[Smt-talk] Fwd: Re: a la mode

Ildar Khannanov solfeggio7 at yahoo.com
Tue Dec 3 08:11:51 PST 2013

Dear Nicolas,

again, very categorical, abrupt statement. "It is counterproductive to believe that diatonic may have kept the same sense from  Greek Antiquity to our days" -- I do not know what is counterproductive, and what is not, before trying it.  On the other hand, in  thephrase that you send to me seven words out of 19 are of Greek origin. These are not only the words, translated from Greek (counter, productive, believe, diatonic, same, sense) but a major categories, discovered by Greek thinkers. In 1930s some in German speaking Europe decided to get rid of the History, with the very sad results. History, our Greek heritage, did not go anywhere. Exactly, the attempts to discard historic continuity--are counterproductive. 

Know it all attitude also fails sometimes. We read in Aristoxenus: intervals smaller than diatessaron are dissonant. More precise translation would be "diatonic", because he distinguishes not dissonance and consonance, but diaphona and synphona. The dissonance in this case must have mean the unstable passing tone. The consonances--symphona, are outer tones of the tetrachord, they comprise the centers of stability. What is ultimately important that the fourth, the fifth and the octave can be kept in attention and memory as two notes simultaneously (syn-phona). That is, perhaps, why the fourth, the fifth and the octave are consonant in both melodic and harmonic form. Even when we sing the two notes of the perfect fourth one after another, we can keep both tones together as if they were staked one above another in a harmonic form. The semitone, tone, semiditone and ditone--are unstable passing intervals that are form by the inner tones of the tetrachord.
 They do not stack up in our attention and memory and can be perceived only as diaphona.

As for the mathematical toys, such as a stack of fifth--since Pythagoras some inquisitive scientists are trying to harness music with the rules of high school math. If you believe that diatonic is the product of the stack of fifth, take a tuning fork and tune your piano by perfect fifth. Gook luck! In Greece there has been a controversy (canoncs vs harmonics), but practical musicians (including Aristoxenus and his father) did not adopt Pythagorean views. They were too abstract and impractical. For example, the Greeks did not use the term "fifth"; it is a chimere, introduced by neo-pythagoreans. There is no fifth as a mathematical relationship of two points in space (1 and 5). Dia-pente--is not a combination of two numbers, but the limit to the neighborhood (using a more up-to-date math). The fifth is always filled melodically and the upper note is always reached by wading the waters of diatonic. This phenomenon is retained in our school book definition
 of hidden fifth: even if there is no actual note in soprano, it is implied by the filling of the ascending leap.

It is rather obvious that in C major there are two tetrachords. The outer limits of the lower one are Do and Fa, the outer limits of the upper--Sol-Do. When in singing the lower tetrachord singers (and instrumentalists when playing scales) make a temporary stop at Fa (because there the tetrachord ends and musician can tie the Fa to lower Do). After that, a quantum leap occurs: to go up from Fa to Sol is a big deal! There is a threshold, or, even, a watershed. Once Sol is taken, the next two notes are diatonic passing tones, until we reach upper Do.

The discovery of tonal function by Rameau had two sources: one, obvious, neo-Pythagorean, and another--hidden but much more powerful, in melodic structure of the system that we inherited from the Greeks. Do, Fa, Sol, Do are the outer limits of the two disjoined tetrachords and, at the same time, three tonal functions. The fact that the Subdominant was given an independent name by Rameau (and not Predominant, as the revisionist theories of the 20th century suggest) is related to its status as the upper limit of the lower tetrachord. And, as implied by melodic theory of tonality (I refer to treatises by Yuri Tjulin and Tatiana Bershadskaya), the Fa can easily roll back on Do. 

Not only in ancient Greek tradition, but in many other non-European traditions the principle of the fixed outer limits and flexible inner tones exists. Maqam and raga both contain fixed steps and spinners. It would be counterproductive to refuse to see it.

Best wishes,

Ildar Khannanov
Peabody Conservatory
solfeggio7 at yahoo.com

On Tuesday, December 3, 2013 8:54 AM, Nicolas Meeùs <nicolas.meeus at scarlet.be> wrote:

Le 2013-12-02 18:24, Ildar Khannanov a écrit : 
Tetrachord, pentachord, heptachord--do not imply dia--. Tetrachord--four strings, nothing else. Trichord--three strings. Where is the dia--? And what does this dia- in this case mean? An easy question? Hmmm... 

The Greek name of the perfect four (tetrachord) is "diatessaron",
      i.e. "through four degrees"; and the perfect fifth is "diapente",
      "through five degrees". The Greek (as many other early theorists)
      were particularly interested in describing the fourth (the
      tetrachord) because it has the capacity of describing the whole
      scale – the "system", the systema teleion, as they named it. Indeed, the system can be described as a concatenation of alternatively conjunct and disjunct tetrachords, much as we describe it as a concatenation of disjunct octaves.

Any scale formed of diatonic tetrachords is diatonic; but the
      tetrachord itself may be STT, or TST, or TTS, and tetrachords may
      be either disjunct or conjunct. This gives a choice of six
      diatonic octave scales. Fa–Sol–La–Ti (TTT) is not a valid
      tetrachord; as a result neither Fa–Sol–La–Ti | Do–Re–Mi–Fa nor
      Do–Re–Mi–Fa | Fa–Sol–La–Ti (that you mentioned) are diatonic
      properly speaking, in Greek terms at least.

It is counterproductive to believe that "diatonic" may have kept
      the same sense from Greek Antiquity to our days. Another
      definition, proposed by François-Auguste Gevaert among others, is
      based on the cycle of fifths: any scale the degrees of which can
      be joined by a cycle of at most 6 perfect fifths (i.e. at most 7
      degrees) is diatonic; any scale that needs between 7 and 11 fifths
      is chromatic; and any scale of more than 11 fifths is enharmonic.
      This definition is not concerned with the number of degrees in the
      scale, but only with the number of steps in the cycle of fifths
      necessary to produce it: the anhemitonic pentatonic scale is
      "diatonic" (less than 6 steps), the harmonic major and minor are
      "chromatic" (more than 6 steps, but less than 11), etc. This
      definition may seem questionable, but it is explicit and

Tonality is a phenomenon of utterly different nature.

Nicolas Meeùs
University Paris-Sorbonne

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