TUNERS OF THE WORLD, UNITE

I got my notice this week that the Annual Convention of the Piano
Technician's guild will be July 8-12 in Provicence, Rhode Island this
year. The only tuning class they're touting in advance is something called
'Temperament Festival' by someone named Skip Becker. If it is just a
chance to rejoice at one's ability to produce identical 12-tet every time,
any other tuners who are likely to be there and I should get together and
picket it...

Judith Conrad, Clavichord Player (jconrad@sunspot.tiac.net)
Director of Fall River Fipple Fluters
Church Musician at First Congregational Church, Bristol, R. I.
Piano and Harpsichord Tuner-Technician

>In traditional 12-eq harmonic theory, with extended
>chords built upwards in 3rds, if the near-quartertone
>identities of 11 and 13 are assumed to be implied
>by the "flat 5th / sharp 11th" and "flat 13th",
>respectively

I think any good jazz musician would assume otherwise.

>, the rational implications of the chord-
>members of a major chord line up pretty much as
>follows:

>27/16 13th
>13/8 flat 13th

>11/8 sharp 11th
>4/3 11th (21/16 is another possibility)

>19/16 sharp 9th (7/6 is another possibility)*
>9/8 9th
>17/16 flat 9th

>15/8 major 7th
>7/4 minor 7th

>25/16 augmented 5th
>3/2 5th
>11/8 diminished 5th (23/16 ???)

>5/4 3rd

>1/1 root

>* See the discussion about these two particular
>intervals between Paul Erlich and and myself
>re: Hendrix Chord [Tuning Digest #1376], as to
>whether 19 is admissible as a rational implication
>of the "sharp 9th".

I think 19 is much more admissible as a rational implication of the
sharp 9th than 11 or 13 are admissible as rational implications of
anything in 12tET (despite what Schoenberg may have had to say).

Let's analyze a few common jazz chords (which are normally played, of
course, in 12tET):

1. 13flat9 (meaning: root, maj .3rd, min. 7th, min. 9th, maj. 13th; the
perfect 5th is often omitted in jazz)

I think this chord derives its flavor from the major triad formed on the
maj.13th with the min. 9th and maj. 3rd. This triad resonates clearly
and forms a polyharmonic structure against the root. Tuning the triad
27:34:40 would destroy this effect and render the chord pretty ugly
(IMO). Hence in this case the maj. 13th should be 5/3 and the min. 9th
should be 25/12.

2. sharp11flat9 (meaning: root, maj. 3rd, min. 7th, min. 9th, aug. 11th)

Here, there is a strongly resonating dominant seventh chord formed on
the aug. 11th with the min. 7th, min. 9th, and maj. 3rd. If we fix the
maj. 3rd at 5/4, then the aug. 11th should be 10/7, the min. 7th should
be 25/14, and the min. 9th should be 15/7. Of course, the fact that the
min. 7th is also close to 7/4 helps to reinforce the root along with the
maj. 3rd. It is very common in jazz to have a maj. 3rd - min. 7th
tritone imply two different roots, one with their traditional roles and
one with their roles reversed. This requires a tuning, such as 12tET,
where the 50:49 "septimal sixth-tone" vanishes.

3. 6/9 (meaning root, maj. 3rd, perf. 5th. maj. 6th, maj. 9th)

Here, we have a chain of 5ths, but also a major 3rd and two minor
thirds. If all these intervals are to be consonant, no just
interpretation will really do. This chord requires a tuning where the
80:81 syntonic comma vanishes.

Other examples can be pointed out, but I think that's enough to make my
point for now.

By the way, the simplest ET consistent in the 19-limit is 80tET.

>Here, 6/5 doesn't figure as an interval that needs to be
>connected directly because it's not a prime axis.

But I need it to be connected directly because it's a consonant
interval.

>I was merely pointing out to Paul that prime factorization
>figures in his own visual representations of pitch resources.
>If one is prepared to argue that we don't hear prime qualities
>in intervals, how can one find a diagram useful which is
>nothing other than the _visual representation of those qualities_?

The reason I like these lattice digrams is that they allow you to see
all the consonant combinations at a glance. According to John Chalmers,
Erv Wilson also uses higher-dimensional triangular lattices like these.

>I agree with you totally here: to my
>ears, it's difficult if not _impossible_ to isolate _exact_
>prime qualities in a _dyad_. But add some more notes, and it
>becomes fairly easy to compare the qualities heard in comparing
>various intervals.

I would agree that stacking 5 notes in consecutive perfect fifths, one
hears a chord with a great deal of perfect-fifthness to it. The 81/64
itself is rather irrelavant. Maybe I picked too tautological an example.
Got a better one?

>(I'd really like to see more feedback about microtonality in
>the blues.)

I think this is a great topic, but I don't think ratios of large numbers
have much to do with it. I do think that many blues performers do
distinguish between 6/5 and 7/6, however, as well as approaching a
pentatonic scale in 7tET.

>> I think the fact that the tritone in major nearly forms a 4:5:6:7
>> with the dominant, and that the tritones in minor nearly form
>> an 8:10:12:14:17 with the dominant, were not inconsequential
>> for the development of tonal harmony.

>I don't think the fact that the tritones are close to septimal
>consonances has anything to do with the development of
>tonal harmony. Tonal harmony, as used during the so-called
>"common practice" period 1600-1900, developed mainly out of
>the fixing of pitches as absolute values (especially in notation),
>the recognition of 5-limit ratios as consonances, and the intuitive
>realization that because ratios have two terms, they are
>related to other ratios in two different ways (Partch's
>"Basic Monophonic Concept #2": "every ratio of a Monophonic
>system is at least a dual identity"; "Genesis", p. 88), giving
>rise to the major/ minor system.

>In fact, assuming a tuning in 5-limit JI (or a meantone
>which approximates 5-ratios well), the tritone could be one
>of four ratios, all of which, while much less dissonant than
>the Pythagorean tritones:
>3^6 729/512 6.12 aug 4th
>3^-6 1024/729 5.88 dim 5th
>were pretty much at, if not beyond, the limit of what could
>be considered consonant:
>3^2 * 5^-2 36/25 6.31 dim 5th
>3^-2 * 5^-1 64/45 6.10 dim 5th
>3^2 * 5^1 45/32 5.90 aug 4th
>3^-2 * 5^2 25/18 5.69 aug 4th

>The 64/45 would normally be considered the tritone which appears
>in the 5-limit Dominant 7th chord of 36:45:54:64. 4:5:6:7 is the
>same as 36:45:54:63.

This whole argument smacks of what I object to in the prime-limit
theoreticians' approach. Harmonically speaking, an interval is never
more likely to be interpreted as a higher-odd-limit, lower-prime-limit
ratio than a higher-prime-limit, lower odd-limit ratio, holding the
approximation error constant. I think the four 5-prime-limit ratios you
listed have very little to do with the way a tritone is heard
harmonically, even if tuned to exactly those ratios. Although I agree
that the Pythagorean tritones, despite their low prime limit, are very
dissonant, I don't think the JI tritones are "much" more consonant. In
fact, they may even be more dissonant, since the Pythagoean tritones
approximate simple 7-limit ratios better.

Anyway, I maintain that the fact that the tritones reinforced the root
of the dominant chord when combined with it harmonically helped to
define major and minor as the "tonal modes", while other modes without
such harmonic-melodic focus fell out of use.

>> This whole argument smacks of what I object to in the prime-limit
>> theoreticians' approach. Harmonically speaking, an interval is never
>> more likely to be interpreted as a higher-odd-limit, lower-prime-limit
>> ratio than a higher-prime-limit, lower odd-limit ratio, holding the
>> approximation error constant.

[Monzo:]
>I'd be interested in seeing some quantifiable info about this.
>Have there been experiments which prove this statement?
>I think this is an important and overlooked aspect of the prime/ odd
>debate in this forum. Lemme see some numbers.

The experiments that Ken Wauchope and I both did (independently) are one
bit of tangential evidence. The problem is that it's hard to come up
with an experimental test of what ratio is implied, acoustically, by a
given interval. But there are mathematical ways of defining such things,
such as which pair of overtones is beating most slowly, etc., and
anything reasonable along those lines will favor something like an
integer-limit rule. Superimposing octave equivalence then brings you
from interger-limits to odd-limits.

>Well, Paul, the reason you say these things is because you favor
>ETs, where you must be concerned with consistency, approximations,
>etc.

No!

>In real just tuning, the differences are quite audible.

I agree that the differences are audible

>The
>5-limit
>tritones provide a biting dissonance in the "dominant 7th" chord
>that *demand* resolution onto the major or minor triad on the "tonic".

I agree that you wouldn't want the 7-limit approximations to be too
smooth -- that would take away from the need to resolve. However, the
tritones are not, in any relevant harmonic sense, 5-limit.

>A perfectly tuned 4:5:6:7 chord is only a tiny bit less consonant
>than a plain old 4:5:6 triad. There's a big difference in the sound
>and feel from the 3- and 5-limit "dominant 7th"s, provided they are
>in perfect tune also. And we're not talking about the tritone dyad
>by itself, we're talking about a 4-part chord.

Right. I think that the dominant 7th arises because it can be formed
from diatonic scale steps. The diatonic scale admits a wide range of
tunings, and the dominant 7th "works" in virtually all of them. Pinning
its tritone down to some complex ratio has no acoustical relevance. Even
contextual relevance is suspect since the diatonic scale in
common-practice music requires the vanishing of the syntonic comma, so
whatever ratio you give, I can probably give you another one, differing
by 81:80, that has as much contextual meaning.