Undertone series

The undertone series is a series of notes that results from inverting the intervals of the overtone series. While the overtone series occurs naturally as a result of wave propagation and sound acoustics, musicologists such as Paul Hindemith consider the undertone series to be purely theoretical. [cite book |last=Hindemith |first=Paul |authorlink=Paul Hindemith |others=translated by Authur Mendel |title=The Craft of Musical Composition |origyear=1937 |edition=revised edition |year=1945 |publisher=Associated Music Publishers |location=New York |pages=78 |quote=It seems to me repugnant to good sense to assume a force capable of producing such an inversion. ... [The undertone series] can never have for music the same significance as the overtone series. ... This "undertone series" has no influence on the color of the tone, and lacks the other natural advantages of the overtone series...] It is believed to exist only as an intervallic reflection of its counterpart series. However, some often debate that the undertone series also exists in nature. Harry Partch, for example, argued that the overtone series and the undertone series are equally fundamental, and his concept of Otonality and Utonality is based on this idea. [cite book |last=Partch |first=Harry |authorlink=Harry Partch |title=Genesis of a Music |origyear=1949 |edition=second edition |year=1974 |publisher=Da Capo Press |location=New York |id=ISBN 0-306-80106-X |pages=89 |quote=Under-number tonality, or Utonality ("minor"), is the immutable faculty of ratios, which in turn represent an immutable faculty of the human ear.]

The misconception that the undertone series is purely theoretical rests on the fact that it does not sound simultaneously with its fundamental tone, as the overtone series does. It is, rather, opposite in every way. We produce the overtone series in two ways—either by overblowing a wind instrument, or by dividing a monochord string. If we lightly damp the monochord string at the halfway point, then at 1/3, then ¼, 1/5, etc., we produce the overtone series, which includes the major triad. If we simply do the opposite, we produce the undertone series, i.e. by multiplying. A short bit of string will give a high note. If we can maintain the same tension while plucking twice, 3 times, 4 times that length, etc. the undertone series will unfold downwards, containing the minor triad. Similarly, on a wind instrument, if the holes are equally spaced, each successive hole covered will produce the next note in the undertone series.

Kathleen Schlesinger, in her 1939 book, "The Greek Aulos", pointed out that as the ancient Greek aulos, or reed-blown flute, had holes bored at equal distances, it must have produced a section of the undertone series. To take from undertones number 8 to 16, for instance would produce a scale, one of the original Greek modes, while numbers 9 to 18, another mode, etc. As the secretive Pythagorean "Harmonists" did not explain this, even the contemporary scholar, Aristoxenos, was confused, mixing it up with the other scale system sketched out in Plato, and musicologists have been misled ever since. She showed that this discovery not only cleared up many riddles about the original Greek modes, but indicated that also many—even most—ancient systems around the world must have also been based on this principle.

In 1868, Adolf von Thimus showed that an indication by a 1st century Pythagorean, Nicomachus of Gerasa, taken up by Iamblichus in the 4th century, and then worked out by von Thimus, revealed that Pythagoras already had a diagram that could fill a page with interlocking over- and undertone series [ Adolf von Thimus: "Die Harmonikale Symbolik des Altertums", Verlag der M. DuMont-Schaubergischen Buchhandlung, Köln, 1868] .This was in turn taken up by Hans Kayser [ Hans Kayser: "Akroasis, The Theory of World Harmonics", Plowshare Press, Boston, 1970; also Lehrbuch der Harmonik, Benno Schwabe Verlag, Basel] in the early 1900’s, who built upon its basis, a theory of aesthetics ranging into geometry, architecture, biology, plant and animal forms, etc.

Ernest G. McClain a little later, and building also on others in between, published two books: "The Myth of Invariance: The Origin of the Gods, Mathematics and Music From the Rig Veda to Plato", (1976) [ Ernest G. McClain (1976). "The Myth of Invariance: The Origin of the Gods, Mathematics and Music From the Rig Veda to Plato", Nicolas Hays: Maine, ISBN 978-0892540129.] and "The Pythagorean Plato—Prelude to the Song Itself" (1978), [Ernest G. McClain (1978). "The Pythagorean Plato—Prelude to the Song Itself", Nicolas Hays: Maine, ISBN 978-0892540105.] He applies von Thimus’ diagram to Plato, showing that often when Plato was ostensibly talking about politics, he was actually talking about music, sharing an inside joke with the knowledgeable Pythagoreans he was addressing, so that some of the outrageous laws he was proposing, full of numerical references, were only musical allegories.

Hugo Riemann, the leading German musicologist of the 19th C, was a great champion of the undertone series, although he applied it in extreme ways. Others also tried to apply it at that time, as there were still unsolved riddles in music theory that Rameau had tackled but not solved—and are still not to this day, although few bother about them now.

The only book, apparently, that pulls all these threads together is the very recent one by Graham H. Jackson, "The Spiritual Basis of Musical Harmony" (2006). [ Graham H. Jackson, "The Spiritual Basis of Musical Harmony", George A. Vanderburgh, Shelburne, ON, Canada, 2006, 196 pp.] . He suggests that the over- and undertone series must be seen as a real polarity, representing on the one hand the outer, material world, and on the other, our subjective, inner world. The overtone series has been accepted because it can be explained by materialistic science, while a conviction about the undertone series can only be achieved by taking our subjective experience seriously. For instance, the minor triad is usually heard as sad, or at least pensive, because we habitually hear all chords as based from below. If we really base our feelings on the high "fundamental" of an undertone series, then descending into a minor triad is not felt as melancholy, but rather as overcoming, conquering something. The overtones, by contrast, are then felt as penetrating into us from outside.

With the help of Rudolf Steiner’s work, he traces the history of these two series, as well as the main other system, our present one, created by the circle of fifths, and shows how in hidden form they are balanced out in Bach's harmony, which is still essentially used today, except when pulled apart by some 20th century music, which has different effects, also described.

Comparison to the overtone series

If we consider C as the fundamental, the first five notes that follow are: C (one octave higher), G (perfect fifth higher than previous note), C (perfect fourth higher than previous note), E (major third higher than previous note), and G (minor third higher than previous note).

The pattern occurs in the same manner using the undertone series. Again we will start with C as the fundamental. The first five notes that follow will be: C (one octave lower), F (perfect fifth lower than previous note), C (perfect fourth lower than previous note), A♭ (major third lower than previous note), and F (minor third lower than previous note).

Overtone vs. undertone

If the first five notes of both series are compared, a pattern is seen:

*Overtone series: C C G C E G
*Undertone series: C C F C A♭ F

The undertone series contains the F minor triad. In this way, some have argued that in fact, the minor triad is also implied by the undertone series and also a naturally occurring thing in acoustics. [cite journal | last = Godley | first = Elizabeth | year = 1952 | month = October | title = The Minor Triad | journal = Music and Letters | volume = 33 | issue = 4 | pages = 285–295 | url = http://www.jstor.org/view/00274224/ap020128/02a00010/0 | format = GIF | accessdate = 2006-09-02 | doi = 10.1093/ml/XXXIII.4.285]

ee also

* Mirror writing (music)
* Musical temperament
* Overtone series

References