Re: [tuning] Harmonic versus subharmonic

Since the perennial topic of comparing the harmonic 5:6:7:9 with the
subharmonic /7:/6:/5:/4 has recently reappeared, I thought I'd give it a
listening test in the context of the so-called "Tristan" chord, that opens
the _Vorspiel_ to Wagner's opera.

"So-called", because there is no scholarly consensus on whether this thing
is actually a chord, or even which particular combination of tones we're
talking about. Is the chord on the downbeat (f3-h3-d#4-g#4) a proper entity?
Does the ascent of the top line from g#4 to a4 on the last eighth of the
measure constitute a resolution, a new harmony, or is it just a passing tone
towards the dominant seven of the next measure? While functionalists in
the Riemannian tradition (tuning scholar Martin Vogel, who wrote an entire
volume on the "Tristanakkord" comes from this tradition) continue to debate
the proper label for the chord, the majority of scholars probably share the
notion that what we're hearing is not so-much a function chordal entity, but
a byproduct of voice leading. In any case, the vertical combination on the
downbeat of the second full measure of T&I is a combination that maps very
closely onto an inversion of either the harmonic 5:6:7:9 or the subharmonic
/7:/6:/5:/4. Naturally, I was curious to hear which mapping was the better
fit, especially since the technology to make the comparison is now readily
available.

Since you might like to duplicate this at home, I've posted a brief midi
file of the pickup note and first three measures from _Tristan_ to the
tuning list web page. I tuned the chord (using WAVMaker to produces WAVs in
very accurate tuning)as 5:7:9:12, as /7:/5:/4:/3, and in 12tet, using sines,
sampled orchestral instruments, and a harmonium-like timbre of my own
device.

I was surprised by the result. I found that the sampled instruments
introduced so much irregularity that it was impossible to compare the
results, while sine waves and the harmonium timbre definitely favored the
two just intonations. The character of the harmonic and subharmonic chords
were equally interesting (the harmonic was "spectral" in character, with a
akin to using a sweeping band pass filter, the subharmonic was formant-like
with the presence of the common overtone). It's basically a toss-up, but I
did, however, find that I liked the _melodic_ intervals of the suharmonic
version someone better.

Daniel Wolf
Budapest

Regarding Daniel Wolf's discussion of the "Tristan" half-diminished chord:

I think Wagner's particular voicing does give the subharmonic tuning quite a
bit of appeal (if you know how to listen to it) when a nice harmonic timbre
is used. However, in a voicing like G#3-B3-D#4-F4, I find 10:12:15:17 to be
the tuning to which the 12-tET version seems to "imply" -- that is,
10:12:15:17 sounds like a "cleaned-up" version of the 12-tET tuning, while
6:7:9:10 and 1/6:1/5:1/4:2/7 both sound "xenharmonic" to me.