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Inharmonicity

  • 1.  Inharmonicity

    Posted 08-12-2019 09:58
    I have explored the mathematics of the theoretical models of various equal temperaments and unequal temperaments.

    However, the biggest drawback of all of these theoretical models is that they do not effectively account for inharmonicity.

    Therefore, I consider inharmonicity to be the final piece of the puzzle.

    I have been asking myself the following question:

    How useful are the mathematical analyses of the theoretical models of various temperaments if they do not effectively account for inharmonicity?


    I believe that inharmonicity should be integrated into the theoretical models mentioned above to make them more useful than they currently are.

    For example, pure 12th equal temperament is based on the pure twelfth (frequency ratio: 3/1). Inharmonicity will cause the 3rd partial of the fundamental frequency to be sharper than the pure twelfth. Therefore, the pure twelfth will need to be sharpened/stretched/widened so that it matches the 3rd partial of the fundamental frequency. This means that pure 12th equal temperament should actually be referred to as equal temperament based on the 3rd partial of the fundamental frequency to account for inharmonicity. Is this correct?

    It is likely to be difficult to effectively incorporate inharmonicity into the numerous theoretical models of unequal temperament since the inequality of their intervals is likely to complicate matters further than they already have been by inharmonicity.

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    Roshan Kakiya
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  • 2.  RE: Inharmonicity

    Posted 08-12-2019 14:05
    When one tunes significantly by ear then there's an automatic correlation between the mathematics and any inharmonicity so that one can achieve valid mathematical result without taking it into account.

    I think that somewhere I've seen references to odd harmonics having different inharmonicity to even harmonics and when observing "easy tuner" one can see sometimes different harmonics travelling at different speeds indicating inharmonicities. Are inharmonicities generally coordinated or correlated among harmonics?

    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

    Registered Piano Technician
    Posted 08-12-2019 15:59

    Maybe Mineral spirits could take it off with 0000 steel wool, but don't do until I ask. We have a Piano Tech meeting tonight here and I'll ask.

     

    Paul

     






  • 4.  RE: Inharmonicity

    Registered Piano Technician
    Posted 08-12-2019 17:32
    Roshan wrote:

    "I believe that inharmonicity should be integrated into the theoretical models mentioned above to make them more useful than they currently are."

    You might look into the work of Alfredo Capurso, who has written extensively about integrating inharmonicity into the theoretical models. To me, his work demonstrates very effectively why a mathematical approach to inharmonicity really cannot work. Mr. Capurso might see it differently.

    Inharmonicity is too inconsistent and random for math. The aural approach that David alludes to is much more useful.

    "For example, pure 12th equal temperament is based on the pure twelfth (frequency ratio: 3/1). Inharmonicity will cause the 3rd partial of the fundamental frequency to be sharper than the pure twelfth so the pure twelfth will need to be sharpened/stretched/widened in order to match the 3rd partial of the fundamental frequency. This means that pure 12th equal temperament should actually be referred to as equal temperament based on the 3rd partial of the fundamental frequency to account for inharmonicity. Is this correct?"

    (That is sort of correct, but your terminology is little strange. "Fundamental frequency" and "first partial" are the same thing; "3rd partial of the fundamental frequency" is a meaningless term because they are two separate things. You would not say "Third partial of the first partial.")

    When a piano tuner speaks of a pure 12th, what is meant is a blended compromise between 3:1 and 6:2 partial relationships that yields a clean effect. Frequency ratio is irrelevant. 

    This is all covered in my PTJ articles on 21st Century Tuning Style, which I invite you again to read.

    Modern pianos and equal temperament both apparently existed before inharmonicity was understood, so early aural tuners just ignored inharmonicity and used the beat rates derived from the theoretical models and did the best they could. By referencing more than one interval to tune each note, they developed something I call the "piano tuner's compromise" which attempts to tune each interval as close as possible to the target beat rate of the model while at the same time trying to tune the other intervals as close as possible to their own targets. The lower partials are not much affected by inharmonicity compared to the higher partials, so this strategy can work quite well.





  • 5.  RE: Inharmonicity

    Posted 08-13-2019 16:30
    "'3rd partial of the fundamental frequency' is a meaningless term because they are two separate things" (Quote by Kent Swafford).

    To what does the 3rd partial relate if it does not relate to the fundamental frequency?


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    Roshan Kakiya
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  • 6.  RE: Inharmonicity

    Posted 08-13-2019 17:02
    Hi,

    As usual these days, I'm late to the parade.

    That said, I find myself wondering if, with this last question, we haven't finally drifted of the edge of the all-too-flat earth into the land of metaphysical niceties.

    If so, is it possible to get things back into the at least partially grounded world of piano work? One in which Decartes and Spinoza can at least meet for lunch and/or coffee?

    Back to the land in which I can deal with the delusions and neuroses with which I am most familiar.

    Kind regards.

    Horace





      Original Message




  • 7.  RE: Inharmonicity

    Registered Piano Technician
    Posted 08-13-2019 17:41
    The fundamental is also called the first partial, for clarity when discussing partials. The fundamental is the mass vibrating at its lowest resonant frequency, as a whole. The mass has partials, the fundamental (or first partial) being one of them. The fundamental does not have partials, the whole does. 

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






  • 8.  RE: Inharmonicity

    Posted 08-13-2019 17:59
    Hi, Mark,

    Precisely so. Well said.

    Thank you very much.

    Kind regards.

    Horace





      Original Message




  • 9.  RE: Inharmonicity

    Posted 08-13-2019 18:25
    Mark,

    I would be grateful if you could illustrate your descriptions with numerical examples.

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    Roshan Kakiya
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  • 10.  RE: Inharmonicity

    Member
    Posted 08-13-2019 18:40
    1, 2, 3, 4, 6, 8. Partials 5 and 7 also exist but are not listed because they are not generally used in partial matching.


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    Jason Kanter
    Lynnwood WA
    425-830-1561
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  • 11.  RE: Inharmonicity

    Member
    Posted 08-13-2019 18:54
    Just try applying your principles to actually tuning a piano and get back to us.

    It boils down to the difference between Theory and Practice.

    In Theory, there is no difference between Theory and Practice
    but in Practice there is.

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    Regards,

    Jon Page
    mailto:jonpage@pianocapecod.com
    http://www.pianocapecod.com
    ------------------------------



  • 12.  RE: Inharmonicity

    Posted 08-13-2019 19:17
    Jason,

    Thank you for providing these figures.

    Are D3, G3, D4, G4, D5 and D6 referred to as "masses" that each contain their own specific set of partials?

    For example, the 1st specific partial of D6 does not match the 2nd specific partial of D5 since each "mass" is affected differently by inharmonicity. Is this correct?

    Is there a standardised set of terms and definitions for the aspects of tuning that are related to inharmonicity?

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    Roshan Kakiya
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  • 13.  RE: Inharmonicity

    Member
    Posted 08-13-2019 19:42
    Roshan, a couple of responses.
    First, in my experience piano technicians do not talk about "masses" but that is a perfectly good physical term for the string that generates partials.
    Second, please don't get tangled in the semantics. The "fundamental" refers to the first partial, yes. The mathematics of tuning tends to think the fundamental is the note itself, but once you start appreciating the ladder of partials and how they are modified by inharmonicity, the "fundamental" loses a lot of its importance. It's only part of the note. That is the distinction Kent was making, but you seem to be focusing on the semantics rather than the meaning.
    Third, Please get yourself some tuning equipment and access to a piano and tune it.

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    Jason Kanter
    Lynnwood WA
    425-830-1561
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  • 14.  RE: Inharmonicity

    Registered Piano Technician
    Posted 08-13-2019 20:27
     Fundamental can keep it's meaning with"equal temperament based on the 3rd partial of the fundamental frequency" nicely changed to "equal temperament based on the 2nd overtone of the fundamental."


  • 15.  RE: Inharmonicity

    Member
    Posted 08-13-2019 20:45
    When the meat goes thru the grinder... the product needs to be as beatless as possible, regardless of which partials are favored. Tuning is an art, not a science.

    Put your pencil away and actually try tuning a piano.  Until then your postulations are irrelevant.
    ------------------------------
    Regards,

    Jon Page
    mailto:jonpage@pianocapecod.com
    http://www.pianocapecod.com


  • 16.  RE: Inharmonicity

    Posted 08-14-2019 10:24
    Jon,

    I am simply trying to gain a better understanding of inharmonicity with regard to the practical side of tuning.

    I do not have enough knowledge about how inharmonicity affects the theoretical models of temperament yet. Therefore, I am not ready to tune a piano yet.

    Tuning is a science as much as it is an art. Mathematics is a powerful tool that can illustrate matters that are related to tuning with more clarity than words can sometimes.

    I will be using the mathematical skills that I have gained in relation to tuning most effectively by focusing on integrating inharmonicity into the theoretical models of temperament. I have learned enough of the mathematics of tuning to be able to attempt to do this now.

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    Roshan Kakiya
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  • 17.  RE: Inharmonicity

    Registered Piano Technician
    Posted 08-14-2019 11:53
    "How useful are the mathematical analyses of the theoretical models of various temperaments if they do not effectively account for inharmonicity?"
    Not very, IMO.

    "I am simply trying to gain a better understanding of inharmonicity"
    Roshan, inharmonicithy is the tendency for the upper harmonics to be out of tune with the fundamental.

    With zero inharmonicity the frequency of the upper harmonics would follow the formula fn = n * f1  where f1 is the frequency of the first harmonic and n is the harmonic number.

    With nonzero inharmonicity the frequency of upper harmonics follows the more complicated formula fn = n * f1 * sqrt(1 + B * n^2)/sqrt(1 + B) where B is an "inharmonicity constant" whose value depends on the string itself. (It's kind of a measure of the "stiffness" of the string, so larger string diameter, shorter string length, and stiffer string material all lead to a higher inharmonicity constant.)

    Here are some sample harmonic frequencies of the note A4 with and without inharmonicity. In this example I have used a value of 6.64E-4 for B. 

    Here's a screenshot of the same table, using Excel's "show formulas" option.

    You can use these formulas to make your own sheet and play around with numbers. The inharmonicity value I used for A4 is fairly typical, but the inharmonicity varies from note to note and from piano to piano.

    I think you'll have to view this in the online version of pianotech to view the images.


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    Anthony Willey, RPT
    http://willeypianotuning.com
    http://pianometer.com
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  • 18.  RE: Inharmonicity

    Member
    Posted 08-14-2019 14:52
    Here are some other views of calculations involving inharmonicity.
    First, from the famed physicist Richard Feynman.
    Second, from Dr. Albert Sanderson.

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    Jason Kanter
    Lynnwood WA
    425-830-1561
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    Attachment(s)

    pdf
    Feynman on tuning.pdf   258K 1 version
    pdf
    Sanderson formulae.pdf   277K 1 version


  • 19.  RE: Inharmonicity

    Posted 08-14-2019 15:58
    Thank you so much Jason for the wonderful link to Feynman's analysis. I'm sure as an undergraduate working out inharmonicity of stiff resonators, or wide organ pipes and like effects with microwave waveguides and fibreoptic cables was very much part of the course but it's interesting that Feynman himself wasn't really able to put his finger on the answer.

    Particularly fascinating is his approach also to distortion in the ear and the rather wobbly graph of the aural tuning looks not a lightyear away from the graphs of my unequal temperament tunings, which as we've discovered are quite difficult to analyse with available mathematical tools.

    I think that the moving soundboard placing the first harmonic node beyond the bridge is a much more relevant cause of inharmonicity than has been given credit for. Feynman picks up on this but theory makes prediction beyond grasp.

    Best wishes

    David P

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





  • 20.  RE: Inharmonicity

    Registered Piano Technician
    Posted 08-14-2019 12:15
    Roshan,

    check out "the craft of piano tuning" by Daniel Levitan

    there is no better book available to offer detailed answers to the questions you've been asking. I think you will get an enormous amount out of reading that book, and I would love to see what questions you have after reading it cover to cover

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    Daniel DeBiasio
    Brooklyn, NY
    646.801.8863
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  • 21.  RE: Inharmonicity

    Registered Piano Technician
    Posted 08-14-2019 17:29
    Rohan, I think it's mistaken to wait till your theoretical understanding is better before you actually try to tune a piano. There are so many countervailing influences that change from instrument to instrument, sometimes in direct opposition to each other, that will always muddy up the math.
    Any tuner will tell you that the best teacher, and ultimate proof, is the piano itself.
    In reality there are too many ambiguities that can't be satisfied theoretically but can be understood through practice i.e. listening, not thinking.

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    Steven Rosenthal
    Honolulu HI
    808-521-7129
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