Synthesizer resolution

On Tue, 28 Apr 1998, Daniel Wolf wrote:
> If I can't have a tuning with the
> accuracy of the Rayna, then I would like the potential deviation from Just
> to be spread around as much as possible in a temperament that represents
> harmonic identities consistently. As long as you are not working with
> sustained textures that really require something like the Rayna, I think
> that absolute frequency resolution will be secondary to consistency.
[snip]
> From my experience,
> 768tet and 1200tet are not good choices but there are ET of this magnitude
> that would be better.

I went through my big consistency table and picked out some of the ETs
between 600 and 1200 that looked good for this purpose. I didn't use
any hard and fast criteria, but just eyeballed ETs that either were
consistent to unusually high odd limits, or had particularly high
consistency levels at lower limits, or some combination. Here are the
results (I left in 768 and 1200 for comparison, but set them off):

| 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31
-----------------------------------------------------------------
*612| 169 21 4 4 2
615| 2 2 . . . . . .
*624| 30 3 2 2 . . . . . . . . .
639| 2 . . . . . . .
643| 3 3 3 . . . . . . .
653| 25 2 2 . . . . . . .
665|7939 6 2 2
*684| 4 2 2 2 . . . .
692| 2 2 . . . . . . .
711| 5 4 3 2 . . .
718| 162 3 . . . . . . . . .
730| 22 22 . . . . .
742| 11 2 2 2 2 . . . . .
764| 5 5 2 2 2 . . .
-----------------------------------------------------------------
768| . . .
-----------------------------------------------------------------
771| 82 2 . . . . . . . .
776| 7 2 . . . . . .
783| 19 7 3 3 .
795| 11 4 2 2 . . .
805| 4 . . . . . .
814| 3 3 2 . . . . .
836| 17 3 2 2 2
848| 10 9 . . . . .
867| 3 3 2 . . . . . . . . .
882| 7 7 3 2 . . . .
894| 11 2 2 2 . . . .
908| 3 . . . . . . . . . .
925| 5 . . . . . . .
935| 8 7 3 3 . . . . . . . . .
954| 9 4 2 2 . . . .
981| 3 2 2 . . . . .
995| 13 . . . . . . .
997| 2 2 . . . . . . . . . .
1012| 27 2 . . . . .
1019| 6 6 . . . . . .
1048| 12 . . . . . .
1065| 33 3 2 2 . . . . . .
1075| 3 2 2 . . . . .
1084| 5 3 2 2 2 . . .
1106| 15 5 4 4 2 . . .
1118| 41 5 . . . . . .
1137| 4 4 3 2 . . . .
1142| 18 . . . . . . . . . .
1171| 56 22 . . . . . . . . . . .
1178| 5 2 2 2 2 2 . . . .
1186| 2 2 . . . . . .
-----------------------------------------------------------------
1200| 11 . . . .

Unfortunately the only ones which are divisible by 12 (which I've marked
with *s) are in the low end of the range, but this was just a rough
pass. Perhaps more careful scrutiny might yield a better compromise.

--pH http://library.wustl.edu/~manynote
O
/\ "Churchill? Can he run a hundred balls?"
-\-\-- o
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Gary wrote,

>I have heard that too, from Ivor Darreg, who did a lot of piano tuning
>in his day. If I remember correctly, he said that it would tend to go out
>of tune more quickly in anything other than equal-temperament.

If the total tension of the strings were held constant, then there is no
reason that tuning to an unequal tuning would lead to less tuning
stability than one had in equal temperament.

Daniel wrote,

>There are arguments about ET and the piano based upon
>the stretched intonation of the partials of the
>rigid piano wire. I have great trouble in supporting
>these arguments historically, however, because the
>early pianos were wired at a much lower tension
>than modern instruments and were considerably less
>stretched. (La Monte Young's _Well Tuned Piano_ =

>deliberately uses a lower tension to reduce, but not
>eliminate the stretching.

That seems contrary to the evidence as well as scientific theory. Check
out

_Change in the Characteristics of Piano Tones under Different Concert
Pitches_
by
Tomoyasu Taguti and Osamu Tokuyama
Konan University, Higashinada, Kobe, Japan

presented at

University of Edinburgh
INTERNATIONAL SYMPOSIUM ON MUSICAL ACOUSTICS
19-22 August 1997:

"This paper reports an acoustical measurement and a listening experiment
on the tones of a piano where A4 is tuned to 436Hz(L:low),
442Hz(N:normal), and 445Hz(H:high)"

"Physically, . . . the inharmonicity of single piano strings . . .
becomes decreased as the pitch is raised. ([This] is implied in fact by
the formula of inharmonicity by Fletcher.)"

"the inharmonicities in cents were {20(L), 17(N), 10(H)} at the 10th
partial, and {45(L), 42(N), 35(H)} at the 20th partial."

On Wed, 29 Apr 1998, Paul H. Erlich wrote:
> Gary wrote,
> > If I remember correctly, [Ivor] said that it would tend to go out
> >of tune more quickly in anything other than equal-temperament.
>
> If the total tension of the strings were held constant, then there is no
> reason that tuning to an unequal tuning would lead to less tuning
> stability than one had in equal temperament.

That makes sense to me--but I'd hate to go against the hands-on
experience of a tuner. Ed Foote, what say you? (IIRC, Ed was the tuner
on the list who was using unequal temperaments on modern pianos, yes?)

> Daniel wrote,
> >There are arguments about ET and the piano based upon
> >the stretched intonation of the partials of the
> >rigid piano wire. I have great trouble in supporting
> >these arguments historically, however, because the
> >early pianos were wired at a much lower tension
> >than modern instruments and were considerably less
> >stretched. [snip]
>
> That seems contrary to the evidence as well as scientific theory. [snip]
>
> "Physically, . . . the inharmonicity of single piano strings . . .
> becomes decreased as the pitch is raised. ([This] is implied in fact by
> the formula of inharmonicity by Fletcher.)"

As I recall, the greater harmonicity that results from greater string
tension is the reason that historical keyboard instruments are scaled so
that the strings are very near their breaking stress, regardless of the
metal used. Occasionally we have discussions on the Harpsichord list
about this. Intuitively, the phenomenon can be explained this way: as
string tension increases, the component of the restoring force in the
vibration caused by string tension (ideal-stringlike behavior)
increases, becoming greater in relation to the component caused by
string stiffness (bar-like behavior), which remains constant--and ideal
strings vibrate harmonically while bars do not.

--pH http://library.wustl.edu/~manynote
O
/\ "Churchill? Can he run a hundred balls?"
-\-\-- o
NOTE: dehyphenate node to remove spamblock. <*>

Daniel Wolf wrote,

>>(La Monte Young's _Well Tuned Piano_ =

>>deliberately uses a lower tension to reduce, but not
>>eliminate the stretching.

I wrote,

>That seems contrary to the evidence as well as scientific theory.

Well, if LaMonte achieved the lower tension not through lower pitch but
through thinner strings, then Daniel may be correct. Thin strings are
less stiff and therefore less inharmonic, although on a given piano the
short length of the high strings contributes about as much
inharmonicity, as compared with lower strings, as their thinness
eliminates. I apologize for the hasty retort. In any case, I agree with
Daniel than stretched partials do not improve the overall
characteristics of ET compared with any other tuning.

>Daniel Wolf wrote,
>
>>>(La Monte Young's _Well Tuned Piano_ =
>
>>>deliberately uses a lower tension to reduce, but not
>>>eliminate the stretching.
>
>I wrote,
>
>>That seems contrary to the evidence as well as scientific theory.
>
>Well, if LaMonte achieved the lower tension not through lower pitch but
>through thinner strings, then Daniel may be correct. Thin strings are less
>stiff and therefore less inharmonic, although on a given piano the short
>length of the high strings contributes about as much inharmonicity, as
>compared with lower strings, as their thinness eliminates. I apologize for
>the hasty retort. In any case, I agree with Daniel than stretched partials do
>not improve the overall characteristics of ET compared with any other tuning.

Jesse Gay wrote: "...constructing a JI piano creates some serious 'mechanical'
difficulties..."

A couple of thoughts: "mechanical" may have been used here in the sense that
trying to get players to accept a piano with (many) more than the usual twelve
keys per octave as would be required for JI was a "mechanical" problem - or
squeezing many more than twelve keys per octave onto a piano in a way
acceptable to performers was an insurmountable "mechanical" problem for the
piano manufacturers.

Also, my piano tuner was unwilling to shift the pitches of some sets of keys
on my rented piano by 40 cents (which I wanted him to do so as to put it into
a limited JI tuning) because he feared that a shift that big might have a
permanent adverse effect on the strings (hurt their tone quality or else cause
them to become unstable and not able to hold their pitches, etc.). Twenty
cents or so was about as far as he was willing to move their pitches. To
build a piano which could withstand frequent large shifts in the pitches of
some notes might present unusual technical difficulties.

This brings me to the following question which a piano tuner on this list
might be able to shed light on: Is it always risky to a piano to occasionally
move all the Abs by 40 cents or all the Ebs by 40 cents so as to move the wolf
in mean tone around?

Good luck.

Dave Hill La Mesa, CA

> Gary wrote,
> > If I remember correctly, [Ivor] said that it would tend to go out
of tune more quickly in anything other than equal-temperament.

Oddly enough, I have found that ET is the most perishable tuning on a
piano, and I don't know why. Perhaps it is that ET is already so "out of tune"
that changes are quite evident, whereas on a well temperament, all the keys
are different anyway,so the change in tuning is only a change in degree. On
equal temperament, a shift in the tuning destroys the only unique thing that
ET has.
Perception aside, well temperaments and meantone tunings on modern pianos
are more stable than ET, after the initial string alterations are relaxed.

> "Physically, . . . the inharmonicity of single piano strings . . .
> becomes decreased as the pitch is raised. ([This] is implied in fact by
> the formula of inharmonicity by Fletcher.)"

Yes, I have read Fletcher, but it doesn't explain why the inharmonicity of
a piano wire measures higher as the tension is increased. Perhaps there is a
point of reversal, inre tension/inharmonicity.
Regards,
Ed Foote
Precision Piano Works
Nashville, Tn

On Fri, 1 May 1998, A440A wrote:
> Oddly enough, I have found that ET is the most perishable tuning on a
> piano, and I don't know why. Perhaps it is that ET is already so "out of tune"
> that changes are quite evident, whereas on a well temperament, all the keys
> are different anyway,so the change in tuning is only a change in degree. On
> equal temperament, a shift in the tuning destroys the only unique thing that
> ET has.
> Perception aside, well temperaments and meantone tunings on modern pianos
> are more stable than ET, after the initial string alterations are relaxed.

Sorry if I seem to be nitpicking, but I want to be clear on this. Ed,
is this a reasonable paraphrase of what you are saying?

(1) The same degree of more or less random detuning that happens with
time would be more noticable with ET than with an unequal temperament,
because you can immediately hear when the ET becomes non-equal.

(2) However, the same degree of detuning does _not_ happen with time;
measured in cents (or whatever), pianos tuned in ET tend to go farther
out of tune in the same amount of time than those tuned in unequal
temperaments (or go out the same distance in less time).

I'm particularly interested in (2); I think I'm fairly clear on (1).

> Yes, I have read Fletcher, but it doesn't explain why the inharmonicity of
> a piano wire measures higher as the tension is increased. Perhaps there is a
> point of reversal, inre tension/inharmonicity.

This is something I've not heard before, and it sounds very interesting.
Could you go into greater depth about this?

--pH http://library.wustl.edu/~manynote
O
/\ "Churchill? Can he run a hundred balls?"
-\-\-- o
NOTE: dehyphenate node to remove spamblock. <*>