Date: Sat, 28 Mar 98 5:04:22 PST
From: "J. Yoshii" <>
Subject: Re: Slotting (was Longish, is definitely long now)



Very interesting observations.  I have a couple of ideas, but perhaps I should insert this (does NOT apply to the quoted bits, only to my drivel!) before proceeding, just to be on the safe side:


BTW, I am using the note-naming convention of c' = first c below treble staff, c''' = first c above treble staff.  (You don't like this?  Too bad.  Any "your note-naming system is wrong" flames I get will go straight to /dev/null .)


> Now comes what, IMHO, is the really interesting part.  If we look at the
> diagrams in the technical books the peaks (slots) are pretty much gone
> around high G.  So how is it that lead players can make the double C
> literally pop out?

IIRC, Arthur Benade's musical acoustics books have some resonance curves (for trumpet with mouthpiece and bell) which look something like

     0) curve for trumpet (with bell) + mouthpiece
    |                 *
    |          *      *
    |          *      *
    |          *      *    *
    |  *       *      *    *   *
    |  *       *      *    *   *  * *
    |__*_______*______*____*___*__*_******__________ frequency
       c'      g'     c''  e'' g''c'''

which I suppose could be interpreted as a sort of "probability that a trumpet can produce this note (assuming no valves are pressed)".

This curve was the superposition of two curves (transfer functions?), one a frequency spectrum (mp + trumpet up until bell?) shaped like

     1) curve for (trumpet (no bell?) (+ mp?) ?)
    |  *
    |  *       *
    |  *       *      *
    |  *       *      *    *
    |  *       *      *    *   *
    |  *       *      *    *   *  *
    |  *       *      *    *   *  * **
    |__*_______*______*____*___*__*_******__________ frequency
       c'      g'     c''  e'' g''c'''

and a curve (from the bell?) which modifies the spectrum and is shaped like

     2) curve for (bell?)
    |           **********************************
    |         *
    |      *
    |   *
    |*______________________________________________ frequency
       c'      g'     c''  e'' g''c'''

Benade used a machine to drive vibrations in trumpets.  The apparatus was some sort of piston driving a membrane attached to the rim of the mouthpiece, I think, and not something resembling human lips which are free to vibrate within the rim and through which air is passing.  Who knows what amplitudes were attained by the membrane's vibrations. Your observation,

> I do a lot practicing on a ring visualizer.  I figured that without the
> horn, I should be able to hit a smooth continuum of notes.  OOPS!  I
> missed something.  As soon as I confine some lip mass in a ring, I have
> a resonant system.  Guess where the fundamental seems to be?  For me its
> right at double C!  I can make double C's pop out on a ring visualizer.

would likely be harder to make with such a piston/membrane device, especially if the membrane was stretchy (i.e., dissipates energy) at double C-ish frequencies.  I do not remember seeing resonance curves for the mouthpiece alone; then again, I have not looked at his books in several years.  Quite possibly he did not do as detailed a study of just the mouthpiece.

Supposing there is actually a mouthpiece curve which should be factored in (and assuming a mouthpiece behaves similarly to a visualizer), it might look something like

     3) curve for mouthpiece only             ^
    |                                         *
    |   mp can be driven at these             * <-- spike for
    |  v  non-resonant frequencies            * resonance at
    |                                         * double c
    |**********************************       *
    |                                  *
    |                                   ******
    |_______________________________________________ frequency
       c'      g'     c''  e'' g''c'''        c''''

In any case, instead of Benade's total resonance curve (no. 0) which dies away above g''', there is now enough of a peak at c'''' for the note to slot.  Multiplying 1), 2), 3) ==>

     4) hypothesized total curve for tpt+mp
    |                 *
    |          *      *
    |          *      *
    |          *      *    *
    |  *       *      *    *   *
    |  *       *      *    *   *  * *         *
    |__*_______*______*____*___*__*_***********_____ frequency
       c'      g'     c''  e'' g''c'''        c''''

Just a hypothesis, but it is consistent with your conclusion,
> IMHO, the slot that we have around double C is by and large brought on
> by the lip mass, the rim and the mouthpiece.  The trumpet isn't supposed
> to resonante up there.

I am not sure how to test this hypothesis, though.  A setup like Benade's might not be able to detect the spike at double c.  Perhaps some sort of artifical lip would be needed.

> QUALIFIER!!  Now there something that still bugs me.  If my little
> theory is true, I should be able to tell no difference between horns
> when I play double C's.  However, I can.  My Shew Horn clearly barks
> much better than my Benge or my Tanabe Horn (sorry Wayne - FWIW my
> Tanabe horn is my first choice for legit B-flat work, but I have a
> tougher time way up there on it).  I don't have a good explanation for
> this.  Any ideas?

Notice that Benade's mouthpiece driver is not affected by what comes after the mouthpiece -- it would not be affected by things like air flow resistance from the trumpet, or fatigue, or the many other factors which influence humans.  Another transfer function needs to be added into the picture, one for the player (dependent on some specific choice of trumpet).

For example, I am lousy at playing below c' and cannot go above d'''. If I had to describe how I affect the sounds coming out of my trumpet , it would be (this is time-, physical_state-, etc.-averaged):

     5) curve which shows how I will degrade the tpt+mp system (4)
    |  **************************
    | *                          **
    |*                             *
    |_______________________________*_______________ frequency
       c'      g'     c''  e'' g''c'''        c''''

So, my probability of playing notes (with no valves pressed) would be the product of 4) and 5),

     6) curve for tpt+mp+me
    |                 *
    |          *      *
    |          *      *
    |          *      *    *
    |          *      *    *
    |  *       *      *    *   *
    |__*_______*______*____*___*__*_________________ frequency
       c'      g'     c''  e'' g''c'''        c''''

which is now attenuated above and below the staff.  This makes sense: I am much more likely to clam (or otherwise sound bad) if I go very much above or below the staff.  Some of the other trumpets I tried, before I chose mine, were much freer blowing above the staff -- curve no. 5 might not die down as quickly in that case (OTOH I did not think they sounded as nice; I do wish my trumpet were as free blowing as that Holton(?) Maynard Ferguson model I tried, though).

Nick, your curve no. 5 would obviously extend out _much_ further, but would be shaped differently depending on the trumpet you were playing: the function's amplitude would be larger, at higher frequencies, for your Shew horn than for the other two.

Now the hard part: how on earth would one test this hypothesis?  No scientific instrument (that I am aware of) can measure how "easily" one can play a note.  It's all very subjective and could be wildly variable (depending on how much playing has already been done that day, one's physical condition, etc.).

I don't know...maybe one could make Benade-style (no. 0) curves for several trumpets; find some players and measure curve no. 3 from their mouthpieces; ask the players to play the trumpets and estimate curves
no. 5 and no. 6 for each one (or else generate an experimental curve no. 6 by having the players attempt to play each note many times, and recording the fraction, #hits/#clams); and then try to find some correlation between experimental curves nos. 6 and theoretical curves obtained by multiplying no. 0 x no. 3 x no. 5?  I should think the
error bars would be pretty horrendous, but maybe some useful information might be found.

> I hope I haven't slipped up here with this cursory coverage.  Perhaps
> Chris Straton could shed some light on this.  His technical ideas are
> always right on.
> I'm sure this more than you bargined for.


Slipped up?  Not at all.  Yours was a nice, thought-provoking posting.  It will be interesting to see what other insights will turn up.

Oh well, back to lurking.

Jean Yoshii