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Just Intonation Dilemma

  • 1.  Just Intonation Dilemma

    Posted 08-21-2018 02:08
    I haven't done much study into Just Intonation, but I have had this dilemma for a while:

    How do I tune the chord with notes 1, 2, 4, 5 so that 1 & 2, 1 & 4, 1 & 5, 2 & 4, 2 & 5, 4 & 5 are all Just ("not beating")? Is it possible? I've been able to generate the frequency of each tone on the software Audacity so that all intervals are pure except for the interval 2 & 4.

    Is the solution to temper the frequency of 2 & 4 as to widen the interval enough so that 1 & 2, 1 & 4, 2 & 4, 2 & 5, 4 & 5 beat equally out of Just?

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    Cobrun Sells
    C.J. Piano Tuner
    www.cjtuner.com
    cobrun94@yahoo.com
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  • 2.  RE: Just Intonation Dilemma

    Posted 08-21-2018 02:39
    Are you using numbered musical notation?

    https://en.m.wikipedia.org/wiki/Numbered_musical_notation#Numbered_notation_described

    According to this system, the notes of your chord would be C-D-F-G. Is this correct?


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    Roshan Kakiya
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  • 3.  RE: Just Intonation Dilemma

    Member
    Posted 08-21-2018 15:39
    It can't be done: 
    You have a perfect C-G fifth.
    From the G you tune a perfect fourth down to D.
    From the C you tune a perfect fourth up to F.
    The D and the F are fixed, and they are -17.6 cents narrow from just.

    Your proposed solution to make them all equally out of tune so that they beat the same was attempted many times in the development of temperaments. Your problem is creating equal temperament inside a perfect fifth. There is currently a very promising movement among piano techs to try to create equal temperament inside the just 12th (perfect 12th).


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    jason's cell 425 830 1561





  • 4.  RE: Just Intonation Dilemma

    Posted 08-22-2018 20:13
    The most pure ratios are:

    9/8 = Major Second.
    6/5 = Minor Third.
    4/3 = Perfect Fourth.
    3/2 = Perfect Fifth.

    Use the ratio 3/2 for C-G (perfect fifth).

    Use the ratio 9/8 for C-D (major second) and F-G (major second).

    This will cause the ratio of C-F (perfect fourth) and D-G (perfect fourth) to become 4/3.

    The ratio of D-F (minor third) should ideally be 6/5 = 315.64 cents. However, this is not possible.

    By using the arrangement I have described above, the ratio of D-F (minor third) would become 4/3 ÷ 9/8 = 4/3 × 8/9 = 32/27 = 294.13 cents.

    D-F (minor third) will be 315.64 - 294.13 = 21.51 cents out of tune. This is called the syntonic comma.

    Therefore, as you have rightly said, the solution is to widen D-F so that all the intervals except for the perfect fifth become equally out of just. Jason has made an interesting point. The aim is to create equal temperament within a just perfect fifth.

    This can be achieved by widening D-F by 2/3 syntonic comma and by narrowing C-D and F-G by 1/3 syntonic comma.

    D-F = 294.13 + 14.34 = 308.47 cents.
    C-D and F-G = 203.91 - 7.17 = 196.74 cents.


    The following interval will be pure:

    Perfect fifth = C-G = 701.95 cents.


    The following intervals will be equally out of just:

    Major seconds = C-D and F-G = 196.74 cents.
    Minor third = D-F = 308.47 cents.
    Perfect fourths = C-F and D-G = 505.21 cents.


    Offsets from just intonation in cents:

    C = 0.00 - 0.00 = 0.00
    D = 196.74 - 203.91 = 7.17
    F = 505.21 - 498.04 = +7.17
    G = 701.95 - 701.96 = −0.01


    Offsets from 12-tone equal temperament in cents:

    C = 0.00 - 0.00 = 0.00
    D = 196.74 - 200 = −3.26
    F = 505.21 - 500 = +5.21
    G = 701.95 - 700 = +1.95

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    Roshan Kakiya
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  • 5.  RE: Just Intonation Dilemma

    Posted 08-27-2018 15:49
    "There is currently a very promising movement among piano techs to try to create equal temperament inside the just 12th (perfect 12th)."

    Jason,

    Does every interval inside the perfect 12th need to beat equally out of just?

    What is the meaning of "just" in this case? Are you referring to 5-limit just intonation?

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    Roshan Kakiya
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  • 6.  RE: Just Intonation Dilemma

    Registered Piano Technician
    Posted 08-27-2018 23:53
    Equally Tempered intervals means the ratios are equal, not the beat speeds. Equal ratios does not result in equal beat speed because the frequency difference is greater between intervals as they move up the compass.

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    Edward McMorrow
    Edmonds WA
    425-299-3431
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  • 7.  RE: Just Intonation Dilemma

    Member
    Posted 08-29-2018 13:36
    "Does every interval inside the perfect 12th need to beat equally out of just?
    What is the meaning of "just" in this case? Are you referring to 5-limit just intonation?"
    .
    It's not about beats.
    .
    Just intonation is a concept that doesn't really work all that well in the world of pianos, because of inharmonicity. Take the octave, which in the world of just intonation is a perfect 2:1 ratio. On a piano, that all depends. Let's take the F3-F4 octave; if you tune a 2:1 ratio, you are matching the second partial of F3 to the first partial of F4. But those two notes also have higher partials that theoretically should align: 4:2 and 6:3 are more prominent than 2:1 in the sound of that octave, and if you tune a perfect 2:1, the 4:2 and 6:3 will not be in tune because those tones are sharper than theoretically just. Piano tuners are not trying to make small ratios, we are trying to make the piano sound sweet, using inharmonicity rather than fighting it or ignoring it. For the F3-F4 octave, we generally strive for a ratio that is between 4:2 and 6:3. 
    .
    The appeal of the "perfect twelfth" is that we can, in fact, in the middle and upper ranges of the piano keyboard, tune a perfect 3:1 ratio for the twelfth, and the octaves sound lovely. This is because the 3:1 provides a degree of "stretch" that conforms very closely to the degree of added sharpness caused by inharmonicity. The octaves are expanded by 1.24 cents but the overall effect is very clean. Listen to this clip: http://tinyurl.com/hcbu3t5

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    jason's cell 425 830 1561

    On Mon, Aug 27, 2018 at 12:48 PM, Roshan Kakiya via Piano Technicians Guild






  • 8.  RE: Just Intonation Dilemma

    Posted 08-29-2018 10:11
    If by chance this may interest anyone, a paper thus concluded:

    "The results of all the experiments indicate that ... beating does not contribute significantly to the percept of "out-of-tuneness", ... these results suggest that the perception of intonation in music is dependent on the acutal interval tuning rather than the concomitant beat rate. ... If beating partials are unimportant vis-a-vis interval tuning, this strengthens the argument for a cultural basis for musical intonation, as opposed to the acoustical basis set forth by Helmholtz and others."
    CENTER FOR COMPUTER RESEARCH IN MUSIC AND ACOUSTICS MARCH 1991
    Department of Music, Report No. STAN-M-70
    PSYCHOACOUSTIC FACTORS IN MUSICAL INTONATION: BEATS, INTERVAL TUNING, AND INHARMONICITY
    Douglas Fleming Keislar

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    Linus Liu
    HONG KONG
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  • 9.  RE: Just Intonation Dilemma

    Registered Piano Technician
    Posted 08-29-2018 14:01
    In my experience, 98% of my clients or so don't judge a piano being in or out of tune by the temperament. They judge out-of-tuneness by unisons. Most people can hear that "this note isn't right; better call the piano tuner."

    Occassionally I'll get a client that can tell the difference in how a temperament was laid. Usually it happens after a certain local "tooner" really screws up the temperamental stretching - i.e. by tuning octaves about 4 cents narrow, rather than expanded by any degree. Even then, only a few clients will tell me that he didn't do it right. Most still can't tell the difference between our tunings. 

    Oh, and most of his clients are happy campers, because his unisons are good. Sad but true.

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    Benjamin Sanchez
    Professional Piano Services
    (805)315-8050
    www.professional-piano-services.com
    BenPianoPro@comcast.net
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  • 10.  RE: Just Intonation Dilemma

    Posted 08-30-2018 12:12
    Thanks. This was for a vocal chorale (rather something similar) so inharmonicity wasn't a major factor in my dilemma.

    However, I am interested in reading that article on out-of-beatness not being the same as out-of-tune.

    I am interested in the pure-12th temperament as well.

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    Cobrun Sells
    C.J. Piano Tuner
    www.cjtuner.com
    cobrun94@yahoo.com
    ------------------------------



  • 11.  RE: Just Intonation Dilemma

    Posted 08-30-2018 22:04
      |   view attached
    "Roshan,

    How would I calculate the frequencies for scale tone 2 and 4 so that all intervals (except the 5th) beat equally?

    Thanks".

    This just intonation dilemma can be solved mathematically by making all the intervals except for the perfect fifth out of tune by the same amount. I have already calculated and posted the value of each interval in cents on one of my previous replies:

    C or 1 = 0.00 cents
    D or 2 = 196.74 cents
    F or 4 = 505.21 cents
    G or 5 = 701.95 cents


    I have converted these figures to frequencies and stored them (as well as the formulas that I have used to perform these conversions) in the Excel file that I have attached to this reply. Middle C = 261.63 Hz.

    I think this solution (making all the intervals except for the perfect fifth out of tune by the same amount) will be successful since you have said inharmonicity is not a major factor in your dilemma.

    Conclusion: The chord is C-D-F-G or 1-2-4-5. All the intervals except for the perfect fifth will be out of tune by the same amount, relative to the ratios of 5-limit just intonation, based on the mathematics I have calculated and posted on this discussion.

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    Roshan Kakiya
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    Attachment(s)



  • 12.  RE: Just Intonation Dilemma

    Posted 07-27-2019 11:07
    Okay okay, another dilemma popped into my head:
    If tuning isn't significantly influenced by beats but rather by the ratios of the frequencies in the intervals then what is the best tuning for the 1 2 4 5 chord?

    The original chord in its inversion in the chorale is 2 4 5 1 (G Bb C F). Should that be tuned as:
    G Bb C F with G as the root and Bb is 6:5 away from G or 19:16 away from G? Both 6:5 and 19:16 are a minor 3rd. 6:5 is a simpler ratio and probably easier to hear the harmonics for (it also depends on the instrument that is playing) but 6:5 produces an undertone of Eb which is not in the original chord. But if the minor 3rd is tuned using 19:16 the undertone would be G which is in the original chord.

    So, 6:5 or 19:16?

    Of course G and C would be 4:3 and G and F would be 7:4.

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    Cobrun Sells
    cobrun94@yahoo.com
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  • 13.  RE: Just Intonation Dilemma

    Posted 07-27-2019 12:22
    Attached is a photograph of a Monochord from the Colt Collection, possibly having belonged to Helmholtz, and a spreadsheet of its measurements

    Best wishes

    David P

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    David Pinnegar BSc ARCS
    Curator and House Tuner - Hammerwood Park, East Grinstead, Sussex UK
    antespam@gmail.com

    Seminar 6th May 2019 - http://hammerwood.mistral.co.uk/tuning-seminar.pdf "The Importance of Tuning for Better Performance"
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    Attachment(s)

    xls
    monochord.xls   23 KB 1 version


  • 14.  RE: Just Intonation Dilemma

    Posted 07-27-2019 13:36
    I do not recommend using just ratios. Tempered intervals will need to be used. This is because just intonation contains commas that cause certain intervals to not have the ratio that they should have. Several just intonation dilemmas can emerge.

    I can illustrate this by using the ratios that you have provided:

    G or 2 = 1/1.
    Bb or 4 = 6/5.
    C or 5 = 4/3.
    F or 1 = 7/4.

    The ratio of Bb-F is 7/4 / 6/5 = 7/4 × 5/6 = 35/24. The ratio of Bb-F should be 3/2.

    The ratio of G-C, a fourth, is 4/3. The ratio of C-F, a fourth, is 7/4 / 4/3 = 7/4 × 3/4 = 21/16. Both fourths are not the same.

    The ratio of Bb-C, a major second, is 4/3 / 6/5 = 4/3 × 5/6 = 20/18 = 10/9. The ratio of Bb-C should be 9/8.


    I recommend using pure octave equal temperament. It solves all the commas by achieving maximum enharmonic equivalency and it is also very easy to use due to its simplicity.

    I recommend using the following values:

    G or 2 = 0 cents.
    Bb or 4 = 300 cents.
    C or 5 = 500 cents.
    F or 1 = 1000 cents.

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    Roshan Kakiya
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  • 15.  RE: Just Intonation Dilemma

    Registered Piano Technician
    Posted 07-27-2019 12:17
    Cobrun --

    As a complete aside, thanks for the tip on Audacity. I love Audacity. I have a project in mind that will require some test tones and I never thought to look for that tool in Audacity. It's there. It works. Thanks!

    Now all I need is a PC based oscilloscope to display the waveform.

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    Geoff Sykes, RPT
    Los Angeles CA
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