(no subject)

Margo Schulter wrote,

>However one chooses to realize the precise mathematics -- and a
>leading-tone of around 28:27, associated with "superwide" cadential
>major thirds at around 9:7 and major sixths at around 12:7 is one
>possibility -- Marchettus is calling for _narrower_ leading-tones and
>_wider_ major thirds and sixths at cadences, the opposite of 5-limit.

>As a curious aside, I might note for Paul's amusement that a literal
>reading of Marchettus's diesis might not be too far from a 22-tet step
>of ~54.55 cents.

22-tET also has superb approximations of 9:7 and 12:7.

>In a system like that of medieval Continental Europe, based on fifths
>and fourths as primary concords, the major second (9:8) and minor
>seventh (16:9) have some "respectability." The first interval is the
>difference between a 3:2 fifth and a 4:3 fourth, while the second is
>the sum of two pure fourths. Likewise the major ninth (9:4) represents
>the sum of two pure fifths.

>Even as bare intervals, M2 or M9 and m7 are held to be somewhat
>"compatible" by various medieval theorists of polyphony ranging from
>Guido d'Arezzo (c. 1030) to Jacobus of Liege (c. 1325).

I would doubt that these intervals were considered consonant outside of their ability to arise from
perfect fifths and fourths in the chordal contexts you mention below:

>Around 1200,
>when Perotin and other composers begin writing regularly for three or
>four voices, these intervals can actually have a relatively blending
>effect when combined with two fifths or fourths.

>Thus Jacobus much likes the "concordant" effect of the major ninth
>when presented in a sonority with two fifths (e.g. G3-D4-A4, 4:6:9),
>or a minor seventh in a sonority with two fourths (e.g. G3-C4-F4,
>16:12:9). In 13th-century practice, a combination of fifth, fourth,
>and major second is also popular (e.g. G3-C4-D4, 6:8:9, or G3-A3-D4,
>8:9:12).

>Such sonorities, also found in many other world polyphonies based on
>fifths and fourths as primary concords, give Gothic music an element
>of color and contrast. These sonorities have a kind of "energetic
>blend" which fits the overall quintal/quartal flavor but at the same
>time brings into play the considerable excitement and tension of the
>major second or ninth or minor seventh. From Perotin to Machaut, these
>relatively blending 3-limit sonorities (along with the unstable but
>relatively concordant thirds) are an attractive feature of the music.

These chords are the 3-limit analogues of the later practice of stacking thirds to create major and minor
"seventh" and "ninth" chords.

>In fact, Jacobus advocates what might be called an ideal of
>"pancompatibility": _all_ the regular Pyhagorean intervals, except for
>outright "discords" (i.e. m2, M7, A4, d5, m6), may be used pleasingly
>in a wide range of vertical combinations, many of which he
>catalogues. Notably included in these essential combinations are M2,
>m7, and M9.

Later medieval works that feature this kind of technique represent an overall level of dissonance that
was not matched again until perhaps Beethoven.

>Here one might raise the question of why English theorists such as
>Theinred of Dover (13th c.?) and Walter Odington (c. 1300) write of
>thirds at or near ratios of 5:4 and 6:5, while the advocacy of such a
>tuning on the Continent only occurs around the time of Ramos
>(1482). To me, an attractive hypothesis would be that tertian or
>"5-limit" styles of harmony are recorded in regions of 13th-century
>England, but only became common in Continental composition during the
>15th century.

Sounds right to me.

>However, we are fortunate to have direct evidence that a Continental
>theorist around 1300 favoring Pythagorean tuning _was_ quite capable
>of describing 5-limit and 7-limit ratios.

Excellent . . . why the "however"?

>Jacobus of Liege, in the Fourth Book of his _Speculum musicae_
>("MIrror of Music") comparing the various concords, remarks that it
>would be quite possible to build instruments using such interval
>ratios as 5:1 and 7:1, but that these intervals would not be properly
>formed in terms of whole-tones and semitones.

>He goes on to demonstrate what we call the syntonic and septimal
>commas: 5:1 or 80:16 is smaller than the regular major third plus two
>octaves at 81:16 by a factor of 81:80; and 7:1 or 63:9 is smaller than
>the regular minor seventh plus two octaves at 64:9 by a factor of
>64:63.

>The inquiring Jacobus also notes the gap between the classic 3-limit
>concords of the fifth (3:2) and fourth (4:3), and the next "useable"
>superparticular ratio (i.e. n+1:n) of the major second (9:8),
>suggesting that the quite "imperfect" quality of the latter concord
>may reflect this gap. Thus he recognizes that the intervening
>superparticular ratios (5:4, 6:5, 7:6, 8:7) are conceivable in theory,
>but don't fit the tuning system used in practice.

>To me, this approach seems much like that of 20th-century authors who
>observe that the harmonic series indeed includes a seventh partial,
>but that it doesn't fit the tuning of classical European music. Those
>who appreciate the musical value of intervals at or near 7:4 may
>rightly emphasize that such authors are hardly exhausting the
>possibilities. However, the issue is one of style rather than of
>mathematics.

Agreed. Monz and Dale Scott, take note.