Re: [tuning] Digest Number 952

I wrote:

>Assume an honest to goodness 12ET on a piano that begins getting its

>octaves stretched upward beginning in the region of middle C (C4).

And Paul asks:

>>You mean don't stretch the lowest octaves at all? Wouldn't that sound

>>horrible, especially given the inharmonicity of the piano?

Greetings,
Forgive the ambiguity. The "stretch" I was speaking of was intentionally
spreading octaves beyond what is necessary to satify ET. All octaves are
stetched somewhat, the pure fifths temperament goes beyond that, so I shall
try to explain.
There is no such thing as a pure octave between strings that are producing
the harmonic series. If the notes agree at their 2:1 ratio, the the common
partials at the 4:2 and 6:3 will be beating. Fix Justness at the 4:2 and the
6:3 will slow donw, but the 2:1 will beat. This is not a problem in the
middle of the keyboard, but the cumulative effect to deal with this shapes a
curve of sharpness or flatness as one heads for the extremes of treble or ba
ss, repsectively.
A careful tuner allows the gradual widening of the octaves to occur so
that the relationships of fourths, fifths, and thirds (or tenths) will
continue to reflect their relationships of the temperament as one progresses
upward. However, by C6 on a piano, if the single octaves are still pure,
the double octaves will sound flat. If the double octave sounds pure, the
single octave beats. So, if the octaves are kept at the 2:1 size, the treble
begins to sound flat, and the bass begins to sound sharp, (which is perceived
as "weak"). "Stretching" solves this.
Tuning a piano's bass section requires differing sizes of octaves. In the
middle of the piano, tuning the octave so that the 2:1 partial correspondance
doesn't beat is normal, though a slight widening makes the unavoidable
expansion later to be more gracefully dealt with. However, as you head
downward, the need for widening the octave shows itself in wider and wider
octaves which favor the 4:2 and then even farther out from center, the 6:3.
To do otherwise would cause the partials from the bass strings to depart from
the fundamentals of the notes far above.
When I described the pure fifths style of tuning, I was describing a
tuning that stretches the octave enough to purify the fifths from where they
were in the temperament octave, making the thirds faster than they would
normally be, and making the octaves in the range of C5 on up beat more than
what has normally been considered acceptable. You will find pure triple
octaves in this tuning,but the smaller spans have quite a bit of "roll" to
them.
There is no need to stretch for pure fifths in the bass, Plomp and
Levelt's critical band stuff explains why we wouldn't hear the difference, so
there are other factors that determine how much stretch works best down
there.
I stretch the bottom octaves more in my non-studio tuning,(classical and
jazz), but in the commercial recording environment, sharpness in the bass
reduces the number of complaints I get from bass players. For a commercial
tuner, this is reason enough to reduce the stetch down there to a minimum.
Good is what doesn't slow a session down, everything else is evil
...........
Regards,
ED Foote
Nashville, Tn.

>All octaves are
>stetched somewhat, the pure fifths temperament goes beyond that,

Thanks for clarifying that. I guess the piano is a poor example to
illustrate the pure fifths temperament.

>There is no such thing as a pure octave between strings that are producing
>the harmonic series.

You mean the inharmonic series?