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Regarding the historical origin and use of the 15ma symbol. It is also important to recognize that the musical language is ever-evolving. An effective symbolic language is most widely useful when it incorporates self-explanatory efficiency. The meaning of 16va/16vb is implicitly understood by anyone familiar with 8va or 8vb. The quindicesema (15ma), like the minim, crotchet, semibreve, and hemidemisemiquaver, is an example of enduring, premodern music terminology.

In jazz harmony, the intervallic series reaches its functional endpoint with the 13th (octave+6th). The intervals of the 9th, 11th and 13th are useful for representing extended triadic chord voicings of the 2nd, 4th and 6th, beyond the octave (root). Within this harmonic context, the interval series is a repeating sequence of: 1, 3, 5, 7, 9, 11, 13; 1, 3, 5… In this modal system, the double octave would be a 15th, but musicians do not use this intervallic value because it is a simple repetition of the octave, just as we don’t refer the interval of an 8th.

15ma” is an archaic symbol not commonly understood even by trained musicians. Its reference is not based on modern associations between musical octaves (2:1; 8×2=16) and octave frequency relationships (f, 2f, 4f), but rather a more esoteric concept of the ‘quindicesima’ (the 15th value in a simple number sequence).

In the diagram above we start with A2=1 (f; do) and A3=8 [8va] (2f; doubled value; do). The modern principle of octave equivalence implies a repeated, logical representation the octave again as A3=1 [8va] (2f; do) and A4=8 [16va] (4f; doubled value; do). It seems that the conceptual error of a quindicesima value of 15 comes from neglecting this repeating interrelationship (between octave note names, solfège and frequency) and instead numbering the intervals in a simple series – detached from their quantified acoustic relationships (hertz, harmonic ratios, cents).

In this context, the 15ma symbol represents an outdated modal note ordering system, analogous to a more modern chromatic system of ordering 128 MIDI note numbers. Yet, even this modern sequence of MIDI note numbers remains detached from inferring to any chromatic intervals as numbers. As such, it remains separate from and inconsistent with modal music theory, while also existing as an essential framework to music technology.

Another example of evolving musical language comes from the practice of microtonality. The historical use of ET (equal temperament), say 36-ET, is less self-explanatory and therefore less efficient as a symbol than 36-EDO (equal divisions of the octave).

The concept of ‘interval’, as described in traditional music theory, retains within its framework a (prechromatic) concept of modal intervals. Within this modal system an octave is represented in intervallic numbers 1, 8, 15 (n+7). Chromatic intervals were added later, through the use of accidentals (b9/#9, #11, b13), for notating the additional intervals without updating the antiquated modal framework.

16va is a representation of a repeating and doubling
relationship inherent to the octave as measured in frequency (100, 200,400 hz). This octave doubling (2:1 ratio) is also the basis of the logarithmic measurement of intervals in cents: octave = 1200 cents; double octave = 2400 cents.

The octave, as an acoustical (physics) relationship, and as explored within audiology (perceptual) and psychophysics (cognitive/psychological), is based on scientific, quantitative principles of the octave – defining it as a doubling/halving of
frequency (1f, 2f, 4f).

My arguments examine the incongruity between a prechromatic, modal music symbol (15ma) and modern systems of representing the octave. Besides an implicit, self-explanatory clarity, the musical symbology of 8va and 16va express a congruity between music theory and our scientific understanding of the octave as a repeating, doubling phenomenon.

It is important to remember that the cents system of interval measurement is logarithmic (exponential), not linear. The basis of both the frequency and the cents measurement is a 2:1 octave relationship. Because pitch measurement systems (linear and logarithmic) are derived from the octave as 2:1, a double octave represented as 16va (16:8 = 2:1) is a notation evolution surpassing music theory symbols based on an outdated, extended modal scale series, like 15ma. In this way, the tradition of describing an octave as ‘8’ (as in 8va) remains, while also integrating the octave’s recurring 2:1 relationship
into music notation as the double octave transposition symbol (16va).

However, the underlying problem remains: of the need to update an antiquated modal framework, both for the chromatic systems of the present and the micro-pitch (meta chromatic; non chromatic) music systems of the future.

Symbols like 15ma are an example of the need to update archaic aspects of music theory with modern scientific understanding of pitch and sound. Especially in an era where technology is assuming an ever-greater impact on the sound arts.

Once the 8va/8vb symbols are understood to mean an octave higher/lower than written, the reference to the Italian definition is no longer relevant to their understanding and use (except as a historical reference). The symbol 16va/16vb gains its meaning by mathematical inference (double octave: 8va x 2 = 16va) and not by any association with an Italian definition (sedicesima).