TARTINI TONES AND MUSIC OF THE SPHERES
INTRODUCTION These pages will be devoted to the development and exposition of a truly comprehensive Theory of Harmony --one that is based on inclusion of all the 'missing' tones of conventional Western harmony, the notes 'between the keys' of the keyboard, the tones that have 'fallen through the cracks' in the design of instruments and the traditional techniques of practice.
Inspired by the explorations of ancient Greek musicians, geometers and mathematicians --who were boldly eager to define, delineate and perform the most subtle distinctions of interval combinations to a degree, both played and calculated, that Western music has consummately neglected over the last two thousand years. Proceeding along a musical vector from the brief but daring overtures into the field of microtones by Debussy, Scriabin, Bartok and others, this site will introduce to the modern musical world 'the rest of the notes' that form a wonderfully rich and expressive extension of harmonic practice for future composition and listening.
The discovery and exhibition of the 'Tartini Tones' several centuries ago served as a launch into this endeavor, as they themselves demonstrated a method for introducing 'off scale' harmonics into the sonorities of sophisticated play, and in fact formed a foundation for the then-budding science of acoustics...
FYI: supplementary posting on
ReplyDeletehttps://classicalpoets.org/2022/12/18/the-tartini-tones-by-joseph-s-salemi/#comment-538350
S.I. Wells
October 3, 2024
I am so glad that Sunday Baroque has brought out this much-neglected facet of music theory and practice. My interest in this matter stems from many years of harmonic study, along with a lot of listening and exploration of tone production by a variety of methods -including ‘off’-tuning of instruments- in order to sample the exotic harmonies they could produce –generally dismissed by mainstream musicians. I was inspired by Scriabin’s venture into the higher harmonics for chord formation, and was excited to learn of Debussy deliberately standing in an isolated area to listen to distant chimes of ‘out of tune’ church bells–hoping to capture inspiration for his own harmonic innovations. The Tartini tones are indeed a complicated subject, as others have remarked, and will likely attract further study well into the future. But I have an answer to their curious antique production and more curious lack of modern production. I think it is in the tuning of the instruments themselves: back in the days of instrument-making in the early 1700s, the generation of the musical modes and scales was undergoing a change from the ancient ‘justly-intoned’ method of tuning, based principally on the cycle of fifths and the natural harmonic sequence to the new ‘wonder’ of modulational versatility: the ‘well-tempered’ scale, popularized famously by one J.S. Bach. Although this new scaling of intervals proved of great utility in tuning instruments and transposing through the entire range of musical keys with ease, it lost something very important in its effect upon the ear:
This was the loss of rational number relations of the frequencies of the musical notes themselves, as developed even more famously by the ancient Greek Pythagoreans. You see, the intervals of the well-tempered scale are all irrational numbers, being multiples of the well-tempered semitone division of the 2:1 octave ratio. That is, the modern semitone forms a frequency ratio of the twelfth root of 2, and all other intervals are powers of this number: i.e., the fifth, with the classic and proper harmonic ratio of 3:2, now is consigned to the ratio of the 12th root of 2 to the seventh power -an irrational number signifying a non-repeating, non-terminating decimal which CAN NOT be represented a ratio of simple whole numbers! I know this is getting technical, but the upshot is that combination tones cannot be produced by irrational intervals –the combination ‘third tones’ have a frequency of zero, so are completely inaudible. I suspect this is why they are seldom heard in modern play –the tuning is irrational, and the players’ ears are trained to reproduce all the other irrational intervals in their performances. Only the really old instruments are apt to convey the true production of ‘undertones’ such as Tartini was hearing. (more found at cywelsian dot blogspot dot com
under the heading Tartini Tones and the Music of the Spheres).