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inharmonicity: bowed vs struck

  • 1.  inharmonicity: bowed vs struck

    Registered Piano Technician
    Posted 09-22-2020 07:16
    Greetings Lists,

    Andrew McIntosh, a composer and violinist on our faculty, writes for bowed piano (among other things). He has observed less inharmonicity in violin strings when they are bowed that when they are plucked, and has floated the notion that piano strings exhibit less inharmonicity when bowed than when struck (possibly due to less displacement/increased tension?). The question is: Is the amount of inharmonicity in piano strings generated, at least in part, by the strings being struck?

    Thanks,

    Alan



    ------------------------------
    Alan Eder, RPT
    Herb Alpert School of Music
    California Institute of the Arts
    Valencia, CA
    661.904.6483
    ------------------------------


  • 2.  RE: inharmonicity: bowed vs struck

    Posted 09-22-2020 07:55
    This really depends upon our models for inharmonicity and really the detail of his observations relating to bowed/plucked inharmonicity differences.

    As far as I can see there are three reasons for inharmonic tones to be generated
    1. behaviour of the string not as a theoretical vibrating line but as a bar with cross-section, from memory examined and analysed by Feynmann and others
    2. relationship of end termination at the bridge for the lower energy harmonics but at a point beyond the bridge at the fundamental, the bridge becoming part of the string's vibration
    3. longitudinal modes of vibration, compression waves within the material rather than transverse waves.

    Bowing would excite mainly modes 1 and 2 whilst striking the string would be more likely to excite mode 3.

    Likewise there is the possibility that plucking a string would, like bowing, be less violent than striking and so cause mode 3 waves to be less relevant.

    However, when we look at for instance the difference between a baby grand and a concert grand one would expect longitudinal waves to be more prevalent in long strings and so to hear more of their effects. I'm not so sure that we do.

    Compression waves through steel travel at much faster speeds than in air. Or put another way, through solids rather than gases. So the longitudinal waves I'd expect to be of far higher frequency and unrelated to the longitudinal wave frequencies. By inharmonicity I generally assume we're looking at deviations away from the expected harmonics of the transverse standing waves. So I would question the observation and seek more detail about how the alleged inharmonicity is being perceived or measured.

    Best wishes

    David P


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    David Pinnegar, B.Sc., A.R.C.S.
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    +44 1342 850594





  • 3.  RE: inharmonicity: bowed vs struck

    Posted 09-22-2020 11:20
    Alan-
    You can test this on a piano by taking inharmonicity readings with CyberTuner or TuneLab, comparing hammer tones with plucked tones. My quick test didn't show a difference between plucked and struck inharmonicity. Three struck tone readings showed a slightly wider range than three plucked tones.

    The piano string is a long, narrow steel rod. My understanding of inharmonicity is that, due to the stiffness of the string material, after one displacement, the higher partials move along the string faster than a pure harmonic. Don Mannino has shown a Kawai slow motion video in which you can see the smaller impulses racing ahead of the main impulse, kinda like porpoises riding the bow wave of a ship.
    A bowed string sound is a series of release-damp-release-damp-release impulses occuring at the frequency of the fundamental as the string is dragged and released by the bow. Thus, the higher partials are started and stopped over and over with the reiterated fundamental, so they won't have the opportunity to race ahead inharmonically.

    I would guess that a single spicatto stroke, applied to an open string and allowed to continue to sound, would produce an inharmonic sound wave.

    ------------------------------
    Ed Sutton
    ed440@me.com
    (980) 254-7413
    ------------------------------



  • 4.  RE: inharmonicity: bowed vs struck

    Member
    Posted 09-22-2020 14:18
    There is not just less inharmonicity when a string is bowed.  There is exactly zero inharmonicity to a bowed string.  The harmonics are perfect harmonics.  The reason was mentioned by Ed Sutton.  It is because each cycle of the fundamental starts over again with all the harmonics in the same phase.  The action of the bow grabbing the string and then releasing it on every cycle ensures that nothing cyclic can happen that is not synchronized with this driving action.  The only reason a struck string can have inharmonicity is that all the energy is imparted at the beginning, and then each resonance is free to cycle at its own natural rate irrespective of what other resonances are doing.  They are independent and not locked together.  But bowing a string forces all harmonics to be synchronized.

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    Robert Scott
    Real-Time Specialties (TuneLab)
    fixthatpiano@yahoo.com
    Hopkins MN
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  • 5.  RE: inharmonicity: bowed vs struck

    Registered Piano Technician
    Posted 09-22-2020 16:47
    Robert,

    Does what you are saying then debunk the notion that piano strings are inharmonic because of their stiffness? And that the inharmonicity is caused solely by being struck (i. e., that same string would exhibit no inharmonicity if excited by bowing)?

    Thanks,

    Alan

    ------------------------------
    Alan Eder, RPT
    Herb Alpert School of Music
    California Institute of the Arts
    Valencia, CA
    661.904.6483
    ------------------------------



  • 6.  RE: inharmonicity: bowed vs struck

    Posted 09-22-2020 21:48
    Alan-
    I believe "inharmonicity" is a characteristic of certain cyclical energetic systems.
    A string in itself is neither inharmonic or "harmonic."
    A steel string under tension with clear terminations which has been set in motion by one impulse (pluck or boink) will exhibit inharmonicity.
    The same string being energized by a rosined bow will not exhibit inharmonicity. Lift off the bow and let the sound continue, and it will begin to exhibit inharmonicity.
    I wonder if any object in the world exhibits numerically perfect "harmonicity?" All objects are made of stuff, and stuff always has something to say about it.


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    Ed Sutton
    ed440@me.com
    (980) 254-7413
    ------------------------------



  • 7.  RE: inharmonicity: bowed vs struck

    Registered Piano Technician
    Posted 09-22-2020 22:42
    I think what Ed has simply and elegantly stated is correct. The last statement regarding the phenomenon of harmonicity in the real world is something I call "the vicissitudes of the physical universe".  Add to that the physical characteristics of our organs of perception and our oft addled brains, simply raising our eyebrows or a turn of the head affects our perception of a sonic event.

    While the math is glorious and I believe that the universe is in a coherent harmonic state across time; within that harmony is a dynamic balance of wobbles, anomalies, and turbulence which in the greater context is superficial and transient. This I call "the spice of life".

    (((((((boiiiing)))))))

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    Steven Rosenthal
    Honolulu HI
    808-521-7129
    ------------------------------



  • 8.  RE: inharmonicity: bowed vs struck

    Member
    Posted 09-22-2020 22:48
    Alan,

    The stiffness of the string does indeed account for the inharmonicity, but only when the string is allowed to ring free.  When a string is bowed, its natural tendency to ring at inharmonic frequencies is thwarted by the action of the bow, which overrides that natural tendency by resetting the phase of each partial on every grab-and-release by the stickiness of the rosined bow.  Because each cycle is exactly the same, the overall waveform is periodic.  A freely ringing string is not periodic.  And periodic waveforms always decompose into a series of perfect sine wave harmonics of the fundamental - the so-called Fourier series.

    ------------------------------
    Robert Scott
    Real-Time Specialties (TuneLab)
    fixthatpiano@yahoo.com
    Hopkins MN
    ------------------------------



  • 9.  RE: inharmonicity: bowed vs struck

    Posted 09-23-2020 09:50
    Page 1.16 (p40) of http://oro.open.ac.uk/65794/1/27777469.pdf really covers issues of unpredictability of inharmonicity.

    The soundboard can alter the effective virtual node or virtual termination of the string at the bridge, causing the length of the string to be effectively lengthened if the bridge is in a spring controlling mode (see p12) or shortened if in a mass controlling mode. But in the latter case whilst the frequency is raised, if part of the mass loading of the string is added to the mass of the string the frequency will be lower.

    The analysis continues to add the variable of the velocity of the wave which is a function of the amplitude divided by the wavelength.

    The bottom line has to be that piano inharmonicity varies and is unpredictable.

    Whether or not it has to be taken into account might be a matter of debate. On an organ with pure tones one is tuning one octave to the other with the ear hearing the octave beating without taking harmonics into account. That can be done on the piano just as much as one might tune an octave above to the 2nd harmonic of the lower note. Propably consistency is the key to a homogenous sound.

    Best wishes

    David P
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    David Pinnegar, B.Sc., A.R.C.S.
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    +44 1342 850594





  • 10.  RE: inharmonicity: bowed vs struck

    Posted 09-22-2020 17:58
    Robert - " The action of the bow grabbing the string and then releasing it on every cycle ensures that nothing cyclic can happen that is not synchronized with this driving action. " This is an important point.

    When a string is hit by a hammer, or plucked an impulse is sent down the string. A starting point to the mathematics might for instance be https://www.animations.physics.unsw.edu.au/waves-sound/travelling-waves/
    For the harmonics to be out of time with the fundamental would require the speed of higher frequency travelling waves to be faster.

    What is really happening is that an impulse is an injection of a whole band of frequencies possible within the time of the point of contact within the string. That smash of frequencies travels in both directions to both agraffe and bridge. That band of frequency pulse is reflected at both terminations. Those reflections interfere with each other and produce a standing wave at whatever points the mode of vibration of the string permits or in mathematical language, however the wave equation can be satisfied. https://newt.phys.unsw.edu.au/jw/sound.spectrum.html indicates that the bowed string has a harmonic spectrum whilst a plucked string has an "almost harmonic spectrum".

    The wave speed depends on tension. This is interesting as a pulse on a string will give a burst of energy which then subsides. The pulse may well increase the angle of deviation of the string to the extent that the tension is disturbed. Accordingly there will be a pitch deviation at high displacement, at the beginning of the vibration, and it's not inconceivable to see that the tension varying during the course of vibration will produce a distortion of the resulting vibration from the fundamental sine wave. The distortion from the sine is perceived as modifying frequencies in terms of introduced harmonics. Some of these issues are explored on https://www.animations.physics.unsw.edu.au/jw/waves_superposition_reflection.htm

    https://newt.phys.unsw.edu.au/jw/strings.html brings forward assumptions about the bending of strings and on account of their stiffness, that the effective length of the string might be less than the real length on account of the stiffness at the ends. https://newt.phys.unsw.edu.au/jw/harmonics.html suggests that with mass placed in the middle of a string if the string is plucked, odd harmonics are flat whereas if bowed all harmonics can be flat.

    Feynmann's analysis was published in Physics Today in 2009 and perhaps someone might have authorisation to http://kirkmcd.princeton.edu/examples/mechanics/feynman_pt_dec_46_09.pdf or another source.

    For me the simpler and more comprehensible answer is that for the low energy harmonics the bridge is the termination of the vibration whilst the bridge is being driven by the fundamental and so placing the virtual node of the vibration beyond the bridge. This will vary from instrument to instrument in string-soundboard interaction.

    Best wishes

    David P

    --
    - - - - - - - - - - - - - - - - - - - - - - - -
    David Pinnegar, B.Sc., A.R.C.S.
    - - - - - - - - - - - - - - - - - - - - - - - -
    +44 1342 850594





  • 11.  RE: inharmonicity: bowed vs struck

    Posted 09-22-2020 18:39
    There is a comprehensive thesis which will be important to all interested in the subject
    The Vibrating String in Keyboard Instruments
    Lincoln, John.Open University (United Kingdom), ProQuest Dissertations Publishing, 1983. 27777469.

    It may also be published elsewhere.

    Abstract:
    In this thesis some aspects of the acoustics of the piano and harpsichord are discussed. Initially the wave equation for the vibrating string is derived. Since the strings in the piano are not perfectly flexible the wave equation is extended to take into account the stiffness of the strings. The wave equation thus modified affects the frequencies of the partials of the strings. The predicted frequencies are compared with the actual frequencies on a piano. The discussion is then continued to show how this affects the actual tuning of the instrument. Computer and programmable calculator programmes are used in these predictions. The theoretical basis for the spectra of piano and harpsichord strings are discussed and brief notes included on the effects of hammer width and string stiffness. The final experimental section deals with the reverberation times for piano and harpsichord strings, taking account of the effects of multiple stringing. Multiple stringing is found to affect the piano strings but not the harpsichord strings reverberation times. No accurate measurements of harpsichord reverberation times could be found in the literature, and the experiments were extended to include the effects of the buff stop. The conclusions contain several suggestions for further study and stress the importance of piano scaling. It is noted that scaling is also of importance to the musician when selecting instruments for the performance of duets, especially with regard to tuning compatibility.  

    An extraordinary paper on analysis of violin inharmonicity is on https://www.jstor.org/stable/23254382








  • 12.  RE: inharmonicity: bowed vs struck

    Posted 09-22-2020 19:13
    Apologies for a further contribution here - http://oro.open.ac.uk/65794/1/27777469.pdf is the open access source for the thesis.

    Best wishes

    David P






  • 13.  RE: inharmonicity: bowed vs struck

    Registered Piano Technician
    Posted 09-23-2020 10:39
    What Robert Scott said is almost exactly what I remember learning in college physics classes. Freely vibrating strings (plucked or struck) will have some inharmonicity that depends on their stiffness and other factors (length, tension, etc.). Vibrating systems that are driven periodically, like with the stick-slip-stick-slip action of a violin bow, won't show inharmonicity because of mode-locking. 

    That said, it wouldn't surprise me if a bowed piano string still exhibited a small amount of inharmonicity. The real world is usually messier than theory.

    ------------------------------
    Anthony Willey, RPT
    http://willeypianotuning.com
    http://pianometer.com
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  • 14.  RE: inharmonicity: bowed vs struck

    Registered Piano Technician
    Posted 09-23-2020 13:21
    Thanks so much for all who have responded on this thread. It has been a real eye-opener concerning inharmonicity in pianos (and corroboration of McIntosh's work).

    I especially want to thank Ed Sutton, RPT, and anyone else who actually measured piano strings excited by various means. This is something that I have not been able to do myself for the past several days.

    Gotta love these lists!

    Alan

    ------------------------------
    Alan Eder, RPT
    Herb Alpert School of Music
    California Institute of the Arts
    Valencia, CA
    661.904.6483
    ------------------------------



  • 15.  RE: inharmonicity: bowed vs struck

    Registered Piano Technician
    Posted 09-23-2020 21:58
    Curious about how he is bowing the piano strings.

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    Cindy Strehlow, RPT
    Urbana, IL
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  • 16.  RE: inharmonicity: bowed vs struck

    Registered Piano Technician
    Posted 09-23-2020 22:34
    Cindy,

    Not sure just how he is bowing the strings. I have invited him to join these lists so he can go through the thread on his own. Perhaps he can explain to everyone without having  to go through me.

    Alan

    ------------------------------
    Alan Eder, RPT
    Herb Alpert School of Music
    California Institute of the Arts
    Valencia, CA
    661.904.6483
    ------------------------------



  • 17.  RE: inharmonicity: bowed vs struck

    Registered Piano Technician
    Posted 09-23-2020 22:44
    I'm not sure how this particular person is doing it, but the musicians that I have seen take small bunches of horsehair, like what would normally go on a bow, thread it down and around a string or strings, hold an end up in each hand, and smoothly lower one hand while raising the other. (It can leave some goopy rosin residue on the strings.)

    Anthony Willey






  • 18.  RE: inharmonicity: bowed vs struck

    Registered Piano Technician
    Posted 09-23-2020 22:54
    In addition to what Anthony relates, short lengths of bow hair can be glued to popsicle sticks and sawed up and down on the strings.I've seen them set up like this with servo motors to control them.
    Bowed piano is a thing, videos on youtube.

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    Steven Rosenthal
    Honolulu HI
    808-521-7129
    ------------------------------



  • 19.  RE: inharmonicity: bowed vs struck

    Registered Piano Technician
    Posted 09-23-2020 22:52
    Without any info on how others may have bowed piano strings, I can say there is (or was) a mechanical device aimed at guitar players, which consists of a rotating wheel/bow that is held in the hand, and placed against the string , acting as a continuous bow. It would be easier to apply than a normal violin-type bow. In case anyone wants to experiment, it is called the Ebow. 

    Mark Schecter
     | |   | | |   | |   | | | 






  • 20.  RE: inharmonicity: bowed vs struck

    Registered Piano Technician
    Posted 09-23-2020 23:00
    I believe the ebow excites the strings electromagnetically.

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    Steven Rosenthal
    Honolulu HI
    808-521-7129
    ------------------------------



  • 21.  RE: inharmonicity: bowed vs struck

    Registered Piano Technician
    Posted 09-23-2020 23:02
      |   view attached
    And then there's this: viola organista
    https://www.merriammusic.com/school-of-music/da-vincis-piano-cello-combo-a-hit/
    https://www.youtube.com/watch?v=gOrn_z9m9lU&feature=emb_rel_pause

    ------------------------------
    Steven Rosenthal
    Honolulu HI
    808-521-7129
    ------------------------------



  • 22.  RE: inharmonicity: bowed vs struck

    Posted 09-24-2020 15:03
    Hello everyone,

    Thank you all very much for your generous answers to Alan's question on my behalf about inharmonicity, and thanks, Alan, for bring the discussion to this forum!

    In answer to the question of how I am bowing the piano, I'm actually using fishing line (not horse hair). I tried using horse hair from an old violin bow at one point, but the length of the hair is shorter than I would like and it necessitates frequent changes of direction, which interrupt the resonance. The horse hair is also not as strong and is more prone to breakage while playing. There is a slight difference in timbre as well between horse hair and fishing line, but despite being a violinist by trade I actually prefer the quality of sound the fishing line produces. Vicki Ray, piano teacher at CalArts, taught me about using small bunches of rosined fishing line, fastened together at the ends by some method (I use a paperclip and electrical tape, and fishing line approximately 3-4 feet long). One end can be fed underneath the string and then pulled back through, and then the string is bowed by holding one end of the fishing line in each hand and drawing it back and forth. 

    What prompted my curiosity about inharmonicity is that I noticed I could isolate individual upper partials in the spectrum of the piano strings by manipulating the speed, contact point, and intensity of bowing (much like one can do on a violin string with a traditional bow), and that those upper partials seemed to be precisely where I would have expected them in a "just" harmonic series. It's always been a slight source of annoyance for me that piano harmonics are sharp compared to where they would be on a bowed cello or bass, due to the inharmonicity, so I was quite excited about the possibility of composing for piano harmonics that were actually in tune with the fundamental of the string, according to the principles of just intonation.

    I knew a little bit about mode-locking from studying string instrument acoustics, and I've been aware for years that string players often tune false fifths by accident when they tune by plucking instead of bowing, presumably due to the inharmonicity of freely vibrating strings, but I wasn't sure of the extent to which the principle of mode-locking still applied to wound piano strings, due to their considerable thickness and stiffness.

    I did some experiments recording different high bowed partials (on a 7-foot Bösendorfer, mostly using the bichords) and analyzing the pitch content using Melodyne software. My initial results were that the inharmonicity of the partials was reduced to within 5 cents of the mathematically correct frequencies all the way up to the 15th partial (and in many cases it was less than 1 cent). Beyond that the partials became quite sharp again. The 16th, 17th, 18th, and 19th partials were all 20-30 cents sharp, and that's as high as I measured. I believe the inharmonicity of partials 16 and above is due to the fact that it took a fair amount of force while bowing to get the string to clearly isolate those partials. 

    Thank you all again for your fascinating responses on this thread! I look forward to reading the many linked articles and learning more.

    Wishing all the best,
    -Andrew McIntosh

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    Andrew McIntosh
    Los Angeles CA
    775-225-7943
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