Genesis1. And they are also not included on numerous copies of “canons” built by Partch disciples and aficionados. Why? Because on all these instruments, a ruler serves no musicalmathematical function.
In the
second edition of Genesis of a Music (Genesis2), published by Da
Capo Press in 1974, pp. 235–242, Partch describes the “reconception” and
“reconstruction” of this instrument but offers no mathematical or musical
reasons why he gave up on the task of building soundboards and bridges that
produce accurately tuned canon strings. On p. 98, Partch acknowledges that on
his canon, “…a high bridge increases both the tension and length of the string.
I have experimented with bridges just high enough to give a good tone, using
guitar first strings, and the results varied greatly from theory.” And on p. 99,
he concludes, “But it must be kept in mind that results on a harmonic canon are
almost always approximations.”
Partch Hoax Doctrine #1.
However, an equally important but less obvious second reason is that Partch never acknowledged tension as a critical constant for tuning open canon strings to a fundamental frequency. Instead, while moving his high bridges back and forth, he simultaneously turned the knobs of his tuners until the string sections on the left and right sides of the bridges produced the frequencies he wanted to hear. With this method, I wonder how many strings Partch broke before he chanced upon his final arbitrary string tensions and equally arbitrary bridge locations.
Fact: Throughout Genesis1 and Genesis2, Partch never discussed — and gave no information about — measured string tensions and measured bridge locations.
Any data on these two properties of Partch’s strings would have immediately revealed that his instrument is not a canon. Contrary to the abovementioned description, it is physically impossible for a bridged canon string to produce ratio 2/1 on one side and ratio 40/21 on the other side unless someone turns the tuner to some arbitrary tension and moves the high bridge to some arbitrary location.
Finally, under these three nonmathematical conditions: (1) unmeasured string tensions, (2) extremely high bridges, and (3) unmeasured bridge locations, the strings of this instrument do not sound the just ratios listed on p. 245; at best, they only produce frequencies that approximate the rational ratios of just intonation. For all these reasons, Partch’s “canons” are in fact zithers, which means that the abovementioned instrument should be renamed “Harmonic Zither II.” Since this will never happen, Partch’s damage to the word ‘canon’ is irreversible and therefore permanent.
Let us
now reexamine the text on pp. 98–99 of Genesis2. Observe that with
respect to tuning, Partch mischaracterized the canon as an inaccurate
instrument. This contrived assessment enabled him to rationalize the
mathematical inaccuracies of his own instruments and tunings. However, if,
according to Partch, canons do not produce accurate tunings, then why did he
repeatedly refer to his zithers as canons? In
Genesis2, Partch included descriptions of five pseudocanons. The Greek word
kanōn and its
definition as a measuring instrument and code of law is at least 3000 years old.
By not adhering to these two timehonored definitions, and by refusing to
explain his nonmathematical interpretation of the word ‘canon’, he appropriated
for himself the historic legacy and prestige of this instrument. In doing so, he
has played all his uninformed and unsuspecting readers for fools. Because Partch
failed to develop soundboards and bridges that facilitate accurately tuned canon
strings, his zithers are not governed by the scientific principles of
predictability and repeatability, and therefore have absolutely nothing to do
with the design and construction of my canon soundboards, bridges, and tunings.
Stated differently, a person cannot be inspired by something that does not
exist.
http://chrysalisfoundation.org/hm_canon.htm
http://chrysalisfoundation.org/bass_canon.htm
In the history of music, these are the first canons that satisfy two musical conditions. Both canons have independently movable bridges that produce mathematically predictable length ratios; and both canons function as fully resonant performance instruments.
Furthermore,
gives a canon building and tuning description by AlJurjānī (d. 1413) that is complete, and therefore true. Based on his knowledge of Euclid (fl. c. 300 b.c.) and Claudius Ptolemy (c. a.d. 100 – c. 165), AlJurjānī’s text specifies the three mathematical principles of accurately tuned canon strings:
(1) “Tension the strings equally so that they all produce identical notes.”
(2) “The top of the bridges must be slightly higher than the semicylindrical [rod].”
(3) “Then determine on the strings the points that correspond to each of the divisions on the ruler. Move the bridges to place them in line with each of these points, so as to make each of the strings produce one of the notes of the desired system.”
If anyone inspired me to resolutely pursue how canons should be built to “…test the rules of music,” it was AlJurjānī!
Finally, in Grove1, Volume 2, p. 128, and in Grove2, Volume 2, p.
547, under the heading “Harmonic Canon,” we find two slightly different versions
of the opening sentence. Grove2 states:
“Here the Arabicized word qānūn does not refer to the modern zither, built in the form of a trapezoid and equipped with strings of different lengths, but rather to the ancient Greek kanōn, described at length in Ptolemy’s Harmonics.”
Summary.
A zither has no neck, and its strings are stretched between two opposite ends of the body, which may or may not function as a resonator. Therefore, all canons are classified as zithers. However, by definition, a canon must be built and tuned according to four mathematical requirements. All open or unbridged strings must be identical, or have the same length, tension, linear density, and fundamental frequency. If such an instrument does not satisfy these conditions, it may be called a zither, but it is not a canon. Why? Because on a stringed instrument without these four constants, the concept of what constitutes a length ratio — namely, a comparison of two measured string lengths — does not and cannot exist. Based on these facts, we conclude that a canon consists of nothing more than a set of identical monochords. However, in this context, the Chinese ch’in is a remarkable exception. Although the ch’in is a zither, it functions exclusively on the basis of length ratios because musicians play its strings as a set of nonidentical monochords. This design requires only one critical constant: all the strings of the ch’in must have identical lengths. Similarly, this is how lutes or instruments with necks, which include sitars, violins, guitars, etc., are built and played: nonidentical monochords with identical lengths are stopped by the fingers according to the principle of length ratios. So, musicians always play the “octave” at the halfway point of the strings, length ratio 2/1, the “fifth” at twothirds of the strings, length ratio 3/2, etc.
Imagine you have a canon where all the strings are identical and 1000.0 millimeters long. The first string is open, and the second string has a low bridge that is only slightly higher than the open string. While playing both strings, slide the bridge back and forth until you hear an unfamiliar interval. This raises the question, “What is the mathematical and musical identity of this interval?” Suppose the longer section on the right side of the bridge is 525.0 mm long. Ratio 1000.0 mm/525.0 mm reduces to length ratio 40/21 [1115.5 ¢], which means that the right section sounds a “sharp major seventh.” Also, since the shorter section on the left side of the bridge is 475.0 mm long, ratio 1000.0 mm/475.0 mm reduces to length ratio 40/19. Because this ratio is larger than an “octave,” it is difficult to comprehend. The solution: lower it by an “octave” so that its quotient is greater than 1, but less than 2. If the numerator is even, divide by 2; otherwise, if the numerator is odd, multiply the denominator by 2. Length ratio 20/19 [88.8 ¢] now identifies the left section, which sounds a “flat semitone” (one “octave” higher). For ratio 40/21, a length of 525.0 mm clearly indicates that the bridge is located near the center of the string. Finally, to determine the frequency of this section, multiply the fundamental frequency of the open string by 1.90476, the decimal ratio of 40/21.
Consider now a photo of the previously mentioned Pollux instrument on p. 243 of Genesis2; it’s the one on the right side of the image. Note that the bridge for ratio 40/21 is not located near the center of String 1. Consequently, on this pretend “canon,” the string sections on the left and right sides of the bridge do not represent length ratios. In fact, all the left/right ratio pairs in Partch’s table have absolutely nothing to do with length ratios. Instead, they are all frequency ratios, like those found on nearly all zithers, built with or without bridges. After turning the knobs of his tuners to some arbitrary tensions, and after moving his high bridges to some arbitrary locations, Partch was unable to analyze the ratio pairs by simply measuring string lengths and constructing length ratios. To identify complicated frequency ratios, he had to play some other instrument on which these ratios already existed (Chromelodeon) or could be realized as length ratios (Adapted Viola).
Finally, if String 1 of Pollux was on a canon that had identical strings and low bridges, then the location of the bridge in the photo indicates that the left string section would sound an interval in the vicinity of a just “fifth,” length ratio 3/2 (or 2/3), and the right string section, an “octave and a fifth,” length ratio 3/1 (or 1/3).
Although the ancient
Greeks and
Arabs had no knowledge of frequency ratios, they were nevertheless able to
accurately determine the mathematical and musical identities of intervals on
their canons and lutes, respectively, through the construction of length ratios.
For a
detailed analysis of this figure, please visit
The numbers in the figure represent cent values. Notice the four zeros in the middle row of this diamondshaped tuning lattice. I call this the neutral axis because it consists of four tones with identical frequencies. In other words, because the tonality diamond contains four crisscrossed diagonals, the neutral axis includes four unisons. If it had five, six, or seven diagonals, a consistent mathematical expansion of this design would require five, six, or seven unisons, respectively. Meyer’s unisons represent a twodimensional interpretation of Rameau’s dualgenerator. As shown in the abovementioned musical illustration, Rameau’s highC dualgenerator has the same musicalmathematical function as the rightmost zero in Meyer’s tonality diamond. In an upward direction and in scale order, this zero generates a major tonality: C0 ¢, E386 ¢, G702 ¢, Bb969 ¢. And by inverting this sequence of intervals, in a downward direction and in scale order, the same zero generates a minor tonality: C0 ¢, Ab814 ¢, F498 ¢, D231 ¢.
Regarding the ascending sequence, the first three cent values represent frequency ratios C1/1, E5/4, G3/2. Rameau demonstrated that the 4th, 5th, and 6th harmonics of the harmonic series of vibrating strings generate the major tonality, expressed as ratios 4:5:6. And regarding the descending sequence, the first three cent values represent frequency ratios C1/1, Ab8/5, F4/3, which is an intervalic inversion of the first sequence. Rameau realized that an inversion of the intervals of the major tonality produces the minor tonality. However, because he was unable to demonstrate that such a sequence occurs as a natural phenomenon of vibrating strings, he eventually conceded that the minor tonality only exists as a manmade or synthetic construct. Approximately one hundred years after Rameau, music theorists began referring to the inversion of harmonics as ‘subharmonics’, to the inversion of socalled overtones as ‘undertones’, and to the inversion of the harmonic series as a ‘subharmonic series’.
If transformed into a
musical instrument, Meyer’s design would require four identical
frequencyproducing sources. For example, on a piano, it would require four keys
and four sets of strings all tuned to the same frequency; on a marimba, it would
require four bars and four resonators all tuned to the same frequency; etc.
Furthermore, as if to denigrate Meyer’s idea, on the cover of Xenharmonikôn 3, Spring 1975, Erv Wilson (1928–2016) published a stellated interpretation of the 11Limit Tonality Diamond in which he eliminated the neutral axis presumably because it constitutes a mathematical redundancy. Unfortunately, in the context of understanding the development of a musically inspired design, belaboring the obvious with nonmusical reductionism contributes nothing but a disservice to the discussion. Wilson gutted the neutral axis to illustrate Meyer’s nonlinear twodimensional lattice as a purely mathematical construct. Although the mapping of Meyer’s lattice ratios over a stellated surface gives the appearance of a twodimensional design, Wilson’s graphic represents nothing more than a linear onedimensional set of symmetrically paired scale ratios arranged in various geometric patterns. The reason for this anachronistic distortion of Meyer’s original design is the position of the first unison, ratio 1/1. Wilson placed it at the center of his diagram, which means he depicted it as the generator of all the other ratios. This, of course, is nonsense because we cannot consistently interpret the other unisons — 3/3, 5/5, 7/7, etc. — as generators. On the contrary. Every unison ratio and every nonunison ratio occupies a unique location and, thereby, serves two uniquely different musical functions in Meyer’s twodimensional diamond. Finally, it is impossible to systematically calculate all the ratios of Wilson’s graphic without ascending and descending in diagonal directions (1) from the neutral axis, (2) through the neutral axis, and (3) to the neutral axis. So, without Meyer’s neutral axis, Wilson’s onedimensional diamond does not exist.
Fact: Wilson’s diagram — entitled Hexadic Diamond on a CenteredPentagon Crystallograph — first appeared as a gift he gave to Partch in 1969, or two years after Meyer’s death in 1967.
This
raises the inevitable question, “What exactly did Wilson give to Partch?” In my
opinion, Wilson devised his figure as a distraction from Partch’s plagiarism.
Wilson’s diagram, which conveys no musical meaning, suggests that anyone with
only a knowledge of ratios and the inversion of ratios could have conceived
of Meyer’s diamond. Proponents of what I call the Musically Absurd
1/1Generated Diamond never acknowledge (1) the natural harmonic series and
(2) the synthetic subharmonic series as two basic requirements for the
construction of all tonality diamonds. These writers maintain that modern
discoveries in the physics of vibrating strings and the musical implications of
these discoveries are irrelevant to a strictly numeric interpretation of this
structure. With Partch in the lead, they advocate the ludicrous possibility that
the ancient Greeks, with their knowledge of ratios, could have conceived of such
a design. From Musical Mathematics, p. 453:
Unfortunately for Partch, Rameau was a famous composer, and unfortunately for
Wilson, Meyer was a practicing musician, and for these two reasons, Meyer’s
design will always represent a stunning synthesis of the major and minor
tonalities of Western music. When Wilson stripped the neutral axis out of
Meyer’s diamond he destroyed the twodimensional integration of these two
tonalities; the result: a Musically Absurd 1/1Generated Diamond that no
one has ever built.
From
Musical Mathematics, pp. 452–453:
At
To further expose the inanity of the latter quotation, consider the following absurd analogy: ‘Although originally invented by Albert Einstein, the equation E = mc2 in now most associated with Robert Oppenheimer.’ Why? The equally absurd response: ‘Because Oppenheimer was the director of the Manhattan Project that actually built the first nuclear weapon.’
Some have tried to defend Partch’s appropriation by claiming that he “borrowed” the tonality diamond from Meyer. If you take something that belongs to another person and (1) you don’t ask for the owner’s permission, and/or (2) you don’t acknowledge the owner’s existence, that’s not borrowing. That’s called stealing. Furthermore, under certain circumstances, it is not possible to “borrow” another person’s property. For example, it would have been impossible for Oppenheimer to “borrow” Einstein’s energy equation because in the 1940’s everyone in the world knew that it belonged to Einstein. So, if Partch had attempted to “borrow” Hermann Helmholtz’s resonator equation (see Musical Mathematics, Section 7.13), all hell would have broken loose. To some, Partch got away with “borrowing” the tonality diamond because in the 1940’s, Meyer was not a wellknown music theorist. I say, all the more reason to give credit where credit is due because — as Walt Whitman reminds us — greatness does not belong exclusively to the famous. Look around. It seldom does.
In
2003, I received an email from Max F. Meyer’s grandson in which he thanked me
for my efforts to restore his grandfather’s legacy and reputation. At that time,
Wikipedia had no webpage on the life and achievements of Max F. Meyer. All the
perpetrators of halftruths have one thing in common: they are oblivious to the
presence of real people and real families who remain forever traumatized by the
theft of their intellectual property.
With the exception of Musical Mathematics and www.chrysalisfoundation.org, Meyer’s diamond appears nowhere in print or on the internet. No wonder, because the systematic eradication of Meyer’s name first began in the unpublished papers, newsletters, books, and journals of “important” contemporary theorists, professors, aficionados, and microtonalists who produced their American wunderkind texts and graphics in the years 1960–2000.
All these intentional omissions — designed to exclude Meyer from the history of just intonation — are too obvious to warrant further comment.
However, the stonewalling of Meyer’s lattice has also caused a deafening silence over independently verifiable dates of discovery, as witnessed by (1) publishers, (2) notaries public, (3) postmasters (who routinely apply dated stamps across taped seams of registered mail envelopes and packages), (4) copyright specialists at the Library of Congress, etc., before 1929.
Fact: With Partch, we get none of the above.
Fact: With Meyer, the provenance of his book is irrefutable because it was published by the Oliver Ditson Company in 1929.
Finally, as curator of the Harry Partch Foundation, I repaired and tuned virtually all the instruments. While I learned a lot, I gave as much as I got. Regarding my efforts to save and restore Partch’s instruments, and all my other contributions to the Harry Partch Foundation, not a single halftruth or wholetruth exists anywhere.
Diamond Marimba I — Pernambuco — 1978 (Rebuilt 2019)
Diamond Marimba II — Honduras rosewood — 1989 (Modified 2008, 2010)
http://chrysalisfoundation.org/diamond_marimba_II.htm
Shortly after completing the first Diamond Marimba, I was perplexed by an unexpected
acoustic phenomenon. Although I had tuned the fundamental mode of vibration of
the seven bars of the neutral axis to the exact same frequency, high G at 784.0
cps, none of the bars’ fundamental frequencies sounded the same. Deeply worried,
I sat at the instrument for days trying to determine the cause. Unless a musical
instrument requires the inclusion of multiple unisons, this phenomenon is
terra incognita to all builders. As if struck by lightning, the answer
suddenly came to me: the higher modes of vibration of the bars — similar to yet
distinctly different from the harmonics of vibrating strings — were influencing
my aural perception of the fundamental frequencies of the bars. From Musical
Mathematics, pp. 163–164:
So, from an intonational perspective, not to mention many other design features as well, my 13Limit Diamond Marimbas are fundamentally different from Partch’s 11Limit Diamond Marimba. In Genesis2, p. 272, Partch gives the following description of this problematic experience:
“However, for many years, I had heard about or read about one strong inharmonic
overtone created by this type of vibrating body. After building the Diamond
Marimba, Bass Marimba, and Marimba Eroica, I still could not say that I had ever
heard this overtone. Finally with the Quadrangularis, I do hear it, in the alto
flanks… Theory finally becomes fact.”
[ 6 ] —
The webpages
http://chrysalisfoundation.org/chrysalis_II.htm
http://chrysalisfoundation.org/New_Chrysalis2013.htm
and Musical Mathematics, Chapter 12, give general and detailed descriptions of the basic components and construction of Chrysalis I and Chrysalis II. Over the past 60,000 years, I know of no acoustic musical instrument built with two soundboards. Meyer’s tonality diamond with its intrinsic neutral axis is a distinctive creation, and Chrysalis I and Chrysalis II built with two soundboards each are also unique.
Furthermore, the intrinsic volume of air between the inner surfaces of the two facing soundboards constitutes a critical factor in sound production. This volume of air functions like a cylindrical resonator that is closed at the two ends and open around the circumference, to my knowledge not previously described in any text. Together with the two soundboards, two circular aluminum bridges, and 82 strings per soundboard, this resonator contributes significantly to the amplitude of the instrument.
