User:Unque/On Imaginary Harmonics
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"Imaginary" numbers, or the square roots of negative numbers, have been attempted to be inserted into and interpreted in terms of Xenharmonics by a number of composers and theorists, with many conflicting concepts existing as to how those imaginary numbers would apply to musical composition and theory. Here, I plan to detail my personal interpretation of how "complex harmonics" might be used in a musical sense.
Defining Imaginary Harmonics
Imaginary and Complex harmonics are quite difficult to define; the imaginary harmonic series has often been interpreted as an axis of a complex plane perpendicular to the real harmonic series, but this does not reflect the definition of imaginary numbers as the square root of a negative value.
Defining imaginary numbers with respect to negative numbers creates more questions than answers, as there is no clear-cut definition of a negative harmonic. A sound wave with a negative frequency is impossible, unless interpreted in one of several ways that sidestep the true mathematical distinction between positive and negative, but most approaches to negative harmonics cite that they are either equivalent to positive harmonics travelling in the opposite direction, or positive harmonics offset by half a wavelength such that each peak becomes a trough and vice versa.
Under either of these definitions of a negative harmonic, the square root of a negative number can be found as two logarithmic divisions of its positive harmonic counterpart; thus, the nith harmonic can be defined as two equal divisions of the (n²)th harmonic. For these purposes, the Imaginary Harmonic Series will be interpreted as the square root of each negative integer, such that the second member of the Imaginary Harmonic Series is sqrt(-2), the third member is sqrt(-3), and so on.
In other words, this definition interprets negative (and therefore imaginary) harmonics as being fundamentally indistinguishable from positive real harmonics. While this may sound unintuitive, dissatisfying, or even devoid of useful application, the presence of the Imaginary Harmonic Series does provide an interesting lens through which to derive a generalization of Hemipyth and irrational number subgroups of Just Intonation.
Complex Diamond Scales
One potential way to derive use from this Imaginary Harmonic Series is via a Complex Diamond. A Complex Diamond can be notated as iD{n} <q>, where n is the prime subgroup and q is the interval of equivalence; if q is not specified, it can be presumed to be 2/1.
To create the Real part of the Complex Diamond, simply take the difference between each distinct value in the subgroup, and reduce it to fit within one equave. The Imaginary part of the scale can be formed by taking the square roots (two equal divisions) of each interval in the scale; finally, the Complex part of the scale is just the Equave Compliment of the Imaginary intervals, or the difference between the Imaginary interval and the equave. These three subscales can now be combined into a single scale, completing the Complex Diamond.
As an example, let's find the simplest possible Complex Diamond, iD{3} <2>. This gives us the real rational numbers 3 and 1/3; reducing these to fit in one equave gives us the intervals 3/2 and 4/3. For the Imaginary part of the scale, we will use the square roots of each of those intervals, which gives us 1\2<3/2> and 1\2<4/3>; finally. for the Complex part of the scale, we will use the difference between each Imaginary interval and the octave, which gives us 1\2<8/3> and 1\2<6/2>. Thus, our final scale is seven notes: 249, 351, 498, 702, 849, 951, and 1200 cents.
Complex Diamonds can also be formed with EDO steps and other arbitrary non-JI intervals; for instance, the Complex Diamond iD{7\12} <2> gives us 7\12 and 5\12 for the real part; 7\24 and 5\24 for the imaginary part; and 17\24 and 19\24 for the complex part.
It should be noted that, because of this scale's regular structure, it necessarily has two periods per equave rather than one; thinking of the scale in terms of this period negates the necessity for the "complex" part of the scale, but requires a bit more calculations concerning which real/rational intervals to keep and which to discard.
External Links
- The Imaginary Harmonic Alliance on Discord
- The complex diamond iD{3} <2> on Scaleworkshop