Tetrachordal scales

Mark: The ancients Greeks in about the end of 5th century BCE analysed the
scales of their own music in terms of two tetrachords and a disjunctive
tone.
If one tunes the diatonic tetrachord (E F G A) to 16/15 x 9/8 x 10/9 or
1/1 16/15 6/5 4/3 as Ptolemy proposed about 160 CE and add another
identical
tetrachord transposed up a 3/2 (B C D e), one obtains the Dorian mode,
the principal mode used by the Greeks. The C or Lydian mode of this scale
is our major mode in JI and it does seemingly have dissimilar tetrachords.
However, if you look at the actual sequence of intervals, you will observe
that their are two 16/15 x 9/8 x 10/9 tetrachords and a 9/8 tone in a
cyclically permuted order.

(To get the natural minor mode, use the tetrachord 10/9 x9/8 x 16/15. In
JI, the A (Hypodorian) mode of the 16/15 x 9/8 x 10/9 tetrachord is not the
same as the natural minor scale as it is in Pythagorean or 12-tet.)

In addition to many diatonic tunings, the Greeks had two other principal
genera in the 4th century BCE. The chromatic genus consisted of roughly
two semitones and a minor third ( E F Gb A), e.g. 16/15 x 25/24 x 6/5 (a
later tuning of Eratosthenes), and the enharmonic genus contained two
microtones and a major third, the most harmonious of these tunings being
that of Archytas who defined the genus as 28/27 x 36/35 x 5/4 (E F- Gbb-
A).

Tetrachordally-analyzable scales are found in Islamic, European, and Indian
musics today.

--John

[Me:]
>> I'll grant that equal temperaments are easy to hear in
>> melodic terms

[Carl Lumma:]
> Whoa! Wait a minute? Where did this come from?

[Paul Erlich:]
> I don't know where it came from, but it's true. Even Mathieu
> admits that a 12-tone chromatic scale is melodically smoother
> in 12tET than in an unequal tuning.

It came from Ben Johnston, "Scalar Order as a Compositional
Resource", published in _Perspectives of New Music_, vol 2 # 2,
Spring-Summer 1964, p. 59:

[Johnston:]
"...Octaves are then divided by a scale of smaller intervals. The
two conflicting criteria which condition this are simplicity and
symmetry: that is, a preference for simplicity or consonance of
harmonic pitch ratios, and a preference for dividing melodic
intervals symmetrically, into "equal" smaller intervals.

"Melodically, the pitch dimension is a linear succession of
octaves, each internally subdivided into smaller intervals which
we speak of adding and subtracting. But this view of pitch offers
no explanation of the common harmonic experience of a gradual
scale of consonance and dissonance. Listening harmonically to
pitch intervals we are actually comparing tempos of vibration...


"Harmonic listening is too easy and too basic to be ignored, even
in purely melodic music. Yet melodic preference for equal scale
intervals is also strong. If a scale is derived harmonically, it
must consist of intervals whose melodic sizes differ by what seems
a negligible amount. What seems negligible depends mostly upon
relative sizes but also upon cultural conditioning and upon "how
good an ear" an individual listener has." [end quote]

Johnston then proceeds to demonstrate how he derived a
53-pitch-class 5-limit JI source scale with an approximately
equal division of the octave. Compare the following:

Interval Prime factors Ratio Cents
-------- ------------- ----- -----
Syntonic Comma 3^4 * 5^-1 81/80 21.5
53-eq "step" size 2^(1/53) n/a 22.6
Pythagorean Comma 3^12 531441/524288 23.5

The fact that the 53-eq step size is almost exactly midway
between the two commas means that 53-eq is good at approximating
all the notes in a 5-limit system (that is, ratios with factors
of 3 and 5), or, conversely in Johnston's case, that a 53-tone
5-limit system that divides the octave as evenly as possible will
approximate 53-eq very closely.

This article is a very lucid exposition of the matter stated in
the title. Johnston describes S. S. Stevens's "four kinds of
scales of measurement: nominal, ordinal, interval, and ratio".
He gives examples of musical use of the four as follows:

nominal formal analysis (ABA, ABACABA, etc.)
ordinal dynamics (pp, p, mp, mf, f, ff)
interval melodic use of pitch (i.e. ET)
ratios harmonic use of pitch & tuning by ear

Johnston's writings get very little citation -- they're well worth
reading.

Joseph L. Monzo
monz@juno.com


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