clarinet fingering and "tritave equivalence"

i just had an interesting thought about learning
the fingering of the notes on the clarinet.

all other woodwind instrument families (flute, oboe,
saxophone, and bassoon) overblow at the "8ve".

"overblow" is a standard wind instrument term
meaning that when the player blows harder the
pitch of the note for any given fingering shifts
to a higher register.

the frequency ratios of the notes in this series
of shifts in pitch follows the harmonic series
from the lowest note -- blown with the softest
breath and loosest embouchure -- thus:

1:2:3:4:5:6:7:8... etc.

on brass instruments, it is standard to use
as many as 8 or more harmonics by tightening
the lips to reach the higher-integer harmonic
ratios.

on woodwinds, it is standard to use the same
fingering for notes in the 1st (lowest) register
and 2nd (next higher) register. then by blowing
a little harder the player can reach the 3rd register,
but here generally there are alterations which
must be made in the fingerings to "vent" other
holes elsewhere along the tube, in order for the
notes to "speak" clearly.

on the bassoon in particular, it is possible to
reach a 4th register with basically the same
fingerings.

but notice that suddenly i've changed my terminology
of the pitch-shift to "register" instead of "8ve".

that's because the clarinet is cylindrical instead
of conical, as the oboe and bassoon, and as a result
it overblows at the "12th" ("8ve" + "5th" ... i know,
traditional musical arithmetic isn't the same as
regular arithmetic ...) instead of at the "8ve".

(the modern flute has been cylindrical in shape since
Boehm's 1847 model, but the acoustical properties of the
flute are a bit different from the "stopped pipe"
clarinet, so it still overblows at the "8ve".)

this means that, because the notes a "12th" apart
have different letter-names (i.e., E and B, F and C,
G and D, etc.), everyone who learns to play the clarinet
learns that the same fingering represents notes which
have these paired letter-names.

this seems to me to be another instance of "tritave
equivalence". (the "tritave" was coined as a name
for the interval of a "12th" in the Bohlen-Pierce
scale, where it really is the interval of equivalence.)

i've mentioned another "naturally ocurring" one before
in the case of a birthday party where everyone, those
who have some musical training and those who have none,
sings _Happy Birthday_, and inevitably some of the people
with no musical training will sing the entire melody
a parallel 5th above the real melody.

so now i'm wondering if there is some special mental
process that occurs in clarinet players which makes it
easier to allow them to accept "tritave equivalence",
because of the connection they must make between the
pairs of notes (letter-names) which have the same fingering.

-monz

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> i just had an interesting thought about learning
> the fingering of the notes on the clarinet.
>
>
> all other woodwind instrument families (flute, oboe,
> saxophone, and bassoon) overblow at the "8ve".
>
>
> "overblow" is a standard wind instrument term
> meaning that when the player blows harder the
> pitch of the note for any given fingering shifts
> to a higher register.
>
> the frequency ratios of the notes in this series
> of shifts in pitch follows the harmonic series
> from the lowest note -- blown with the softest
> breath and loosest embouchure -- thus:
>
> 1:2:3:4:5:6:7:8... etc.
>
> on brass instruments, it is standard to use
> as many as 8 or more harmonics by tightening
> the lips to reach the higher-integer harmonic
> ratios.
>
> on woodwinds, it is standard to use the same
> fingering for notes in the 1st (lowest) register
> and 2nd (next higher) register. then by blowing
> a little harder the player can reach the 3rd register,
> but here generally there are alterations which
> must be made in the fingerings to "vent" other
> holes elsewhere along the tube, in order for the
> notes to "speak" clearly.
>
> on the bassoon in particular, it is possible to
> reach a 4th register with basically the same
> fingerings.
>
> but notice that suddenly i've changed my terminology
> of the pitch-shift to "register" instead of "8ve".

Not really -- it was only an octave when going from 1 to 2. Going to
the 3rd register from either is not an octave shift, even on these
instruments.

> that's because the clarinet is cylindrical instead
> of conical, as the oboe and bassoon, and as a result
> it overblows at the "12th" ("8ve" + "5th" ... i know,
> traditional musical arithmetic isn't the same as
> regular arithmetic ...) instead of at the "8ve".
>
> this means that, because the notes a "12th" apart
> have different letter-names (i.e., E and B, F and C,
> G and D, etc.), everyone who learns to play the clarinet
> learns that the same fingering represents notes which
> have these paired letter-names.

Only in the particular case of the lowest two registers. The next
register shift on the clarinet takes you up another major *6th*, 3:5
ratio.

> this seems to me to be another instance of "tritave
> equivalence". (the "tritave" was coined as a name
> for the interval of a "12th" in the Bohlen-Pierce
> scale, where it really is the interval of equivalence.)
>
> i've mentioned another "naturally ocurring" one before
> in the case of a birthday party where everyone, those
> who have some musical training and those who have none,
> sings _Happy Birthday_, and inevitably some of the people
> with no musical training will sing the entire melody
> a parallel 5th above the real melody.

Monz, this here is not an example of tritave equivalence, or tritave
anything.

The perfect fifth (3/2) is no more an interval of equivalence under
tritave (3/1) equivalence than it is under octave (2/1) equivalence.

Under octave (2/1) equivalence, the perfect fifth (3/2) is equivalent
to the perfect twelfth (3/1) but neither is an equivalence interval.

Under tritave (3/1) equivalence, the perfect fifth (3/2) is
equivalent to the sub-octave (1/2) but neither is an equivalence
interval.