Frequencies

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Frequencies

Closing and opening reeds


Blow reeds are riveted to the upper reed plate, draw reeds to the lower plate (holding your harp in the usual way). Normal blow notes are mainly generated by blow reeds, normal draws by draw reeds.
Imagine that you blow very gently into a harp channel - so gently, that there is no sound. The blow reed will all the same move a tiny little bit towards the reed plate, thereby tending to close the gap in the plate. The latter will also be true for the draw reed if one draws very gently. This common feature of both kinds of reeds will be described by the notion of a closing reed (see the table).
At the same time the second reed in the channel will respond reversed. Blowing very gently will cause the draw reed to move away from the reed plate, thus opening the gap a little bit more. The same is true for the blow reed if one is drawing. We will call the respective reed an opening reed (see the table).

Playing frequency


The red dots in the diagram depict the movement of the tip of an oscillating harmonica reed. Each red dot represents a snapshot of the reed tip. Looking from the left to the right and combining all snapshots to a movie (a GIF-animation), one can  "see" the reed tip move up and down.



A sustained note with constant loudness implies reed movements  up - down - up - down ... . A normal draw note on channel #4 of a C-harp corresponds to 600 repetitions of the pattern up-down-up (or down - up - down) per second.

This amounts to a playing frequency f of 600 Hz (Hertz). One single oscillation lasts a time which is called oscillation period T. If there are 600 oscillations per second this oscillation period T will be the 600th part of one second, i. e. 1/600 seconds = 0.00167 seconds. Or, the other way round: If one oscillation lasts 0,00167 seconds, you will get the overall number of oscillations per second (i. e. the frequency = "how many Hertz") by dividing 1 (second)  by 0,00167.


James Antaki's turboharp ELX makes those reed movements audible (listen to the audio clip).


Eigen frequency


Plucking a reed you can hear its eigen frequency - the pitch of the decaying sound.


The reed tip still moves up and down, but the movement weakens from one "up" to the next. It makes still sense to talk about oscillation period and frequency (although the oscillation is not periodic in a litteral sense). The notion eigen frequency comes from the German word eigen which means own. I have depicted a realistic waveform together with an idealized picture of tip motion. The recording lasted for 0.2 seconds, so there are about 120 (invisible) periods up-down-up. The decrease in the diagramm (red dots) is greatly exaggerated for clarity.

(For experts: What I call eigen frequency is strictly speaking a damped eigen frequency. For harmonica reeds there is practically no difference, so I use the shorter notion.)


This audio clip presents the sound of a plucked D-reed (hole #4 of a C-harp). A tuning device or an appropriate smartphone app will demonstrate that this eigen frequency lies above the playing frequency recorded in the audio clip above.
One can actually hear that playing frequency of a normal note is below eigenfrequency. It sounds deeper than the plucked reed (this is one reason why harp tuning is not quite as simple as tuning for example a guitar string).
Playing experience with all conceivable kinds of air-driven instruments as well as theoretical work by several physicists can be summarized in a simple rule of thumb:
Playing frequencies of closing reeds are lower as eigen frequency, and playing frequencies of opening reeds are higher.

This holds for all kinds of notes on the harmonica, also for bend and overbend notes. Saxophones sound below the eigenfrequency of the (closing) saxophone reed. Trumpets sound above the eigen frequency of the player's lips (opening "reed").


There are always two
Up to now we always considered one single reed. But there are two of them in each channel: a draw reed and a blow reed. Whether you're blowing or drawing: there will always be one closing and one opening reed.

The following recordings (again with the help of a turboharp ELX) prove that both reeds in the channel are collobarating - no matter, what kind of note you are playing.

In the first three audio clips I am starting with a normal draw D on channel #4 of a C-harp and then bending it down as far as possible. You can first listen to the motion of both reeds (black), then exclusively to the blow reed (yellow), then only to the draw reed (magenta).
The recordings can only give a rough impression how strong the reeds actually oscillate. We are for example more sensitive for low volumes (we hear logarithmically, as experts say).

Measurements confirm: Normal notes are mainly generated by the closing reed, but the opening reed is also involved. If the closing reed moves 1mm up and down, the opening reed moves approximately 0.1mm.

Bending reduces the movement of the closing reed and augments the contribution of the opening reed. Bend notes with the lowest attainable pitch are mainly generated by the opening reed.


The next three audio clips present a normal draw B and successively lower draw bends on hole #3 of a C-harp. The lowest draw bend note Ab is mainly generated by the blow reed.




Next, there is a normal blow C and an overblow (near Eb) on hole #4 of a C-harp. The overblow is mainly generated by the draw (!) reed D.




Finally, a normal blow E on hole #8 of a C-harp which is bent down. The lowest blow bend is nearly exclusively generated by the draw reed.


A physical point of view


The diagram below shows all notes attainable for me on channel #4 of a C-harp. I worked with single reeds (all other reeds tightly taped) and with both reeds in the channel. From left to right: single blow reed, single draw reed, both reeds. The green lines mark the eigen frequencies of the reeds (plucked C, plucked D).

Numeric indications like 100ct  (cent) measure how different intervals are appreciated by us (we hear not only volume, but also frequencies "logarithmically"). Tuning devices show 0 (cents) if we meet the right pitch of a note, deviations are shown in cents. The difference between two seminotes is 100ct, so an octave sums up to 1200ct. In the diagram 0 (zero cents) stands for the playing frequency of a normal C.



Let's interpret the diagram! Remember that 100ct correspond to one halftone. See also the note names at the right margin. Each single reed can be a closing or an opening reed, which is indicated bei red and blue.

The eigen frequency of the draw reed D lies above the eigen frequency of the blow note C. This is true for holes #1 to #6 and is reversed for holes #7 to #10. This is a decisive observation. For holes #7 to #10 there should be an extra diagram (you may work out one for your own)!

Single reed

  • A closing reed will sound below its eigen frequency. This happens playing normal notes or bends. Mind that it is possible to blow bend or draw bend each  reed on the whole harmonica if (!) the respective other reed plate is taped so that one is playing with single reeds.
  • An opening reed will sound above its eigen frequency, which allows for overblows and overdraws. It is possible to bend such overbends to higher pitches.

Single reeds are perfectly described by the above rules of thumb. The frequency bands below resp. above eigen frequency allow for continuously bending down the normal note resp. bending up the lowest possible overbend.

Two reeds

Although it was emphasized above that always both reeds are collobarating, the diagram shows only major contributions for the sake of clarity. We will nevertheless always discuss the role of both reeds.

With one exception (no blow bend!) the diagram for coupled reeds looks like the "sum" of the two diagrams for single reeds. Similar diagrams exist for channels #7 to #10. The physically motivated idea of closing and opening reeds allows for a unified and simple discussion of normal blows, normal draws, blow bends, draw bends, overblows, and overdraws!

  • Normal notes come mainly from the closing reeds, which sound somewhat below its respective eigen frequencies. For a normal D, the second reed (C) is opening and thus can oscillate with the higher frequency. For a normal C, the second reed (D) should be opening, so it should not join an oscillation with lower frequency. Actually measured oscillations are not very strong, but they exists. We must accept that this observation shows the limitations of our simple physical assumptions and models.

  • Draw Bends can be explained with ease: The upper reed D is closing, so it can sound with lower frequencies. At the same time the lower reed C is opening, so it can join oscillations with higher frequencies. Within the region of the two overlapping frequency bands one can bend normal D continuously down. One further remark: It is obviously only a rule of thumb that the pitch of the lowest draw bend is one half step above the pitch of the lower reed - this depends on the instrument and on the player.

    Comparing with the diagrams for the two single reeds, one might be tempted to describe a draw bend on coupled reeds as draw bending the draw note and overdrawing the blow note. This is especially true for the lowest draw bend which comes mainly from the blow reed.

    It is not as easy to bend blow notes on the upper channels #7 to #10 continuously down. Playing the normal note as a starting point with small volume between teeth and tongue may help.

  • Overblows and Blow bends: The lower reed C is closing, so one should expect low blow bends (as for a single reed). Instead the upper reed D "wins" in playing an overblow (or an upward bended overblow) as an opening reed with higher frequency.  There is no explanation for this preference of overblows within our physical model. Nor can the small (but existing) oscillations of the lower (closing) reed above its eigen frequency be explained.

    Starting with a normal note C and playing a smooth transition up to Eb is impossible, because closing reed C has to sound below normal C, while opening reed D has to sound above normal D. Therefore overblows seem to "pop up".

    A look at the diagramm shows that normal C lies a scant below plucked C, plucked D lies two semitones above plucked C, and closing reed D sounds a scant above eigen frequency of D. Summing up results in a scant minor third, so the player has to bend this overblow note a little bit to play an Eb.
Summary

For a physical description of normal blow notes, normal draw notes, blow bends, draw bends, (bended) overblows and (bended) overdraws one pair of notions, closing and opening reeds, together with a rule of thumb suffice largely (with some phenomena remaining which cannot be explained within this simple assumptions.)

  • Normal notes come mainly from closing reeds with playing frequencies somewhat below eigen frequency.
  • Both reeds work together in generating bend notes. Playing frequency lies lower than eigen frequency of the closing reed and higher than eigen frequency of the opening reed. One can bend the normal note with the higher pitch gradually down because there are two overlapping frequency bands. Lower bends come mainly from the opening reed, so one might interpret them as "overbends".
  • (Bended) Overbends are created by an opening reed, sounding higher than the normal note associated with this reed. This explains why overbends seem to pop up at least a minor third above the pitch of the closing reed in the channel. Bending overbends up is possible within the allowed frequency band above eigen frequency.
 
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