The Pentadecaphonic system - an 'equal tempered scale' with interesting properties

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The pentadecaphonic system: an 'equal tempered scale' with interesting properties

INDEX

1. Introduction

2. Pentadecaphonic system construction

3. Scales in the pentadecaphonic system

4. Pentadecaphonic system properties

5. Simulation software

6. The pentadecaphonic guitar

7. Links

1. Introduction

The history of music and the related idea of sound of, at least, the last two centuries (but the ancient Greeks have still some influence) looks at the world with the Fourier's eyes and considers physical quantities and signals as the superposition of 'harmonics' or 'modes', related together in such a way that there should be a sort of 'natural notes', 'simple' relationships between sounds and a 'harmony' that Nature prefers, versus other sound combinations.

Our question is: are we sure that this is a real behaviour of Nature and not simply intrinsic to the way we represents Her? In other words: in the representation of reality by 'harmonics', how much is due to the observed objects and how much is due to the observers? Could it be a human need the tendence to find 'simple' relationships between complex phenomena? It is not casual that nowadays the Fourier's interpretation of reality is in crisis and new conceptions and representations of the world have been proposed, after the revolution of the Computer Science (for example the Mandelbrot's Fractal Theory, the Chaos Theory or the Cellular Automata).

Therefore, we are going to the conclusion that the musical taste, for which some sounds are 'consonant' together and some others are 'dissonant', is more related to the cultural education and to the phycophysics of the ear (as the Tonotopic Theory proposes) than to a physical basis. For this reason, still the 'natural notes' are important, but not because they are the 'true' representation of the world, but because they are deep-seated in the western music taste (two thousand five hundred years is a long long time in human terms...).

We also are starting to believe that the evolution of Music will not move to the 'atonal' direction in the short term, but more probably it will gradually widen the range of sounds available, with musical systems that should not destroy the wonderful results of the past, but include them within increased and new possibilities (concrete well known examples are the Blues and in general the 'Ethnic' or the 'New Age' Music, that try to play 'in the cracks' of the keyboard).

With this vision in mind, we have explored simple extensions of the 'equal tempered dodecaphonic scale', with the objective to add only few new sounds to the musical system, but in such a way that it is possible to compose new melodies . The idea is not to destroy the existing 'Western musical taste', but just to work for a smooth evolution of it.

As a consequence of this conservative approach, the working hypotheses have been:

to analyse other 'equal tempered scales', for having the advantages of the 'equal temperament' (i.e. a complete freedom of modulation from any one key to any other and the convenience in the practical use of keyboard instruments, tuned differently, to play together without retuning).
to maintain as much as possible the 'true' intonation, i.e. to be closest as much as possible to the traditional 'natural' notes (see [1]):

Note Freq. ratio

minor third 6 / 5

Major third 5 / 4

perfect forth 4 / 3

perfect fifth 3 / 2

minor sixth 8 / 5

Major sixth 5 / 3

Octave 2

This last constraint is assumed, not because we are convinced that the 'true' intonation has an absolute relevance to characterize a 'good' musical system (for a kind of 'natural imprimatur'), but because it is still a significant aspect of the current musical taste.

Looking at the literature on the subject (see for example [2] ), we have analyzed the following 'equal tempered systems':

Octave from A4 = 440 Hz to A5=880 Hz

-----------> frequency in [Hz]

- tridecaphonic system (obtained by dividing an octave into 13 semitones)

- tetradecaphonic system (obtained by dividing an octave into 14 semitones)

- pentadecaphonic system (obtained by dividing an octave into 15 semitones)

- esadecaphonic system (obtained by dividing an octave into 16 semitones)

We have also performed the Consonance Analysis, because it is an example of alternative approach to classify 'consonant' couple of sounds, without refferring to the 'harmonic' representation (see [3]). Applied to the pentadecaphonic and the esadecaphonic systems, in comparison with the dodecaphonic one, it gives the following results:

in our opinion, the tri- and the tetra- decaphonic systems do not offer significant advantages (only one or two sounds more) and the notes obtained are significantly different from the dodecaphonic ones, in such a way the major and minor triads are not very close to the 'natural notes'.
in the esadecaphonic system the difference between two sounds, in the range of the tone, begins to be not enough significant (the tone axis intersects twice the discrimination frequency curve) for the human ear; in addition the note sequence is not particularly close to the 'natural notes'
the pentadecaphonic system shows, on the contrary, very interesting properties (among which, together with the dodecaphonic system, a quite good approximation of the 'natural notes'), derived also by a numerical basis (12 and 15 can be both divided by 3), to be considered the best candidate for the evolution of the dodecaphonic system.

The pentadecaphonic system is a musical system, able not only to contain almost all the effects and sounds of the dodecaphonic system (the most used triads, chords and scales), but also (we hope), due to the increased number of notes, able to inspire new sonorities and music styles.

In the following notes, we will limit our investigation on the way the pentadecaphonic system can produce the same or better results than the dodecaphonic system, within the current concept of Western Music.