Werckmeister and maximal proximity of intervals

Now that all 39 intervals are worked out in Werckmeister chromatic (sometimes
called Werckmeister III), it is amazing to discover that every interval is 6
cents apart from its closest adjacent intervals. This is the same interval
between the two sizes of Perfect Fifth (696 and 702 cents). There are 5
regions of 78, 84, and 90 cents which are not further divided, but they are 6
cents apart.

With the conservative literature pointing to the 5-cent limitation for the
average musical ear listening melodically, Werckmeister has elegantly
utilized maximal proximity of the intervals by planning them equidistantly by
6 cents. Of course, this is quite buried in the wash of the music. It seems
a different way of perception for the music involved.

Surprised, Johnny Reinhard

This is not really amazing. You can verify that each tuning with
two sizes of fifth has this property.
So Werckmeister didn't have to plan anything besides using pure
and one size of tempered fifths.

What did you mean by "chromatic content of a piece"?

Manuel

On 2/19/02 1:15 PM, "Afmmjr@aol.com" <Afmmjr@aol.com> wrote:

> This seems to make for a well-temperaments in 200-tET. This seems to be the
> set of all Werckmeister intervals, based on the 6 cent increments. Manual, do
> you have examples for Vallotti or Kirnberger? Thanks, Johnny
>
The increment isn't exactly 6 cents though. It's 5.865 cents, which is
between 1/204 and 1/205 octave. Like I've said, 200 and 188 can *map* to
Werckmeister well because of the derived Pythagorean comma being 4 notes
wide. But if you've ever heard 200, the fifth gets a little tedious. And
188, being twice 94, can be a bit alienating. In 212, you have the fifth
from 53 and the 1/53 comma cut into four parts, since 212 = 53 * 4.

If you're using 1024 note per octave based equipment, you have the 1024
fifth which lies between that of 306 and 359... And luckily the Pythagorean
comma is 20 notes wide, making it easily divisible by 4.

On 2/19/02 11:13 AM, "Afmmjr@aol.com" <Afmmjr@aol.com> wrote:

> Now that all 39 intervals are worked out in Werckmeister chromatic (sometimes
> called Werckmeister III), it is amazing to discover that every interval is 6
> cents apart from its closest adjacent intervals.
>

The Pythagorean comma is about 24 cents. Werckmeister cuts that up into
four parts and scatters it. Which means every interval is going to change
in about 6 cent increments. Umm. What do you find so amazing about this?
The discovery that the fluctuation of intervals was gradual?

> This is the same interval between the two sizes of Perfect Fifth (696 and 702
> cents). There are 5 regions of 78, 84, and 90 cents which are not further
> divided, but they are 6 cents apart.
>
This is true of every interval. Given the size of the Pythagorean interval,
I'll show you the deviations in rounded off cents. Well actually since the
deviations are 5.865, 11.73, 17.595 and 23.46, the progression would be 6,
12, 18, 23 if I rounded them off; just so it looks consistent I'll leave it
6, 12, 18, 24.

There are:

2 fifths (0, -6)
3 major seconds (0, -6, -12)
4 major sixths (0, -6, -12, -18)
4 major thirds (0, -6, -12, -18)
4 major sevenths (0, -6, -12, -18)
5 tritones (0, -6, -12, -18, -24)
4 minor seconds (0, +6, +12, +18)
4 minor sixths (0, +6, +12, +18)
4 minor thirds (0, +6, +12, +18)
3 minor sevenths (0, +6, +12)
2 fourths (0, +6)

...and there's your 39.

I'll admit occasionally filling in commutative blanks longhand produces
unthought of results. One instance of this is the corollary I saw in the
relative error theorem in which it not only tells you how many intervals in
one temperament are accurate, but by the same formula, the same number
represents to what multiple of the temperament the exact interval is still
accurate.

But that's just it. It's only a corollary. When you were first talking
about finding out how many intervals there were in W3, I knew it would be
some kind of distribution like this, in 6 cent increments, I just didn't
know exactly how it would be distributed.

> With the conservative literature pointing to the 5-cent limitation for the
> average musical ear listening melodically,
>

SIDE NOTE...!

Okay.

Again I hear a tangential reference to this. I think it's amazing that
people come up with the same cutoff point - 5 cents, which is exactly 240
notes per octave. People from both ends seem to concur on this, the cents
people about how under 5 cents is below the common ear, and the high
temperament people about how over 240 notes an octave smears together.

> Werckmeister has elegantly utilized maximal proximity of the intervals by
> planning them equidistantly by 6 cents.
>
From what I've heard about how people in that era were going back and forth
about how to cut up the Pythagorean comma, into how many different parts,
and where to distribute the off intervals, "planning the intervals
equidistantly" as I said is just a corollary, a symptom of the fact that
you're applying a gradual tempering in the first place.

In other words, in a well temperament, if you cut up the Pythagorean comma
into ANY number of parts and distribute them ANY way you can imagine, the
variation in intervals is always going to be multiples of the fractional
comma.

Werckmeister, like 29, sounds sweet to me, at first.
Werckmeister, like 29, after awhile, gives me an annoying headache.

Marc

--- In tuning@y..., "Orphon Soul, Inc." <tuning@o...> wrote:

> Again I hear a tangential reference to this. I think it's amazing that
> people come up with the same cutoff point - 5 cents, which is exactly 240
> notes per octave. People from both ends seem to concur on this, the cents
> people about how under 5 cents is below the common ear, and the high
> temperament people about how over 240 notes an octave smears together.

Does that mean that 224 and 270 would be good choices as equal temperaments? Maybe you should try them next--they go nicely with the 11-limit temperaments we've dubbed "octoid" and "hemiennealimmal" respectively, though since I don't know how you use these microtemperaments that may not matter to you.

On 2/21/02 2:22 AM, "genewardsmith" <genewardsmith@juno.com> wrote:

> Does that mean that 224 and 270 would be good choices as equal temperaments?

Oh of course. If you're asking, because of the threshold, if anything over
240 won't matter, of course not. Even though the actual chromatic notes
smear, the geometries created by higher temperaments still create
differences which aren't even really that subtle.

Anyone who was playing my early 1990s fretboards observed, that as long as
you feel comfortable with playing in 20 or 30 notes per octave, that things
pretty much feel the same up until the early 3 digits.

The big difference was between 217 and 323. We didn't have any boards in
between and we didn't have the facility to make them at the moment.

Playing the 217 board wasn't really much different than playing the 171,
118, etc, in terms of feeling your way around. But the 323 had this spooky
way of just having that extra level of attention you needed. Which was
almost a Zen thing, as I mentioned which, my friend Ken tapped into and ran
rings around it, very spaced out - so spaced out that he never remembered
playing it.

After the consensus was in that there was a big difference there, we tried
to narrow down, between 217 and 323 where this smearing would kick in. We
did some random tones and slides with sine waves and midi, and it seemed to
be in the mid 240s. The actual turning point we decided would never be
significant. Calling it 244 was a joke because we through experiments and
from raw high temperament theory narrowed it down to "from 240 to 248". At
240 it still sounds like notes, at 248 it doesn't.

But as far as what you asked, yeah, 224 and 270 I might have some notes on.
I think we did some midi retuning to those. Yeah actually a few times. I
only listed the ones I'd worked with enough that I recalled them instantly.

Marc

Marc, it remains amazing to me--and many musicians I've shared this info
with--that there are exact increments of (slightly under) six cents between
each interval available in a well-temperament. Having played Werckmeister
III (the "chromatic") tuning a number of times, and having produced it in
others, I never noticed.

It has been amazing enough to play Werckmeister in an ensemble. It felt like
a 12-note blues. To think, only 12 exact notes and I had 39 intervals? This
was not apparent through these experiences.

And so it seems that the 696 to 702 cents is likely the largest of the
distances between 2 fifths. Does this mean that the 5.865 cent increments
available in Werckmeister are the largest increments? It appears to me that
there is greater distinction between intervals when they have a larger
distance between them.

Thanks for the feedback. Musician and musicologist friends threaten
implications resulting from distributing information that Bach would use up
to 39 intervals with only 12 notes. One implication might be future analyses
of Bach based on 39 interval usage (as opposed to key analysis).

Best, Johnny Reinhard

>Marc, it remains amazing to me--and many musicians I've shared
>this info with--that there are exact increments of (slightly
>under) six cents between each interval available in a
>well-temperament.

Johnny, it is not very hard to understand why it is not
amazing that this is the case with two sizes of fifth.
For the sake of simplicity, let's take the semitone as
generating interval instead of the fifth. This also visits
all tones in the scale and we don't need to take wrapping
around the octave into account.
Say we have two semitones of X and X+C cents, for example
96 and 96+6=102 cents.
Then each interval consists of some number of semitones of
size X and some of size X+C, say nX + m(X+C). Rewrite this
into (n+m)X + mC. Then n+m is the interval class. Then
you see that for a particular interval class n+m is constant
and m=0,1,2,etc. which is 0x6, 1x6, 2x6, etc. cents,
the difference between the interval sizes is a multiple of
6 cents.

Manuel

--- In tuning@y..., Afmmjr@a... wrote:

> And so it seems that the 696 to 702 cents is likely the largest of
the
> distances between 2 fifths.

the largest of the distances . . . what do you mean? in Werckmeister
III, these are the only two sizes of fifth . . .

> Does this mean that the 5.865 cent increments
> available in Werckmeister are the largest increments?

they're the *smallest* differences between different intervals in
Werckmeister III.

confused,
paul

> And so it seems that the 696 to 702 cents is likely the largest of
the
> distances between 2 fifths.

the largest of the distances . . . what do you mean? in Werckmeister
III, these are the only two sizes of fifth . . .

Paul, what I mean is that the distance may be greater for this first well-temperament than it is for later published well-temperaments. 6 cents (rounded off)is likely the largest. 703 and 695 don't happen do they?

Johnny

--- In tuning@y..., Afmmjr@a... wrote:
> > And so it seems that the 696 to 702 cents is likely the largest
of
> the
> > distances between 2 fifths.
>
> the largest of the distances . . . what do you mean? in
Werckmeister
> III, these are the only two sizes of fifth . . .
>
>
> Paul, what I mean is that the distance may be greater for this
>first well-temperament than it is for later published well-
>temperaments.

oh, so that's what you meant!

>6 cents (rounded off)is likely the largest. 703 and 695 don't
>happen do they?

the largest difference is probably in kirnberger ii, where d-a and a-
e are each flattened by 1/2 pythagorean comma, which is 12 cents,
which is the difference between the different sizes of fifth (and
hence everything else) in kirnberger ii . . .