NU not TU, and unequal tones in 1/6-comma

Using 'TU', every temperament you try to write down involves a lot of
large numbers - but most of the digits involved are musically
insignificant in almost all cases. For Werckmeister, the interval C-G
could be "1^" meaning 'make it a bit flat'. In TU this would become
-180 --- 180 of what? Well, 180 of something many times smaller than
you could possibly hear. The units could be 10 times larger without
losing anything audible, which would make it easier both to write down
(lose all those zeros!) and to read. TU are hyperinflation.

12 of these large numbers representing tiny intervals go into a
circulating temperament, leading to a forest of zeros as here:
www-personal.umich.edu/~bpl/larips/comparison.html
where I can't see anything which corresponds to the musical effect of
the tunings - apart from, possibly, whether they have wolves or not.

Since Margo's synthesizer uses units of 1/1024 octave, I think this is
a logical choice of unit for her to calculate in - 'Yamaha Unit'? It
has the nice feature that 20 of them are almost exactly a Pythagorean
comma. Practically, your expected error on any desired interval is
about 1/2000 octave; in order to avoid rounding errors you can
calculate down to 0.1 YU.

The formula for a 7/6 is just 1024*log(7/6)/log(2), which gives 227.7
YU. You can choose between 227 and 228 synth steps for the pure
interval. If one chooses 599 steps as a 'pure' fifth, three of them
give a third of 251 YU, thus the total widening of the fifths for a
pure 7/6 comes to 24 or 23 synth steps. Margo's choice of 5+6+6
widening gives a total of 17, which is an error of 6.3 steps or about
1/3 comma.

Now if you *really* want to do theoretical work - i.e. to be
mathematically exact - why not use the exact values of intervals?
Units always imply some approximation. Or at least use units which
allow you to write down the sizes of intervals: TU are only intended
to measure the deviation from some pure 5-limit interval, which is
very restrictive for general theory.

Anyway, about unequal tones. Here's how a (hypothetical)
1/6-comma-like meantone harpsichord tuning can lead to a melody
instrument using not only unequal semitones, but unequal tones - as
implied by Galliard, if we think this feature of his theory was
intended to have acoustic application.

Each fifth of the harpsichord is tempered by 2 'Neidhardt units'
(actually 2 cents each). Then the fifths are about 349 NU, a pure
fifth being 351 NU. A pure major third is then 193 NU. To complete our
list of consonances, the minor third is 158 NU, giving sixths of 407
and 442 NU.
The major scale of the harpsichord is:
C D98 E196 F251 G349 A447 B545 C600

'Neidhardt units' are the most economical way to write down these
interval sizes (and those of most circulating temperaments) to the
desired accuracy.

Now consider a bass viol which plays exactly in tune with this scale,
and a violinist who makes pure intervals above the bass viol.
(Consider the similarity in timbre and sustain.)
Suppose that the viol plays the bass line in unison with the harpsichord:
C B545 C A447 E196 D98 G349 C

and the violinist plays the scale C D E F G A B C above this bass.
What pitches will the violinist ideally hit? I find the violin to play
as follows:
C D103 E193 F254 G354 A449 B542 C600

The tones and semitones are then:
C-D 103
D-E 90
E-F 61
F-G 100
G-A 95
A-B 93
B-C 58
Compare these to 9/8 (102 NU), 10/9 (91 TU), and 16/15 (56 NU).
Even with a quarter-comma (3 NU) detuning fudge factor, the tones are
definitely unequal!

With a 1/6-comma meantone tuned harpsichord, if the sustaining
instruments strive to be as in-tune as possible (the viol in unison
with the harpsichord, and the violin with the viol), u n e q u a l
tones naturally occur in melody. Where does this leave the
Chesnut/Haynes/Duffin/Lehman approved idea that 1/6-comma meantone was
a standard of 'melodic purity' - whatever that means? It is precisely
in the melody that purity of intonation leads to significant
deviations from meantone.

It would be interesting to repeat this exercise assuming various
different regular harpsichord tunings and various different bass lines...
~~~T~~~

Hi Tom,

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@...> wrote:

> Since Margo's synthesizer uses units of 1/1024 octave,
> I think this is a logical choice of unit for her to
> calculate in - 'Yamaha Unit'?

If you just came up with that on your own, it's an
interesting coincidence, because i've had this webpage
online for several years, right down to the same
abbreviation you used:

http://tonalsoft.com/enc/y/yamaha-unit.aspx

The whole point of TU (at least as i know it, which
is 1/720 pythagorean-comma) is to provide integers
which can describe the size of various different
temperaments. It was not designed to represent JI
intervals.

The TU was designed to represent 5ths narrowed by
some fraction of either the syntonic or pythagorean
comma. If that fraction is divisible by any
combination of factors contained in 2^4 * 3^2 * 5
then it will give an exact value.

But certainly i think it makes sense to use the
tuning resolution of your hardware to calculate
your tunings. Back in the early 1990s i used a
lot of Yamaha equipment, and calculated tons of
tunings in yus.

Another important logarithmic division is 768,
because this is the resolution of most other MIDI
hardware. A unit of 768-edo has been dubbed the
"hexamu":

http://tonalsoft.com/enc/number/6mu.aspx

However, i have found by my own experiments that
the hardware may divide the octave into either 768-edo,
or a 768-note subset of 1200-edo (i.e., cents resolution).

Aside from that consideration, my vote for nice
integer values to represent tunings still goes
to the tina:

http://tonalsoft.com/enc/t/tina.aspx

I admit that tinas are not perfect for temperaments,
the integer values approximating some very well
but others only so-so. But it's a fantastic unit
of measurement for 31-limit JI, which covers an
awful lot of ratios.

-monz

email: joemonz(AT)yahoo.com
http://tonalsoft.com/support/tonescape/help/tonescape-overview.aspx
Tonescape microtonal music software

Hi Tom,

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@...> wrote:

> Each fifth of the harpsichord is tempered by
> 2 'Neidhardt units' (actually 2 cents each).

Note that there is already a unit of measurement
of that size, and it already has two different
names (not including NU), "centitone" or "iring":

http://tonalsoft.com/enc/c/centitone.aspx

Also note that there are two other units which
are close in size to this: the "sk" of 612-edo
and the "grad", which Werckmeister himself used
and which is 1/12 pythagorean-comma.

You can find all of these listed in my webpage:

http://tonalsoft.com/enc/u/unit-of-interval-measurement.aspx

And Manuel also has a page like this:

http://www.xs4all.nl/~huygensf/doc/measures.html

-monz

email: joemonz(AT)yahoo.com
http://tonalsoft.com/support/tonescape/help/tonescape-overview.aspx
Tonescape microtonal music software

(...)
> With a 1/6-comma meantone tuned harpsichord, if the sustaining
> instruments strive to be as in-tune as possible (the viol in unison
> with the harpsichord, and the violin with the viol), u n e q u a l
> tones naturally occur in melody. Where does this leave the
> Chesnut/Haynes/Duffin/Lehman approved idea that 1/6-comma meantone was
> a standard of 'melodic purity' - whatever that means? It is precisely
> in the melody that purity of intonation leads to significant
> deviations from meantone.

Oh, come on. Your example imposes a double standard upon different
members of your hypothetical ensemble. You're constraining your bass
viol player to hit exact unisons with the harpsichord. That's one
standard. But, you're compelling your violinist to IGNORE everything
the harpsichordist's right hand is doing with chords, melodies, and/or
any other doubling...instead of trying to match pitch with any of the
exemplary notes therein. That's a second standard for your violinist.

Why not just flip it around, equally arbitrarily, and compel your
violinist to play as well "in tune" as possible with the right-hand
chords he's hearing from the harpsichord, and your bass viol player to
supply whatever exact bass pitch sounds best under this?

Why not have both your string players adhere to the same standard as
one another...either both ignoring the harpsichord, or both striving to
match it?

And what about the violinist's line having melodic integrity all by
itself, with evenly spaced steps, as at least some violinists would try
to do if they're not obsessed with creating momentary just-intonation
5:4, 4:3, 3:2, or 6:5 above some bass?

Your "purity of intonation" phrase here clearly means TO YOU that the
guy should be striving for 5:4, 4:3, 3:2, 6:5, etc over wherever the
bass notes happen to be. But, what if "purity of intonation" meant TO
THOSE MUSICIANS that they're hitting the marks of the system standard
(in this case 55-ET) dead-on, and ignoring such opportunities to do
what you would wishfully compel them to do with vertical beatless
consonances?

=====

Your problem gets worse if you go to other species of counterpoint,
instead of note-against-note as in your example. Try the species where
your two lines are supposed to move in syncopation against one another,
taking turns with motion. Have your bass start on C, and have the
treble leap from E to A and hold the A. Then, have your bass move to D
while that A is still being held; and over that D the treble comes down
to G. Under that G the bass now moves to E...and then have your treble
leap up to C. Is that C supposed to be anywhere near 2:1 from the bass
line's initial C, or have their micro-adjustments for just intonation
already screwed up things too badly in merely three bars of music? If
these players aren't leaping to hit marks (with or without any doubling
from the harpsichord or any other fixed standard), and not changing
their own pitch slightly while the other player moves, either, how do
you reconcile this fantasy?

This same latter example, with the bass line playing Do-Re-Mi and the
treble playing Mi-La-Sol-Do, reckoned in frequencies:

Give your bass Do an arbitrary starting frequency of 144 (easily
divisible by lots of things). The treble's Mi, if he's doing a 5:4, is
then 180; and he leaps to La at 240 to make a pure 5:3 over the Do.
Under that 240 the bass moves to Re at 160, so it's 3:2. Over the 160
the treble builds his Sol at 4:3 or 213.3_. The bass player makes a
6:5 Mi-Sol here, so he has to do 177.7_. (Hey, this is already lower
than the 180 that the other guy just played!) And over the 177.7_, the
treble builds an 8:5 up to a new Do at 284.4_. And already within
these three bars of music, the old Do was 144 and the new Do has sunk
to 284.4_ instead of 288, which would have been a pure 2:1. Why should
music this simple need to have (already) two different pitches for Mi,
and two for Do?

All this junk goes away if they'd simply hit the PURE positions Do/Ut,
Re, Mi, Fa, Sol, and La that the regular system has prescribed for
them, where all four of the Ut-Re-Mi and Fa-Sol-La tones are the same
size as one another. Hence, its usefulness not only for pedagogy but
also for practice. And Fa-Ut, Ut-Sol, Sol-Re, Re-La, and La-Mi are all
the same size as one another, too: slightly narrow from 3:2. And Ut-Mi
and Fa-La are the same size as one another: slightly wider than 5:4.
Those "impurities" (in a different sense of "purity"!) are a small
price to pay, in the trade-off that the musical passage can be as long
as one likes and still end up at the same pitch level where it
started.

All those pitches happen to match your (hypothetical) "1/6-comma
meantone tuned harpsichord" exactly; fancy that. Both the bass violist
and violinist could stay right on pitch by simply hitting all the
harpsichord's exemplary notes, and adhering to the same intonation
standard as one another, too. They're as in-tune as possible with the
p u r i t y o f t h e s t a n d a r d .

Brad Lehman