Re: [tuning] Re: tuning system of the 65 Zeng bells from 433 B.C.

Carl:

> Martin wrote...
> >> No, the probability is actually very high. Without a section of
> >> an approximately harmonic series we could not hear pitch.
>
> Actually, I'm not sure this is true. Check out this site with
> me (actually, the like may have come from one of these lists)...
>
> http://www.hibberts.co.uk/index.htm

Thanks, Carl. This is a good site. I did not see anything, however, that
would question conventional wisdom on the pitch of bells. Bill Hibberts
refers to established psychoacoustics on this issue.

The pitch of inharmonic sounds has been a favorite in psychoacoustic
research
for a long time. The subject sounds good, and there are many figures in it
;-)

As a rule of the thumb, one can say that slight inharmonicity does not knock
out pitch perception, but strong inharmonicity does. So we have bells with
one pitch, with multiple pitch, or with no pitch. All can be good bells, but
in recent centuries the demand in European church bells has focused on
harmony and one-pitch bells. The bell casters had learned to make bells with
a sufficiently harmonic section in the series of partials.

Martin

>As a rule of the thumb, one can say that slight inharmonicity does
>not knock out pitch perception, but strong inharmonicity does.

In the strong case, if there is one partial stronger than the others,
doesn't this partial tend to get heard as the pitch of the bell?

-Carl

--- In tuning@yahoogroups.com, "Martin Braun" <nombraun@t...> wrote:

> bells, but
> in recent centuries the demand in European church bells has focused
on
> harmony and one-pitch bells.

they don't sound like one-pitch bells to me (as would be expected
since only *some* of the partials fall into a single harmonic
series), i definitely hear more than one pitch in each bell, and as a
result most multi-voice music not specifically arranged for carillon
sounds awful to me on carillons. but not as awful, surely, as if the
partials had not been carefully tuned, in which case you would hear
even more pitches, second-order beating, etc.

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >As a rule of the thumb, one can say that slight inharmonicity does
> >not knock out pitch perception, but strong inharmonicity does.
>
> In the strong case, if there is one partial stronger than the
others,
> doesn't this partial tend to get heard as the pitch of the bell?
>
> -Carl

not really, one of the main pitches of the bell (perhaps the one
treated as "the" pitch for registration purposes? i don't know), the
so-called "hum tone", does not correspond to a physically present
partial at all, but rather is the implied fundamental corresponding
to several harmonically-related partials.

yes, i consider the carillon bell to be "strongly inharmonic" -- but
if you rule out the presence of even *sections* of harmonically-
related partials (as you have in the carillon bell), then strong
inharmonicity leads to a significant lack of pitch clarity, and a
perception instead as "noise". witness drums and cymbals. while it's
sort of possible to tune a membrane to a certain pitch (typically the
fundamental mode of vibration is used), the perception of pitch is so
weak that almost no drummer ever bothers tuning to the key that the
bass, guitar, keyboards, etc. are playing in. it's basically "noise",
not "pitch".

then again, if you're talking about a bell-like instrument (say a
tuning fork) where the vast majority of the energy is in a single
partial (the lowest one), then yes, this partial will be heard as the
pitch of the instrument. but most western idiophones are designed
carefully, their series of partials being bent, ideally, all the way
toward harmonic-series locations, to produce the clear-pitch effect
martin mentions. the sound of a single partial carrying an
instrument's pitch is the sound of a sine wave, which is considered
quite "unmusical" compared to sounds with nice harmonic overtone
content.

which effect, by the way, is pretty much the whole underpinning of
harmonic entropy.

>then again, if you're talking about a bell-like instrument (say a
>tuning fork) where the vast majority of the energy is in a single
>partial (the lowest one), then yes, this partial will be heard as the
>pitch of the instrument. but most western idiophones are designed
>carefully, their series of partials being bent, ideally, all the way
>toward harmonic-series locations, to produce the clear-pitch effect
>martin mentions. the sound of a single partial carrying an
>instrument's pitch is the sound of a sine wave, which is considered
>quite "unmusical" compared to sounds with nice harmonic overtone
>content.

I was talking about arbitrary scraps of metal.

-Carl

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

>
> I was talking about arbitrary scraps of metal.
>
> -Carl

that falls under the "drums and cymbals" part of my explanation,
especially cymbals . . . can you determine the pitch of each of jon
fishman's cymbals?

>> I was talking about arbitrary scraps of metal.
>>
>> -Carl
>
>that falls under the "drums and cymbals" part of my explanation,
>especially cymbals . . .

How 'strongly' inharmonic would you call a series of partials
at 210, 310, 410 and 510 hz.?

>can you determine the pitch of each of jon fishman's cymbals?

No, but cymbals are too high freq. to have a pitch, I think.

-Carl

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> No, but cymbals are too high freq. to have a pitch, I think.

You should come to my garage. I could pull at least 4 or 5 from the rack that you can definitely discern a core pitch; when choosing a good match for a pair of symphonic crash cymbals, they are paired anywhere from a major 2nd to a minor third apart in pitch. This allows for a full bloom in the sound of the crash, and does not happen if they are pitched too close together.

On the other hand, for many musical situations it is *not* recommended to use a cymbal with an identifiable pitch, but to pick one that has many frequencies present in a broad sprectrum.

This is just what are 'commonly' known as cymbals. There are also crotales, which are sometimes known as "antique cymbals" (as Debussy used them in "Prelude to the Afternoon of a Faun", in E and B) They are made from the same metal formulation as traditional 'crash' cymbals, but are cast in thicker and smaller plates, and can be tuned to a very exacting pitch. This is why they are sold in one- and two-octave sets.

[the preceding "factual and informative" data was supplied by Jon 'Positive Contribution' Szanto]

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> I was talking about arbitrary scraps of metal.
> >>
> >> -Carl
> >
> >that falls under the "drums and cymbals" part of my explanation,
> >especially cymbals . . .
>
> How 'strongly' inharmonic would you call a series of partials
> at 210, 310, 410 and 510 hz.?

not very inharmonic at all -- the virtual pitch is about 103 hz
(frequency of a sine wave that seems to be at the same pitch).
>
> >can you determine the pitch of each of jon fishman's cymbals?
>
> No, but cymbals are too high freq. to have a pitch, I think.

listen to them slowed down and tell me what you think.

>> How 'strongly' inharmonic would you call a series of partials
>> at 210, 310, 410 and 510 hz.?
>
>not very inharmonic at all -- the virtual pitch is about 103 hz
>(frequency of a sine wave that seems to be at the same pitch).

How'd you calculate that?

-Carl

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> How 'strongly' inharmonic would you call a series of partials
> >> at 210, 310, 410 and 510 hz.?
> >
> >not very inharmonic at all -- the virtual pitch is about 103 hz
> >(frequency of a sine wave that seems to be at the same pitch).
>
> How'd you calculate that?
>
> -Carl

i've seen a lot of data on experiments like these.

210 is the second harmonic of 105, and 510 is the fifth harmonic of
102. the other "implied fundamentals" are in this range. so whatever
kind of weighted average you take, you'll predict a fundamental
around 102 to 103, and you'll be about right!

when i get to work, i can record a few examples comparing this sound
to various sine waves and we can decide which matches the pitch best.

Paul:

> > in recent centuries the demand in European church bells has focused
> > on harmony and one-pitch bells.
>
> they don't sound like one-pitch bells to me (as would be expected
> since only *some* of the partials fall into a single harmonic
> series), i definitely hear more than one pitch in each bell, .....

Then you hear more than most of us. In psychoacoustics, people call this
analytic hearing. Like most abilities in hearing, it is probably partly
innate and partly acquired.

Do you hear multiple pitch per bell strike in the sound sample of my report?
And how would you compare the sounds with those of carillons?

Martin

--- In tuning@yahoogroups.com, "Martin Braun" <nombraun@t...> wrote:
> Paul:
>
> > > in recent centuries the demand in European church bells has
focused
> > > on harmony and one-pitch bells.
> >
> > they don't sound like one-pitch bells to me (as would be expected
> > since only *some* of the partials fall into a single harmonic
> > series), i definitely hear more than one pitch in each bell,
.....
>
> Then you hear more than most of us. In psychoacoustics, people call
this
> analytic hearing. Like most abilities in hearing, it is probably
partly
> innate and partly acquired.

no, i'm talking about two *virtual* pitches, not two *spectral*
pitches.

> Do you hear multiple pitch per bell strike in the sound sample of
my report?

please refresh my memory -- url?

Paul:

> > Do you hear multiple pitch per bell strike in the sound sample of
> > my report?
>
> please refresh my memory -- url?

http://w1.570.telia.com/~u57011259/pics/bells3.mp3

Martin

--- In tuning@yahoogroups.com, "Martin Braun" <nombraun@t...> wrote:
> Paul:
>
> > > Do you hear multiple pitch per bell strike in the sound sample
of
> > > my report?
> >
> > please refresh my memory -- url?
>
> http://w1.570.telia.com/~u57011259/pics/bells3.mp3
>
> Martin

i think so, but it's kind of hard to know exactly what the bells are
doing, given all the other instruments in this (beautiful) sample.

Paul asked, on June 19:

"do you have any data on the series of partials for each, or any, of the
bell tones? such data would have some bearing on the tuning issue, since
certain tunings sound better
with certain series of partials than others."

I could now run spectral analyses of the two tones from each of six of the
original bells.

Thanks to Carl for pointing to the site of Bill Hibbert, and thanks to Gabor
for recommending Bill's "Bell Waveform Analysis Program: Wavanal". It is
excellent, the same as Bill's site.

Spectrograms and sound samples for one bell are now published in detail. It
turned out that the spectral charateristics of these bell differ strikingly
from those of church bells and carillons. Some of the implications of these
differences are also discussed.

http://w1.570.telia.com/~u57011259/Bellspectra.htm

Now, what would these results mean for the art and science of tuning?

Martin

Paul, you wrote:

> "do you have any data on the series of partials for each, or any, of the
> bell tones? such data would have some bearing on the tuning issue, since
> certain tunings sound better with certain series of partials than others."

So which "bearing" do the spectral data of the bells have "on the tuning
issue"? I think I have now provided all information that you asked for. Or
do I see this wrong?

http://w1.570.telia.com/~u57011259/Bellspectra.htm

Martin

Dave Keenan wrote (July 22):

> I also believe you when you say that the inscriptions do not explain
> the tuning.

Dear Dave -

That's good. In exchange I can say that I follow, without further checking,
your analysis of the meaning of the occasional extra characters that are
added to the tone names on the bells. Your result that C ("gong") probably
is the principal tone name of the tone-name system that was applied in the
bell inscriptions agrees with what I read in the literature.

I am glad that you agree that this fact need not be relevant for the tuning
of the bells. One can have a bell ensemble with D as the main tone, and F as
the norm tone, in a C-based name system. It is like having a clarinet in Bb,
tuned to a norm tone A, playing in a C-based name system.

Perhaps we can come to an agreement as to your other questions as well.
1) Is there a standard scale?
2) If yes, is there a main tone of the scale?
3) Is there a norm tone?
4) Is the tuning fifth/fourth based or octave based?
5) Is one of the three sections of the 33 melody bells (10+12+11) a
multi-octave instrument (2.5 or 3 octaves), or is it an additive
single-octave instrument.

1) Is there a standard scale?
DEFGAC is repeated 8 times. No other possibility comes close to this.

2) If yes, is there a main tone of the scale?
For physical and physiological reasons, the lowest tone of a scale has a
natural dominance. It is D in this case.

3) Is there a norm tone?
If a fixed standard tone is used to tune the instruments of an ensemble, the
least tuning variance in the ensemble must occur at this tone. It is F in
this case. Because the small variance at F is also statistically
significant, on has to conclude that a norm tone was most probably used and
that it was F.

Additional evidence on issues (2) and (3):
- The interval D-F clearly is the one that is most precisely tuned (see
table B).
- D is the middle of CGDAE. The "middle" (zhong) is one of the main terms in
Chinese thinking. It is the fifth geographical direction, and it appears in
one of the two characters for the Chinese name of China (zhong guo), which
means "land (or realm) of the middle".

4) Is the tuning fifth/fourth based or octave based?
A fifth/fourth based tuning, also called Pythagorean, is easy to detect
because of the terrible thirds (408 and 294 Cent). There is no trace of it
in the bell data. The tables show that the tuning is octave based and that
within the octaves the tuning focus is on thirds.

5) Is one of the three sections of the 33 melody bells (10+12+11) a
multi-octave instrument (2.5 or 3 octaves), or is it an additive
single-octave instrument.
Acoustically, bells are not useful for fast, dancing, or party music, where
players could be permitted to jump around between the bells. But they are
very useful for serious music, where players have to stay put at their
position. In Gamelan music players sit, and they do not rock around to reach
far-off keys. As I saw in film material of a test performance with the
original bells, one non-jumping player can best handle three of the melody
bells, which in most cases amounts to one octave. Also note that the bells
have to be struck far more precisely than Gamelan keys. If one player can
best handle an octave section of an instrument, the instrument is an
additive single-octave one. This is what we see with Gamelan instruments in
Indonesia and perhaps also with marimbas in Central America.

Martin

--- In tuning@yahoogroups.com, "Martin Braun" <nombraun@t...> wrote:
...
> Perhaps we can come to an agreement as to your other questions as well.
...

Dear Martin,

I would prefer if you would respond to all the actual questions I
asked and points I made in
/tuning/topicId_5844.html#45708
rather than selecting a few and attributing to me some others that I
did not ask.

Regards,
-- Dave Keenan

Dear Dave,

I had summed up your questions, because multiple interleaving becomes
difficult to read. But if you like it, here it is:

My replies are marked by ***......***:

>--- In tuning@yahoogroups.com, "Martin Braun" <nombraun@t...> wrote:
>David Keenan wrote (July 18):
> > 2. Even if they realise that you only mean "tempered by the same
>amount",
> > this is not true. You are using pitches averaged over all octaves of all
> > sets, while failing to take into account the fact that the octaves are
> > stretched on average by 13 cents.
>
>Octave stretching exists only in some cases. It is in no way as consistent
>as
>in Gamelan tuning.

Let's face it, nothing is very consistent in the tuning of these bells,
however you seem quite happy to find significance in mean fifth sizes, why
not mean octave sizes?

*** Too small and too inconsistent to be relevant to the examined
questions.***

23 of the octave intervals are wider than 1200
cents, while 10 are narrower. The mean size is 1213 cents.

> Therefore it seems appropriate to average across
>octaves.
>Further, octave stretching would not affect the size of the fifths in
>question.

Of course it will affect them, and does. You have calculated mean sizes for
the fifths in FCGDAE by first reducing all the pitches to the same octave
while assuming 1200 cent octaves, then finding the mean pitches, then
finding the fifth sizes between those mean pitches.

I have not made any assumptions at all about the octaves, but have found
the sizes of the actual fifths (not fourths or twelfths) between the actual
pitches (not mean pitches, and only when those pitches are in the same
set), and then I have found the mean size of all the F:C fifths and the
mean size of all the C:G fifths etc. The results are quite different from
yours which I find quite misleading.

F 696 C 700 G 705 D 704 A 706 E

The habit of assuming 1200 cent octave equivalence is a hard one to break,
but we must do so when analysing a tuning from scratch like this.

When I do the same with fourths I get

E 504 A 505 D 503 G 511 C 515 F

> > These stretched octaves also render invalid your claims about F being a
> > "norm tone" or tuning standard. When you do not assume any intention to
> > tune octaves to a precise 1:2 ratio and compare, between sets, only the
> > same note in the same octave, you find that there is no clear winner for
> > the title of "norm tone". Of those pitches which occur in all three
sets,
> > G4 has the lowest variance.
>
>Well, I did assume the intention to tune 1:2 octaves, because they had
>large
>zithers, which yield very precise octaves.

I don't see what that has got to do with it. Perhaps they did not _want_
1:2 octaves for the bells. Perhaps they found that the bells sound better
if you tune the octaves a little wider than the octaves obtained using
string harmonics, as is the case with gamelan. Indonesians have stringed
instruments too but choose to widen the octaves for their metallophones.

But as I pointed out, there is no need to assume any particular intended
octave size, just don't octave reduce anything before looking at means and
variances.

> > Given the names inscribed on the bells, it seems likely to me that any
> > standard tuning note (norm tone), if it exists, would be one we've
> > translated as C, G, D, or A since these are the names that do not
>consist
> > of a suffix added to the name of another note.
>
>Other scholars were convinced that C was the main tone of the system,
>relying on music theoretical fragments.

Now that you have sent me the scanned tables showing the Chinese characters
for the notes, it is also reasonably obvious to me that C is the main tone.
i.e. it is the most common tonic (or ending note). This doesn't necessarily
make it the tuning standard. But that would seem most likely, if there
_was_ a tuning standard. But I see no evidence for a tuning standard in the
pitch data.

The Chinese pitch names are not as simple as you have indicated in your
article. There are 3 other characters which apparently indicate the
register or octave in terms of the physical size of the bell. Two of them
mean "small" (the TF character) and "smaller than small" (the penguin
skiing character) which seem to be used for progressively higher octaves
(but not in the same pitch class), and the other apparently means
"opposite" (the square root of X-bar character) and is less clear. These
are used in conjunction with the single characters for CGDA. They are not
used with F, or any other note name that already has two characters, with
one exception. This is presumably because 3 characters would be
unacceptable. 3 character note names do not occur anywhere in the scanned
tables.

The exception is E. Even though the chinese name for E already has two
characters, meaning "major-third from C", in four cases the octave
modifiers are added to it and the character referring to C is dropped. So
they effectively read "octave above major third" and "two-octvaes above
major third". But an octave above a major third above what? I figure the
"C" can only be omitted because it is the tonic.

The other thing is that the character for C literally translates as
"palace". The palace is the centre of the kingdom, it sets the standards
(makes the laws), all roads lead there.

On that basis, I suspect the primary pentatonic mode was CDEGAC.

I'm unsure about some of the other characters, because the printing or
scanning is unclear. It would be good to have actual photos of the
inscriptions. But I can't find anything among their possible literal
meanings that is even remotely as suggestive of a tonic as is "palace",
except perhaps "to weave or knit" for G, although the zhi character with
that meaning was not the best match. The one with the "fine, delicate"
meaning looks closer.

I have found the following possibilities at
http://www.zhongwen.com/

C gong palace
G zhi to weave or knit, lance+sound, watchtower, fine, delicate
D shang discuss, trade, business, commerce, merchant
A yu feather, wing

> It is very common, however, that
>names and theory differ from practice (just think of the "western" seven
>letters for 12 tones).

I don't understand your point here. Clearly the main tone in western
tunings _is_ one that is named with a single character with no modifiers
(sharp or flat), as the Chinese name for F does not.

> I tried to write down musical practice as
>visible in
>the data.

This is a laudable aim. But I hope you understand now that you have made an
unwarranted and unnecessary assumption and so your calculations are not
valid.

> > By the way, you should have translated gong zeng, zhi zeng, shang
>zeng as
> > Ab, Eb and Bb, not G# D# and A#, so that all the nominal(in chinese)
>thirds
> > would also be spelled as such in western notation (i.e. as thirds, not
> > diminished fourths).
>
>"Diminished fourths" only occur in the theory of the so-called "tonal"
>style of European music.

This is not relevant.

> In contexts of 12-tone descriptions it is now well
>established to use only the # for the five "accidentals".

This is only true of 12-tone _equal_tempered_ situations when there is no
particular tonal centre and no consonant intervals being discussed. And
even then it is also quite common to use EbBbFCGDAEBF#C#G# as the default
rather than FCGDAEBF#C#G#D#A#. That is not the case here. Correct spelling
of intervals is important.

For you to use G# D# and A# for what, in the Chinese, are clearly "major
third below" (or "minor sixth above") C G and D, is a travesty. It amounts
to an unwarranted assumption of 12-tone-equal-temperament before you even
start.

> > I find no evidence there that fifths are narrowed, or fourths
>widened, to
> > favour thirds.
>
>Well, the fourths according to your method are widened.

The amount is not significant given the huge variance.

> I rejected your
>method, however. I knew it from Andr� Lehr in the Netherlands. The
>bells do
>not represent a system that is based on intervals that cross the octave
>borders, like our piano. They are for octave confined music, just like the
>gamelan instruments,

What evidence do you have for this?

*** See yesterday's post.***

And if it is so, which octaves are we confined to, G to G, C to C, D to D?
And how do you know.

*** See yesterday's post. The backbone octave of the ensemble is a
non-closed one. It is D to C.***

>because one player only has one octave in the reach of his hands.

What does it matter if it is played by one player or three, what has that
got to do with the issue of how to calculate mean interval sizes?

*** See yesterday's post. A melodic phrase can best be played by one player,
rather than by several interleaving players. What one player has in his/her
hands tells us which intervals are most relevant to examine.***

*** See yesterday's post. I considered the intervals in the DEFGAC frame,
because it is the obvious backbone of the ensemble. Concerning all other
intervals, we only know that they must have been of secondary importance, if
they were of any importance at all.***

If I am including intervals that cross these imagined octave boundaries,
then so are you.

It seems from the photographs that a single player could easily span any
fifth with two hands. But I do not see this as relevant anyway. Both notes
of a fifth may never be played simultaneously and this would not affect my
point about how to calculate their mean size.

> Here, as in Gamelan, we have within-octave intervals and
>across-octave chroma consistency. Circles of fifths and fourths, or
>parts of
>it, do not seem appropriate to describe these instruments and their music.

How do you know this?

*** See yesterday's post on the absence of Pythagorean tuning.***

It certainly isn't in the data or photographs.

>The term mean-tone tuning only makes sense in a 12-tone instrument.

Perhaps that is so for the literal meaning of "mean tone", but not for the
tempering principle that has come to be known as meantone temperament. This
can be applied to a chain of only four fifths (5 notes) and is simply the
principle of narrowing the fifths from 2:3 to bring the major and minor
thirds closer to 4:5 and 5:6. This is what you are claiming the Zeng bells
are doing for CGDAE.

>What the
>Chinese had 2000 years before the Europeans was a tuning focus on the
>thirds at the cost of the fifths.

I disagree. I don't see any cost to the fifths, if anything the cost falls
on the octaves and fourths. But the tuning errors are so huge it is
ridiculous to draw any such conclusions.

> > How one can conclude anything from a nominally 12-note tuning where
> > interval sizes vary +-50 cents, I'm not sure. But if it can be said to
> > approximate any kind of regular tuning, I'd have to say it looks like a
> > incredibly badly tuned 12 note equal division of a stretched octave.
>
>They are "incredibly badly tuned", if you apply this standards of
>multi-octave string instruments. If you apply the standards of
>metallophones, with their non-harmonic - but potentially very
>interesting -
>sound spectra, and the standards of musical practice of Gamelan
>instruments,
>the bells represent sufficiently balanced tuning compromises between scale
>precision and pleasant timbres.

But aren't you claiming that when struck in the right place there is very
little inharmonicity in the bell tones?

*** No. When struck in the right place, A and B tones are well separated,
but the spectrum of each of them always has both harmonic and non-harmonic
components. See the spectral analyses concerning the original bells on my
web site.***

Even if the inharmonicity _is_ responsible, surely you're not seriously
claiming that those +-50 cent errors are _meant_ to be there. I'm afraid
this just doesn't hold water. How do you explain the enormous variations
between what are presumably bell cast from the same mold, in different sets?

*** I know of no evidence that they were cast from the same molds. Some
researchers think that they were cast at different times, or even at
different places (as is possible with different Gamalan instruments of one
ensemble, as well). But, obviously, they were integrated in one unitary rack
and reflect the same tuning objectives.***

I suggest that either their casting was very inaccurate and they had no way
of tuning the bells after casting, or 2,400 years has taken its toll (pun
intended).

Either way, I think the claims you are making cannot be supported by the
data, because of the uncertainty caused by these huge variations.

> > You say "it is obvious that the tuning focus was on pure thirds, not on
> > pure fifths". But I'd say it was treating thirds and fifths
>approximately
> > as equals. However, given that the bells tend to have an approximate
>third
> > among their partials, tuning these so that an individual bell sounds
> > harmonious when both of its notes sound simultaneously will force them
> > towards either a 4:5 or 5:6 ratio, and so even if the emphasis for the
> > tuning _between_ bells is entirely on the fourths and fifths, you would
>end
> > up with a tuning that appeared to favour minor and major thirds as
>much as
> > fourths and fifths.
>
>In octave confined music you simply have more thirds than fourths and
>fifths. The data reflect that the bell casters cared more about the thirds
>in the octave frame.

I'll await evidence of this "octave frame" before I comment on this.

>I think it should be clear from the context of the paper that we are
>dealing
>with a material 6-tone scale, not with a 6-tone scale in musical
>practice.

I'm afraid I did not find that clear. But I'm glad you agree. However there
is still the possibility that the F was not used any more often than say
the B, and just happens to occur in every octave because you have to put
_something_ on the D bell.

>I wrote about this at length in a reply to one of Kraig's posts.

I'm sorry I missed that.

> There is a
>possibly corresponding case in Gamelan pelog of Java, where the material
>scale always has seven tones in the octave, whereas the three modes
>that are
>used in pelog only select five of them, and different ones depending on
>mode.

Yes.

> > Notice that the pentatonics can be played with pitch strictly increasing
> > from right to left, whereas the hexatonic would require a
>counter-intuitive
> > reversal of direction between E and F.
>
>Good point. But for settings with one player per octave (mostly only 3
>bells), such a reversal would not have been much of a problem.

What evidence have you for one player per octave?

*** See yesterday's post.***

> > Your data does show a bell with 342 cents between its two notes (the
>D5-F5
> > bell in set I) but interestingly, on your data tables page you do not
> > suggest that either of its two notes are mistuned.
>
>Good point. In the table only assumed mistuning re ensemble is marked, not
>bell-internal one. The bell possibly was accepted and not melted down and
>re-cast, because it still fitted into the chroma scale of the ensemble, as
>shown in data table A.

Possibly.

> > With regard to what is mistuned and what isn't. The only place that we
> > agree that something is very wrong is in the low octave of set two,
>where
> > we find only 14 cents separating Bb from B.
>
>This supports the view that the six "accidentals" were second-class
>passengers in the ensemble.

It only supports the view that Bb was a second-class passenger, or that
2,400 years underwater and underground is a long time.

***The time was physically irrelevant for the state of conservation. The
crystalline structure of bronze does not change between a few years and
thousands of years. The bells were as good as new when they were dug up.***

Regards,

***Martin***

Dear Martin,

The main remaining issue I have with your paper is one that I have raised
several times but you have not really addressed, except to say:

"I rejected your method, however. I knew it from André Lehr in the
Netherlands. The bells do not represent a system that is based on intervals that cross the octave borders, like our piano. They are for octave confined
music, just like the gamelan instruments, because one player only has one
octave in the reach of his hands. Here, as in Gamelan, we have
within-octave intervals and across-octave chroma consistency. Circles of
fifths and fourths, or parts of it, do not seem appropriate to describe
these instruments and their music."

However, even if I assume, for the sake of argument, that you are correct
in this regard, and the scale is primarily an octave-confined DEFGAC, then
I still find a serious problem with your method of calculating mean
interval sizes by first calculating mean octave-reduced pitches while
assuming 1200 cent octaves. I find it extraordinary that you assume an
intention to tune 1200 cent octaves based on no evidence whatsoever. And
you ignore the contrary evidence of the gamelan practice of stretching the
octaves, while being quite happy to cite other gamelan practices as
evidence for other hypotheses.

I must insist that the only meaningful way to calculate mean interval sizes
is not to assume _anything_ about the intended octave size, but to first
calculate the actual interval sizes _within_ each octave of each set, and
then calculate their mean.

The following is a complete list of the nominal intervals within your
proposed scale, that do not cross your proposed octave boundaries:

Qty Mean Std-dev Qty on a single bell
(cents) (cents)
semitones E:F 8 120 26
wholetones D:E 8 200 23
F:G 8 184 12
G:A 9 202 30
minor thirds D:F 8 320 14 8
E:G 8 303 27 6
A:C 9 310 15 6
major thirds F:A 8 384 27
fourths D:G 8 503 8
E:A 8 504 24
G:C 9 511 29
fifths D:A 8 704 33
F:C 8 696 28
minor sixths E:C 8 815 22
minor sevenths D:C 8 1015 32

Do you dispute these results? Some of them are very different from those
you base your claims on.

We should really look at _all_ the intervals that occur a significant
number of times, but for now I'm going along with your octave-confined
assertion.

Your basic argument seems to be that the intervals or pitches which are
tuned most consistently, in all octaves of all sets of bells, are likely to
be the most important ones, and the ones the tuning is "based on". Also
that those with the smallest deviation are most likely to be the closest to
their _intended_ tuning. I don't have a problem with these assumptions,
only with your failure to carry them through consistently by not assuming
an octave size that is not given to you by the data.

The above results do not show "a norm tone of F4 ~ 345 Hz".

We see that the interval with the lowest deviation by far is the D:G
fourth, suggesting that one of these two notes is most likely to be the
starting point for the tuning. F is only third in line, on this analysis.
Your low deviation for the F "chroma" is merely an artifact of your
unwarranted assumption of a 1200 cent octave. Of the octave-specific
pitches that occur in all three sets, G4 has the lowest standard deviation
(4 cents).

The above results do not show "a third-oriented tuning with equally
tempered fifths (~696 Cent) in the series CGDAE".

I'm not sure how to make _any_ sense of this claim when you deny the
relevance of the fifths C:G, G:D and A:E (because they cross the octave
boundary), except to suppose that you are so immersed in the assumption of
1200 cent octave-equivalence that you consider these to be equivalent to
the remainder out of 1200 cents, of the fourths G:C, D:G and E:A. But such
"remainder fifths" do not actually exist anywhere in the bell tuning. They
are, like the F4 norm tone, an artifact of your unwarranted assumption of
1200 cent octaves.

Here are the actual C:G, G:D and A:E fifths, which cross your octave boundary,
Qty Mean Std-dev Qty on a single bell
(cents) (cents)
fifths C:G 6 700 20
G:D 6 705 15
A:E 6 706 28

The results suggest equal importance for (very approximately) justly
intoned minor thirds and fifths, and provide no evidence whatsoever for any
tempering of the fifths. There is however some evidence that some octaves
and fourths are stretched by approximately 11 cents, thereby distributing
the syntonic comma.

Here's how it works for one octave of the hexatonic (dual overlapping
pentatonics).

--------1211------
/ \
| E4--509--A4--509--D5
| \ / \ / \
| --702-- --702-- \
| / 316 / 316 316
| / \ / \ \
D4--509--G4--509--C5--509--F5

Here's a start at what I find to be a plausible explanation of how the
tuning has resulted. The individual bells are tuned by aiming for a 5:6
ratio between their two main frequencies (E:G, A:C, D:F). Then the tuning
_between_ bells is aimed at 2:3 ratios for C:G, G:D, and D:A, the four
notes that are always named without reference to any others (gong, zhi,
shang, yu).

This automatically results in a 4:5 ratio for the F:A and C:E major thirds.

It is more difficult to explain how the tempering was acheived, but it was
probably aimed at approximately equalising the wholetones D:E, G:A, C:D and
F:G at around 193 cents rather than the fourths at 509 cents or the octaves
at 1211 cents. But once the syntonic commas are distributed over these
wholetones, the stretched fourths and octaves will be the automatic result
(given the Just minor thirds and fifths).

Perhaps a more complete description will explained why some of the
wholetones are closer to a 9:10 (182 c) minor wholetone and some are closer
to an 8:9 (204 c) major wholetone, but given the deviations, the suggestion
of a target of 193 cents for all of them seems quite reasonable.

Here's the wholetone we haven't looked at so far, because it crosses your
octave boundary.

Qty Mean Std-dev Qty on a single bell
(cents) (cents)
wholetone C:D 6 196 20

Is this anything like how André Lehr describes the tuning?

Regards,
-- Dave Keenan
Brisbane, Australia
http://dkeenan.com/

Dave Keenan wrote:

> Dear Martin,
>
> The main remaining issue I have with your paper is one that I have raised
> several times but you have not really addressed, except to say:
>
> "I rejected your method, however. I knew it from Andr� Lehr in the
> Netherlands. The bells do not represent a system that is based on
intervals
> that cross the octave borders, like our piano. They are for octave
confined
> music, just like the gamelan instruments, because one player only has one
> octave in the reach of his hands. Here, as in Gamelan, we have
> within-octave intervals and across-octave chroma consistency. Circles of
> fifths and fourths, or parts of it, do not seem appropriate to describe
> these instruments and their music."
>
> However, even if I assume, for the sake of argument, that you are correct
> in this regard, and the scale is primarily an octave-confined DEFGAC, then
> I still find a serious problem with your method of calculating mean
> interval sizes by first calculating mean octave-reduced pitches while
> assuming 1200 cent octaves. I find it extraordinary that you assume an
> intention to tune 1200 cent octaves based on no evidence whatsoever. And
> you ignore the contrary evidence of the gamelan practice of stretching the
> octaves, while being quite happy to cite other gamelan practices as
> evidence for other hypotheses.

Dear Dave,

there is sufficient evidence to assume that the bell casters worked from a
concept of an octave of 1200 Cent.

1) As opposed to the Gamelan practice, the apparent octave stretching in the
bell tuning is small and inconsistent. The most cautious assumption that is
possible from the data is that all deviations from an octave of 1200 Cent,
in either direction, are accidental and not tuned on purpose.

2) The scale DEFGAC of the bell ensemble obviously is a tempered one. It
reveals a balancing of thirds, fourths, and fifths. A temperament of this
type, for an ensemble of 66 tones, can only be achieved, if there is a
tuning template. It must have been realized either on a string instrument or
on
an instrument that had been tuned from a string instrument. Only string
instruments allow the repeated alteration between exact and detuned thirds,
which is necessary for this type of temperament. Octaves on string
instruments, then, are very close to 1200 Cent.

> I must insist that the only meaningful way to calculate mean interval
sizes
> is not to assume _anything_ about the intended octave size, but to first
> calculate the actual interval sizes _within_ each octave of each set, and
> then calculate their mean.
>
> The following is a complete list of the nominal intervals within your
> proposed scale, that do not cross your proposed octave boundaries:
>
> Qty Mean Std-dev Qty on a single bell
> (cents) (cents)
> semitones E:F 8 120 26
> wholetones D:E 8 200 23
> F:G 8 184 12
> G:A 9 202 30
> minor thirds D:F 8 320 14 8
> E:G 8 303 27 6
> A:C 9 310 15 6
> major thirds F:A 8 384 27
> fourths D:G 8 503 8
> E:A 8 504 24
> G:C 9 511 29
> fifths D:A 8 704 33
> F:C 8 696 28
> minor sixths E:C 8 815 22
> minor sevenths D:C 8 1015 32
>
> Do you dispute these results?

No. I did not check them, though.

> Some of them are very different from those you base your claims on.

They are not comparable with my results, and therefore not in conflict with
them.

> We should really look at _all_ the intervals that occur a significant
> number of times, but for now I'm going along with your octave-confined
> assertion.
>
> Your basic argument seems to be that the intervals or pitches which are
> tuned most consistently, in all octaves of all sets of bells, are likely
to
> be the most important ones, and the ones the tuning is "based on". Also
> that those with the smallest deviation are most likely to be the closest
to
> their _intended_ tuning. I don't have a problem with these assumptions,
> only with your failure to carry them through consistently by not assuming
> an octave size that is not given to you by the data.
>
> The above results do not show "a norm tone of F4 ~ 345 Hz".

Because you analyzed intervals, instead of distribution of frequencies of
single tones.

> We see that the interval with the lowest deviation by far is the D:G
> fourth, suggesting that one of these two notes is most likely to be the
> starting point for the tuning. F is only third in line, on this analysis.
> Your low deviation for the F "chroma" is merely an artifact of your
> unwarranted assumption of a 1200 cent octave. Of the octave-specific
> pitches that occur in all three sets, G4 has the lowest standard deviation
> (4 cents).
>
> The above results do not show "a third-oriented tuning with equally
> tempered fifths (~696 Cent) in the series CGDAE".
>
> I'm not sure how to make _any_ sense of this claim when you deny the
> relevance of the fifths C:G, G:D and A:E (because they cross the octave
> boundary), except to suppose that you are so immersed in the assumption of
> 1200 cent octave-equivalence that you consider these to be equivalent to
> the remainder out of 1200 cents, of the fourths G:C, D:G and E:A. But such
> "remainder fifths" do not actually exist anywhere in the bell tuning. They
> are, like the F4 norm tone, an artifact of your unwarranted assumption of
> 1200 cent octaves.
>
> Here are the actual C:G, G:D and A:E fifths, which cross your octave
boundary,
> Qty Mean Std-dev Qty on a single bell
> (cents) (cents)
> fifths C:G 6 700 20
> G:D 6 705 15
> A:E 6 706 28
>
> The results suggest equal importance for (very approximately) justly
> intoned minor thirds and fifths, and provide no evidence whatsoever for
any
> tempering of the fifths. There is however some evidence that some octaves
> and fourths are stretched by approximately 11 cents, thereby distributing
> the syntonic comma.
>
> Here's how it works for one octave of the hexatonic (dual overlapping
> pentatonics).
>
> --------1211------
> / \
> | E4--509--A4--509--D5
> | \ / \ / \
> | --702-- --702-- \
> | / 316 / 316 316
> | / \ / \ \
> D4--509--G4--509--C5--509--F5
>
> Here's a start at what I find to be a plausible explanation of how the
> tuning has resulted. The individual bells are tuned by aiming for a 5:6
> ratio between their two main frequencies (E:G, A:C, D:F). Then the tuning
> _between_ bells is aimed at 2:3 ratios for C:G, G:D, and D:A, the four
> notes that are always named without reference to any others (gong, zhi,
> shang, yu).

It would have been a very poor strategy first to tune the bells internally
(5:6 ratio) and then change their tones again to reach further tuning aims.
All filing on the bells had to be minimized as much as possible in order to
keep a well-sounding bell, or one that sounded bell-like att all. As I
explained above, a tuning template for the complete scale of DEFGAC was
mandatory.

> This automatically results in a 4:5 ratio for the F:A and C:E major
thirds.
>
> It is more difficult to explain how the tempering was acheived, but it was
> probably aimed at approximately equalising the wholetones D:E, G:A, C:D
and
> F:G at around 193 cents rather than the fourths at 509 cents or the
octaves
> at 1211 cents. But once the syntonic commas are distributed over these
> wholetones, the stretched fourths and octaves will be the automatic result
> (given the Just minor thirds and fifths).
>
> Perhaps a more complete description will explained why some of the
> wholetones are closer to a 9:10 (182 c) minor wholetone and some are
closer
> to an 8:9 (204 c) major wholetone, but given the deviations, the
suggestion
> of a target of 193 cents for all of them seems quite reasonable.
>
> Here's the wholetone we haven't looked at so far, because it crosses your
> octave boundary.
>
> Qty Mean Std-dev Qty on a single bell
> (cents) (cents)
> wholetone C:D 6 196 20
>
> Is this anything like how Andr� Lehr describes the tuning?

His analysis was much less sophisticated than yours. I only mentioned it,
because he treated the intervals irrespective of the ensemble's scale,
something you did in your first analysis as well.

Best,

Martin

At 11:08 PM 4/08/2003 +0000, you wrote:
>Dear Dave,
>
>there is sufficient evidence to assume that the bell casters worked from a
>concept of an octave of 1200 Cent.
>
>1) As opposed to the Gamelan practice, the apparent octave stretching
>in the
>bell tuning is small and inconsistent.

Dear Martin,

But _any_ tempering we may find in the bell tuning will be small and inconsistent, so how can that be considered evidence for 1200 cent octaves?

I have provided an argument as to why the octave tempering is small -- because it is approximating half of a syntonic comma, which allows the thirds and fifths to be pure while still allowing for a minor third to be equivalent to three fourths less an octave, and for a major third to be equivalent to four fifths less two octaves, and other similar relationships.

These meantone-like relationships are not found or expected in gamelan tunings. The syntonic comma (80:81, ~21.5 c) is irrelevant in gamelan. The much larger "major limma" (128:135, ~92 c) may however play a similar role in pelog tunings.

You have correctly observed that the bell tuning is tempered to distribute the syntonic comma, but you are quite mistaken about _where_ the pieces of the syntonic comma have been put, or in other words _which_intervals_ are tempered. Because you chose, right from the start, to blind yourself to the one possibility which turns out to be true.

> The most cautious assumption
>that is
>possible from the data is that all deviations from an octave of 1200 Cent,
>in either direction, are accidental and not tuned on purpose.

No! The most cautious assumption is no assumption at all! Or to put it another way: Even if that _were_ the most cautious assumption, it is not necessary to make it.

>2) The scale DEFGAC of the bell ensemble obviously is a tempered one.

Yes. We certainly agree on that. We even agree on the comma that is being distributed.

>It reveals a balancing of thirds, fourths, and fifths.

Yes. And more specifically it reveals the tempering of fourths (or wholetones or octaves) in favour of pure thirds and pure fifths.

In your abstract you describe the tuning as "third-oriented" and in your conclusion you write "a preference of pure thirds over pure fifths".

But there is no evidence whatsoever in the data, of "a preference of pure thirds over pure fifths". I say again, this is merely an artifact of your unnecessary assumption of 1200 cent octaves. There is however evidence of a preference for pure thirds over pure fourths (and octaves).

> A temperament of this
>type, for an ensemble of 66 tones, can only be achieved, if there is a
>tuning template.

I can go along with that.

> It must have been realized either on a string
>instrument or
>on
>an instrument that had been tuned from a string instrument. Only string
>instruments allow the repeated alteration between exact and detuned
>thirds,
>which is necessary for this type of temperament.

I wouldn't go so far as to say the template can "only" have originated on a string instrument. I can imagine an iterative process of casting and filing and recasting so as to require less filing, and filing, and recasting to require even less filing and so on. However I agree that it would be much easier to first produce the temperament on a string instrument, and so I agree that is probably what happened.

> Octaves on string
>instruments, then, are very close to 1200 Cent.

This makes no sense to me whatsoever. Octaves on a string instrument can be anything you want them to be. Just as you describe detuning thirds above, so the octaves can also be detuned.

However, I don't understand why you talk of detuning thirds. I thought we both agreed that the bell tuning appears to be aiming for pure thirds.

...
> > Do you dispute these results?
>
>No. I did not check them, though.
>
> > Some of them are very different from those you base your claims on.
>
>They are not comparable with my results, and therefore not in conflict
>with them.

You claim that the mean sizes of the fifths F:C and D:A are 687.9 c and 697.8 c respectively. I claim that they are 696 c and 704 c respectively (to the nearest cent). How is this "not comparable". Yours are respectively 8 and 6 cents narrower than mine (to the nearest cent). Similar differences apply to the other fifths and other intervals you give on your data page.

> > The above results do not show "a norm tone of F4 ~ 345 Hz".
>
>Because you analyzed intervals, instead of distribution of frequencies of
>single tones.

And why is one method more valid than the other? Possibly because one method does not make any assumptions about intended octave size.

However, I _did_ also analyse the distribution of frequencies of single tones, and reported my result: "Of the octave-specific pitches that occur in all three sets, G4 has the lowest standard deviation (4 cents)." Again I did not assume any particular intended size for the octaves.

So F4 loses out by two independent methods when no particular octave size is assumed.

> > The results suggest equal importance for (very approximately) justly
> > intoned minor thirds and fifths, and provide no evidence whatsoever for
>any
> > tempering of the fifths. There is however some evidence that some
>octaves
> > and fourths are stretched by approximately 11 cents, thereby
>distributing
> > the syntonic comma.
> >
> > Here's how it works for one octave of the hexatonic (dual overlapping
> > pentatonics).
> >
> > --------1211------
> > / \
> > | E4--509--A4--509--D5
> > | \ / \ / \
> > | --702-- --702-- \
> > | / 316 / 316 316
> > | / \ / \ \
> > D4--509--G4--509--C5--509--F5
> >
> > Here's a start at what I find to be a plausible explanation of how the
> > tuning has resulted. The individual bells are tuned by aiming for a 5:6
> > ratio between their two main frequencies (E:G, A:C, D:F). Then the
>tuning
> > _between_ bells is aimed at 2:3 ratios for C:G, G:D, and D:A, the four
> > notes that are always named without reference to any others (gong, zhi,
> > shang, yu).
>
>It would have been a very poor strategy first to tune the bells internally
>(5:6 ratio) and then change their tones again to reach further tuning
>aims.
>All filing on the bells had to be minimized as much as possible in
>order to
>keep a well-sounding bell, or one that sounded bell-like att all. As I
>explained above, a tuning template for the complete scale of DEFGAC was
>mandatory.

Yes. You're quite right about that. Please re-read my description (above and below) as applying not to the bells, but to a string instrument for the setting up of the template, and change my "then" to "also" so as not to imply any particular sequencing between the tuning of the pure minor thirds and the tuning of the pure fifths.

> > This automatically results in a 4:5 ratio for the F:A and C:E major
>thirds.
> >
> > It is more difficult to explain how the tempering was acheived, but
>it was
> > probably aimed at approximately equalising the wholetones D:E, G:A, C:D
>and
> > F:G at around 193 cents rather than the fourths at 509 cents or the
>octaves
> > at 1211 cents. But once the syntonic commas are distributed over these
> > wholetones, the stretched fourths and octaves will be the automatic
>result
> > (given the Just minor thirds and fifths).
> >
> > Perhaps a more complete description will explained why some of the
> > wholetones are closer to a 9:10 (182 c) minor wholetone and some are
>closer
> > to an 8:9 (204 c) major wholetone, but given the deviations, the
>suggestion
> > of a target of 193 cents for all of them seems quite reasonable.

There is a very simple way to decide which of our two models of the bell tuning more closely aproaches the reality.

You take your F4-19.5c norm tone, your 1200 cent octaves and your 696.2 c fifths and generate an "ideal" version of the tuning for the DEFGAC hexatonic over the three and a half octaves from G3 to C7. Then find the sum of squares of the errors between this "ideal" tuning and the actual data over all 51 notes.

I'll do the same with my G4-46.3c norm tone, my pure fifths and minor thirds and my half-syntonic-comma-wide octaves.

I suspect you can already guess the result.

-- Dave Keenan
Brisbane, Australia
http://dkeenan.com/

Here's a better diagram of my model of the Zeng bells tuning. [You can convince the Yahoo web interface to ungarble it by choosing "Message Index" and "Expand Messages".]

F6--C7
| |
F5--C6|-G6|
| | | | |
F4--C5--G5--D6--A6
| | | | | |
C4|-G4|-D5|-A5--E6
| | | | |
G3--D4--A4--E5
| |
A3--E4

The horizontal lines are pure fifths and the vertical lines are pure minor thirds (usually between two notes on a single bell). This can be seen as two independent JI lattices overlaid.

F6--C7
| |
| |
| |
F4--C5--G5--D6--A6
| | |
| | |
| | |
G3--D4--A4--E5

and
F5--C6--G6
| | |
| | |
| | |
C4--G4--D5--A5--E6
| |
| |
| |
A3--E4

Each of which would be easily tuned by ear (on strings). The C:E and F:A major thirds, diagonals of the above rectangles, are also pure and could be used to help tune these lattices.

But any interval that goes _between_ these two lattices is a tempered interval. Such as the octaves, and the fourths and wholetones shown below. The fourths and octaves are widened. The wholetones are narrowed (relative to 8:9).

F6 C7
/ \ /
F5 C6 G6
/ \ / \ / \
F4 C5 G5 D6 A6
/ \ / \ / \ / \ /
C4 G4 D5 A5 E6
/ \ / \ / \ /
G3 D4 A4 E5
\ / \ /
A3 E4

For complete symmetry, all these tempered intervals would be tempered by a half syntonic comma, however the possibility exists that the tempering is asymmetric so that half of these intervals are close to pure and the other half are tempered by close to a full syntonic comma.

When I get time I intend to implement the model in a spreadsheet, with this asymmetry as a parameter, and see what value minimises the sum of squares of the errors between the model and the data.

It is also possible that the different bell sets are deliberately designed to distribute the commas differently. This is worth investigating.

-- Dave Keenan
Brisbane, Australia
http://dkeenan.com/

I'm not sure if anyone but me is still interested in the Zeng Bells,
but I figure I should keep posting my results just in case.

I set up in a spreadsheet,
(a) Martin's model which has 1200 c octaves and 696.2 c fifths, and
(b) my model which has half-comma wide octaves (1210.8 c) and pure
fifths and thirds (both major and minor), and compared them both
against the actual tuning data by calculating the RMS error between
the model and the data for all 51 tones (FCGDAE only, not the
"accidentals").

I included a parameter in both models to allow all the pitches of the
model to be shifted to minimise the error.

As expected, my model had the lowest error, but I must admit to being
surprised at how small the improvement was. Martin's model differs
from the data by 19.7 c RMS while mine differs from the data by 17.7 c
RMS.

I conclude from this that it may not be possible to determine, from
the tuning data, whether there was any intention on the part of the
bell designers to _systematically_ distribute the comma.

In fact, I can go further and say that it may not be possible to
determine, from the tuning data, whether there was any intention to
temper the scale _at_all_.

In my previous post I explained how my model of the tuning can be seen
as two subgraphs, each one connected by pure intervals, but only
tempered intervals going between the subgraphs. I speculated on a
possible asymmetry in the tempered intervals. In fact it is possible
to push this asymmetry to the extreme in either direction so that half
the fourths and octaves are pure and the other half are a full comma
wide (and half the tones are major and half are minor). Doing this
makes the entire graph connected by pure intervals alone, and
therefore untempered and completely tunable by ear without any
sophisticated tricks.

It turns out that both of these justly intoned models do better than
Martin's model but worse than my symmetrically tempered model, having
RMS errors of 18.2 c and 18.8 c.

It may well be that there is some other justly intoned model that does
even better than my tempered model, at matching the data.

So Martin, I must now retract my earlier agreement that the data shows
a tempered scale.

Dave Keenan wrote:

> I'm not sure if anyone but me is still interested in the Zeng Bells,
> but I figure I should keep posting my results just in case.

Dear Dave,

thanks for posting your results. Thanks in particular, because they did not
show what you might have expected.

> I set up in a spreadsheet,
(a) Martin's model which has 1200 c octaves and 696.2 c fifths, and
(b) my model which has half-comma wide octaves (1210.8 c) and pure
fifths and thirds (both major and minor), and compared them both
against the actual tuning data by calculating the RMS error between
the model and the data for all 51 tones (FCGDAE only, not the
"accidentals").

> I included a parameter in both models to allow all the pitches of the
model to be shifted to minimise the error.

> As expected, my model had the lowest error, but I must admit to being
surprised at how small the improvement was. Martin's model differs
from the data by 19.7 c RMS while mine differs from the data by 17.7 c
RMS.

This difference is so small that both models would have to be regarded as
equally appropriate, provided all assumptions for the models were equally
realistic. This, however, is not the case. The model that takes into
consideration the facts that the bell ensemble is an instrument in D and
that one chime player has an octave under his hands remains more realistic
to me.

> I conclude from this that it may not be possible to determine, from
the tuning data, whether there was any intention on the part of the
bell designers to _systematically_ distribute the comma.

I never suggested that the bell casters might have done this. To my
knowledge the comma was unknown to the Chinese in those days. This does not
mean that they were unable to tune tempered scales, though.

> In fact, I can go further and say that it may not be possible to
determine, from the tuning data, whether there was any intention to
temper the scale _at_all_.

> In my previous post I explained how my model of the tuning can be seen
as two subgraphs, each one connected by pure intervals, but only
tempered intervals going between the subgraphs. I speculated on a
possible asymmetry in the tempered intervals. In fact it is possible
to push this asymmetry to the extreme in either direction so that half
the fourths and octaves are pure and the other half are a full comma
wide (and half the tones are major and half are minor). Doing this
makes the entire graph connected by pure intervals alone, and
therefore untempered and completely tunable by ear without any
sophisticated tricks.

> It turns out that both of these justly intoned models do better than
Martin's model but worse than my symmetrically tempered model, having
RMS errors of 18.2 c and 18.8 c.

The terms "better" and "worse" would only be justified, if the differences
were significant.

> It may well be that there is some other justly intoned model that does
even better than my tempered model, at matching the data.

> So Martin, I must now retract my earlier agreement that the data shows
a tempered scale.

I don't think you have to. It might be enough, if you widened your concept
of temperament. The bell casters obviously focused on the thirds, and
obviously at the cost of the primitive Phytagorean tuning. Such a strategy,
however mathematically imperfect it may be, should - at least in my view -
be called a strategy of temperament.

Martin

--- In tuning@yahoogroups.com, "Martin Braun" <nombraun@t...> wrote:
> Dear Dave,
>
> thanks for posting your results. Thanks in particular, because they
> did not show what you might have expected.
>
> > I set up in a spreadsheet,
> (a) Martin's model which has 1200 c octaves and 696.2 c fifths, and
> (b) my model which has half-comma wide octaves (1210.8 c) and pure
> fifths and thirds (both major and minor), and compared them both
> against the actual tuning data by calculating the RMS error between
> the model and the data for all 51 tones (FCGDAE only, not the
> "accidentals").
>
> > I included a parameter in both models to allow all the pitches of
> the model to be shifted to minimise the error.
>
> > As expected, my model had the lowest error, but I must admit to being
> surprised at how small the improvement was. Martin's model differs
> from the data by 19.7 c RMS while mine differs from the data by 17.7
> c RMS.
>
> This difference is so small that both models would have to be
regarded as
> equally appropriate, provided all assumptions for the models were
> equally realistic.

Dear Martin,

I think the best way to decide whether an assumption is realistic or
not is to see if it makes the model agree well with the data, however
I will go along with this for the sake of argument.

> This, however, is not the case. The model that takes into
> consideration the facts that the bell ensemble is an instrument in D and
> that one chime player has an octave under his hands remains more
> realistic to me.

I think that my model is no less disposed to this than is yours
(although I would hesitate to call it a "fact"). What aspect(s) of my
model do you find to be at odds with this? Did you see my preceeding
two posts?

/tuning/topicId_45096.html#46210
/tuning/topicId_45096.html#46224

> > I conclude from this that it may not be possible to determine, from
> the tuning data, whether there was any intention on the part of the
> bell designers to _systematically_ distribute the comma.
>
> I never suggested that the bell casters might have done this. To my
> knowledge the comma was unknown to the Chinese in those days. This
does not
> mean that they were unable to tune tempered scales, though.

I agree that one does not have to understand the mathematics of (or
even have to name) a comma in order to do tempering. But the tempering
that we are talking about here does involve distributing the syntonic
comma, whether you know you are doing it or not.

When you add, "At this point in history, China was 2000 years ahead of
Europe, ... in musical acoustics", you clearly imply that there was an
intention on the part of the bell designers to _systematically_ temper
-- that the tempering was not a mere accident of trying to do the
impossible (make octaves, fifths and thirds all pure) -- that they
really understood the problem and had a system for solving it by
temperament, rather than by simply deciding which intervals to attempt
to make pure, and abandoning the rest to whatever fate awaited them.

I have shown that there is no evidence for (or against) this in the
tuning data.

...
> > It turns out that both of these justly intoned models do better than
> Martin's model but worse than my symmetrically tempered model, having
> RMS errors of 18.2 c and 18.8 c.
>
> The terms "better" and "worse" would only be justified, if the
differences
> were significant.

Very well. The justly intoned (untempered) models matched the data
just as well as the tempered models did.

...
> > So Martin, I must now retract my earlier agreement that the data shows
> a tempered scale.
>
> I don't think you have to. It might be enough, if you widened your
concept
> of temperament. The bell casters obviously focused on the thirds, and
> obviously at the cost of the primitive Phytagorean tuning. Such a
strategy,
> however mathematically imperfect it may be, should - at least in my
view -
> be called a strategy of temperament.

No. The concept of temperament cannot be widened that much without
breaking (becoming utterly useless). The standard justly-intoned
diatonic major scale (1/1 9/8 5/4 4/3 3/2 5/3 15/8 2/1) focusses on
thirds at the expense of Pythagorean tuning, but no-one who knows
anything about tuning would ever call it tempered. In this case the
fifth D:A (and fourth A:D) have simply been abandoned (as pure
consonances).

If the bell designers were not aware of the commas, and did not
distribute them in any way*, then this cannot by any stretch of the
imagination be called "a strategy of temperament". *I have shown that
at least two models in which the commas only appear whole (which are
called Just tunings, not temperaments), are essentially as good as our
tempered models at matching the data.

In your conclusion you wrote:
"The 65 Zeng bells prove that about 2500 years ago the Chinese had
fifth generation, fifth temperament, ... At this point in history,
China was 2000 years ahead of Europe, not only in bell casting, but
also in musical acoustics."

I have shown that they prove no such thing. Of course it may well be
true, but any evidence for it is completely lost in the noise of
general mistuning. We should have realised that there could be no
evidence for or against tempering when the standard deviation of most
pitches and intervals is of a similar size to (or larger than) the comma.

Dear Martin,

I understand you have an aversion to octaves other than 1200 c in this
context. So let's assume pure octaves everywhere and look at the most
obvious justly intoned tuning siggested by the bells themselves,
namely the one where all the minor thirds that occur between the two
notes of a single bell are intended to be pure (D:F, E:G, A:C) and
where there are pure intervals between the low notes of the bells (and
therefore parallel pure intervals between the high notes of the
bells). i.e. without crossing the boundaries of your favourite octaves
we have a pure fifth from D to A and a pure fourth from E to A, giving
an 8:9 major tone from D to E.

Taking D as 1/1 this is the untempered scale 1/1 9/8 6/5 27/20 3/2 9/5.

This can be diagrammed as the following octave-equivalent lattice.

D---A---E
\ / \ / \
F---C---G

The RMS error between this scale and the data is 19.8 cents.
Essentially as good as any of the other proposals so far. Please check
it for yourself.

Then, I hope you will understand that the bells provide no evidence of
tempering?

What they do provide is evidence of their designer's and maker's
inability or unwillingness (we can't tell how much of each) to tune
them very accurately.

Of course, the less harmonic and the more rapidly decaying a timbre
is, the harder it is to tell if it is mistuned and therefore the less
need there is to tune it accurately.

I respectfully suggest that you remove the current misleading version
of your article from the web. I look forward to reading a corrected
version in future.

Regards,
-- Dave Keenan

Dave Keenan wrote (46210):

>> Octaves on string instruments, then, are very close to 1200 Cent.

> This makes no sense to me whatsoever. Octaves on a string instrument can
be
anything you want them to be. Just as you describe detuning thirds above,
so the octaves can also be detuned.

Of course, they can. But musicians don't do it, because they gain nothing by
it while losing beauty of sound.

and in 46253:

> I'm afraid you can't simply redefine tempering so that it includes justly
intoned scales!

I am afraid I not only can, but I have to. Just intonation in non-electronic
instruments, let alone idiophones (!), is a myth.

and in 46254:

> Then, I hope you will understand that the bells provide no evidence of
tempering?

> What they do provide is evidence of their designer's and maker's
inability or unwillingness (we can't tell how much of each) to tune
them very accurately.

I agree. But we also see that the thirds are much better than could be
expected from a concept of Pythagorean tuning. This alone proves tempering
efforts on the side of the bell casters, however you like to describe this
in terms of tuning maths.

> Of course, the less harmonic and the more rapidly decaying a timbre
is, the harder it is to tell if it is mistuned and therefore the less
need there is to tune it accurately.

I agree. But the sound samples in the spectral analysis section of the
article show that the decay times are long enough and the fundamentals are
dominant enough to have a clear effect on our perception of consonance and
dissonance of intervals.

> I respectfully suggest that you remove the current misleading version
of your article from the web. I look forward to reading a corrected
version in future.

I'll correct the article, as soon as I see an error.

Martin

Dave Keenan:
> > I'm afraid you can't simply redefine tempering so that it includes
> > justly intoned scales!

Martin Braun:
> I am afraid I not only can, but I have to.

Thanks for a good laugh, Martin. I suggest you stick to
neurophysiology or whatever it is you claim to do. Although even there
I worry that you may be too fond of making sensational claims to allow
you to draw only careful conclusions from the data.

I won't spend any more time on this. I think I have taken it far
enough so that the situation is clear to everyone else.

Regards,
-- Dave Keenan

--- In tuning@yahoogroups.com, "Dave Keenan" <D.KEENAN@U...> wrote:
> Dave Keenan:
> > > I'm afraid you can't simply redefine tempering so that it
includes
> > > justly intoned scales!
>
> Martin Braun:
> > I am afraid I not only can, but I have to.
>
> Thanks for a good laugh, Martin. I suggest you stick to
> neurophysiology or whatever it is you claim to do. Although even
there
> I worry that you may be too fond of making sensational claims to
allow
> you to draw only careful conclusions from the data.
>
> I won't spend any more time on this. I think I have taken it far
> enough so that the situation is clear to everyone else.
>
> Regards,
> -- Dave Keenan

thanks for pursuing this, dave. it seems both you and martin agree
that the tuning reflects an attraction toward the 5-limit, and
doesn't simply represent the stereotypically chinese 3-limit or
"pythagorean" tuning. i note that neither of you seemed to make a
distinction between *standard deviation* and *standard error*. in my
line of work, at least, it's the *standard error* and not the
standard deviation that quantifies the uncertainty associated with
the _mean_ of a statistical sample.

--- In tuning@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> thanks for pursuing this, dave. it seems both you and martin agree
> that the tuning reflects an attraction toward the 5-limit, and
> doesn't simply represent the stereotypically chinese 3-limit or
> "pythagorean" tuning.

No Paul, I'm afraid the data doesn't even support _that_ assertion.

I had not previously tested the fit of a pythagorean tuning (with pure
octaves) for FCGDAE, against the bell tuning data. Now that I have, I
can report that the error is 21.9 cents RMS.

I remind you that Martin's tempered model with pure octaves and 696.2
cent (meantone) fifths has an error of 19.8 c RMS, and the model with
all pure fifths, but half-comma wide octaves (and hence wide fourths)
has an error of 17.7 c RMS. Now Martin says (and I agree) that the
difference between 17.7 c and 19.8 c is not significant, so he must
also agree that the difference between 19.8 c and 21.9 c is not
significant.

The difference between 21.9 c and 17.7 c might be considered
significant, but the change that occurs here is one of widening the
octaves (or equivalently the wholetones or the fourths) while the
fifths remain pure.

Assuming it is significant, the reason for widening the octaves would
seem to be to improve the thirds in the scale, however this need not
be the case. Given the size and inconsistency of the errors it would
be difficult to even recognise the impossibility of having all
thirths, fourths, fifths and octaves pure. The tuning may well have
resulted from attempting to tune a Pythagorean scale with pure fifths
as usual, while simultaneously making the individual bells sound as
good as possible (which requires something like pure minor or major
thirds between their two "fundamentals").

The distinction I'm trying to make here is between caring which thirds
approach pure based on where they are in the scale, and caring which
approach pure based only on whether they occur on a single bell.

i.e. if there is a change towards pure thirds it may well have been
"technology driven" rather than "thought driven". As an example of
this, I just learnt that all those famous ancient greek bronze statues
of the human form that proliferated around 450 to 400 BCE, that were
previously thought to represent a major leap forward in artistic
_thought_ towards greater realism, probably represent "only" a major
leap forward in _technology_, for making molds from actual live humans!

Apparently someone intimately familiar with both human anatomy and the
technology of mold making and casting noticed the incredibly accurate
detail on the _underside_ of the feet. However, there was still
apparently some artistry involved in "improving" the resulting wax
positive.

Yet another possible model for the bell tuning is

C---G---D---A---E
\ /
F

making the CGDAE pentatonic strictly Pythagorean (with pure octaves)
while the F's just come along for the ride on the D bells. After all,
the D bells might as well have _something_ as their second note, and
best to make it as close to a pure interval as possible so it
interferes least with the D.

This model has an RMS error of 20.6 cents.

> i note that neither of you seemed to make a
> distinction between *standard deviation* and *standard error*. in my
> line of work, at least, it's the *standard error* and not the
> standard deviation that quantifies the uncertainty associated with
> the _mean_ of a statistical sample.

Thanks Paul,

I must admit to being statistically unsophisticated. I just copied
Martin's usage, thinking it would be one less brick wall to bang my
head against. ;-)

Having looked it up in a textbook, I now understand that the standard
deviation estimates the accuracy of the population from the sample
while the standard error measures the accuracy with which the sample
mean can be expected to approach the population mean.

standard_error = standard_deviation/sqrt(n) where n is the sample size.

Given that the sample size is either 8 or 9 for all the pitches and
intervals under discussion (FCGDAE), dividing by sqrt(n) wouldn't
change the relative sizes very much. Do you see that this, or their
absolute sizes, might change any of my conclusions?

Do you agree that this question of standard error versus standard
deviation has no bearing on the RMS numbers I'm quoting above? I'm
simply measuring the difference between the model and the data by
summing the squares of the differences between the predicted and
actual pitch in cents for all 51 pitches, then dividing by 51 and
taking the square root. I adjust a single offset for all notes in the
model until I get a minimum in this RMS value (a best fit). Then I
quote this minimum RMS value.

Regards,
-- Dave Keenan

--- In tuning@yahoogroups.com, "Dave Keenan" <D.KEENAN@U...> wrote:

> > i note that neither of you seemed to make a
> > distinction between *standard deviation* and *standard error*. in
my
> > line of work, at least, it's the *standard error* and not the
> > standard deviation that quantifies the uncertainty associated
with
> > the _mean_ of a statistical sample.
>
> Thanks Paul,
>
> I must admit to being statistically unsophisticated. I just copied
> Martin's usage, thinking it would be one less brick wall to bang my
> head against. ;-)
>
> Having looked it up in a textbook, I now understand that the
standard
> deviation estimates the accuracy of the population from the sample
> while the standard error measures the accuracy with which the sample
> mean can be expected to approach the population mean.
>
> standard_error = standard_deviation/sqrt(n) where n is the sample
size.
>
> Given that the sample size is either 8 or 9 for all the pitches and
> intervals under discussion (FCGDAE), dividing by sqrt(n) wouldn't
> change the relative sizes very much.

a factor of 3 isn't very much?

> Do you see that this, or their
> absolute sizes, might change any of my conclusions?
>
> Do you agree that this question of standard error versus standard
> deviation has no bearing on the RMS numbers I'm quoting above? I'm
> simply measuring the difference between the model and the data by
> summing the squares of the differences between the predicted and
> actual pitch in cents for all 51 pitches, then dividing by 51 and
> taking the square root. I adjust a single offset for all notes in
the
> model until I get a minimum in this RMS value (a best fit). Then I
> quote this minimum RMS value.

what i'm saying is that one can state the standard error of the
observed data, and then make a statistical judgment as to whether
certain theoretical tuning models can be rejected as falling outside
a reasonable probabilistic confidence interval from the data. though
i haven't examined the data in detail, i expect that more of the
tuning models that you guys have proposed can be rejected if one uses
the standard error instead of the standard deviation. if we assume
the errors were random deviations from an underlying model, you'd
expect the predictions of the model to fall within one standard error
of the data 68% of the time.

--- In tuning@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> --- In tuning@yahoogroups.com, "Dave Keenan" <D.KEENAN@U...> wrote:
> > Given that the sample size is either 8 or 9 for all the pitches and
> > intervals under discussion (FCGDAE), dividing by sqrt(n) wouldn't
> > change the relative sizes very much.
>
> a factor of 3 isn't very much?

I said "relative" sizes. I meant sizes relative to each other, not
relative to the means. Sqrt(8) isn't very different from sqrt(9).

--- In tuning@yahoogroups.com, "Dave Keenan" <D.KEENAN@U...> wrote:
> --- In tuning@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> > --- In tuning@yahoogroups.com, "Dave Keenan" <D.KEENAN@U...>
wrote:
> > > Given that the sample size is either 8 or 9 for all the pitches
and
> > > intervals under discussion (FCGDAE), dividing by sqrt(n)
wouldn't
> > > change the relative sizes very much.
> >
> > a factor of 3 isn't very much?
>
> I said "relative" sizes. I meant sizes relative to each other, not
> relative to the means. Sqrt(8) isn't very different from sqrt(9).

oh, i didn't know some were 8 and some were 9. but the absolute sizes
of the confidence intervals, important for hypothesis testing, is
what i was really thinking about.