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 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 fading sound.


The reed tip still moves up and down, but the movement weakens from one "up" to the next. It still makes sense to talk about oscillation period and frequency (although the oscillation is not periodic in a literal sense). The notion eigen frequency is derived 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 term.)


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 clearly hear that playing frequency of a normal note is below eigen frequency. 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).

There are always two
The following recordings (again with the help of a turboharp ELX) prove that both reeds in the channel oscillate - 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 tone pitches (marked in color) which I am able to play on channel #4 of one of my C-harps (both reeds oscillate, right half of the diagramm). I was also interested in the question which pitches are attainable with single reeds (all other reeds tightly taped, left half of the diagram).

The green lines mark the eigen frequencies of the reeds (plucked C, plucked D). Like on a tuning device there are no frequencies (in Hertz) specified, but differences in the perceived pitch (in cent). The reason is that we hear not only volume, but also frequencies "logarithmically". The origin (0ct = zero cent) can be arbitrarily chosen, in the diagramm 0ct corresponds to a normal blow note C. The difference between two seminotes is 100ct, so an octave sums up to 1200ct.

For channels #1 to #6 there would be similar diagrams and explanations. Channels #7 to #10 would  require to switch the roles of blowing and drawing (for example blowbend instead of drawbend).


Playing properties 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 than eigen frequency, and playing frequencies of opening reeds are higher.

Saxophones sound below the eigenfrequency of the (closing) saxophone reed. Trumpets sound above the eigen frequency of the player's lips (opening "reed"). We want to apply this rule of thumb to the blues harp. Referring to the diagram we are going to discuss which constraints resp. extensions may be necessary in doing so.

Single reed

  • Single reeds obey the rule of thumb without limitation.
  • Playing with relaxed embouchure and soft breathing, only the closing read will sound. Among all possible playing frequencies the resulting frequency will be next to the eigen frequency of this reed.
  • All other playing frequencies (bends and overbends) require specific configurations of the oral cavity. 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

Our sound samples demonstrate that always both reeds in the channel of a blues harp oscillate, and with common frequency. Hence the rule of thumb can't be valid unrestrictedly. Playing for example a blow note on channel #4, the draw reed is opening, but it oscillates below eigen frequency.

The rule of thumb for single reeds is still very useful also for two reeds in the channel - it just has to be applied to the reed which oscillates more strongly (in fact playing normal notes or overbends one reed oscillates considerably more than the other one, and for bend notes there is no collision with the rule of thumb). In the right half of the diagram (coupled reeds) the colors red and blue refer to the more strongly oscillating reed (resp. to both reeds for bend notes). With one exception (no blow bend!) the diagram for coupled reeds looks like the "sum" of the two diagrams for single reeds.

  • Normal notes: Playing with relaxed embouchure and breathing softly, mainly the closing reed is oscillating. Playing frequency is somewhat below the respective eigen frequency.

  • Draw Bends: For draw notes both frequency bands are overlapping. The higher pitched reed D is closing, so it can oscillate with lower frequencies. At the same time the lower pitched reed C is opening, so it can join oscillations with higher frequencies. Within the region of the two overlapping frequency bands one can bend the normal draw note D continuously down. 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: For blow notes the two allowed frequency bands are mutually exclusive. Bending the closing reed C down would still be conceivable, the opening reed D oscillating only weakly (mind that during bended overblows closing reed C has to oscillate weakly with "forbidden" frequencies, so for blow notes there is always one reed which has to break the rule of thumb). Instead, trying to bend with an appropriate embouchure (playing for example a draw bend in the first instance) one effects no reasonable sounding note at all or an overblow, which one can in addition bend upwards. Now the higher pitched reed D oscillates as an opening reed above its eigen frequency as expected. But our rule of thumb is unable to explain why the overblow is always "winning".

    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 above eigen frequency of D. Summing up this results in a scant minor third. Starting with a normal blow note C, the overblow seems to "pop up". In order to play an Eb with accurate intonation the player has to bend  the overblow somewhat up.
Summary

One pair of notions, closing and opening reeds, and a rule of thumb - closing reeds oscillate below and opening reeds oscillate above eigen frequency - suffice to organize the diversity of notes on the blues harp: normal blow notes, normal draw notes, blow bends, draw bends, (bended) overblows and (bended) overdraws.

Additional assumptions: Playing "normally" on the harp excites mainly oscillations of the closing reed. For normal notes and overbends the rule of thumb applies only to the (substantially) stronger oscillating reed.

  • 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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