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Fundamental Frequencies vs Coincidental Partials

  • 1.  Fundamental Frequencies vs Coincidental Partials

    Posted 6 days ago
    Should inharmonicity be ignored within the central octaves of a piano's keyboard?

    Should intervals within the central octaves of a piano's keyboard be tuned without stretch?

    What are the benefits of a stretched framework?

    What are the benefits of an unstretched framework?


    An example of a stretched octave:

    If the 1st partial of the 2nd note is matched with the 2nd partial of the 1st note, the result will be a stretched octave due to the sharpening of partials caused by inharmonicity. In this case, coincidental partials will have been matched.


    An example of an unstretched octave:

    If the fundamental frequency of the 1st note is 440 Hz and the fundamental frequency of the 2nd note is 880 Hz, the result will be an unstretched octave which has a frequency ratio of 2/1 (880 Hz / 440 Hz = 2/1). In this case, fundamental frequencies will have been arranged in a way that causes them to have a specific frequency ratio.

    ------------------------------
    Roshan Kakiya
    ------------------------------


  • 2.  RE: Fundamental Frequencies vs Coincidental Partials

    Posted 6 days ago
    It depends on what you want the tuning to do and to achieve . . .

    Where one's dealing with a variation of size of thirds, using stretching can sound fine on one instrument but harsh on others so there can be no hard and fast rule. No-stretch in the central octaves solves that problem and this is a region where in reality little stretch is required by the Railsback curve, so the departure from the norm here does minimal "damage" to our conventional perceptions. Anthony Willey very kindly wrote to me with regard to implementation with his Easy Piano Tuner app and if he's able to achieve control of the stretch to allow conventional harmonic stretching above and below whilst keeping the central three octaves "pure", this will open up reliable use of unequal temperaments and help people possibly to avoid bad experiences which might have given unequal temperaments a bad name in the past.

    This recording is the 1859 Hallé Broadwood tuned to Kirnberger III accompanied by an 1885 Bechstein in Kellner. Both had not been tuned for three months so are not perfect, but I think demonstrates an atmosphere of the Victorian piano -
    https://www.youtube.com/watch?v=I-W8oRAd6Bw

    Meanwhile in contrast, the Bechstein being more stable than the indisputably historic instrument
    https://www.youtube.com/watch?v=7dDo1GBMspE
    https://www.youtube.com/watch?v=vkuPMACOOLk
    https://www.youtube.com/watch?v=5xhRPI1LlO0
    demonstrates reasonably well what such tunings can achieve.

    Best wishes

    David P

    Best wishes

    David P

    ------------------------------
    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"
    ------------------------------



  • 3.  RE: Fundamental Frequencies vs Coincidental Partials

    Posted 6 days ago
    David, Please share Anthony's instructions, so that I can try your Kirnberger III the way you do it. (Hopefully).

    ------------------------------
    Richard Adkins
    Piano Technician
    Coe College
    Cedar Rapids, IA
    ------------------------------



  • 4.  RE: Fundamental Frequencies vs Coincidental Partials

    Registered Piano Technician
    Posted 6 days ago
    Roshan,
    Since all pianos have inharmonicity, some more than others, the benefits of a stretched framework, if I understand that term correctly, are intervals that sound better. Your question about stretched vs. unstretched is a little like this: if I start my car engine and prepare to drive off without buckling my seat belt, the car makes an annoying beeping sound. If I’m only going to drive 500 ft., what are the benefits of fastening my seatbelt or not fastening it? It save time if I don’t fasten it and there is much less chance of an accident than if I were driving, say, one mile. The advantages of fastening it: no annoying beeping, reenforcing a good habit, i.e., fastening the seat belt every time I prepare to drive, and being ready for the small chance of an accident in 500 feet.

    All intervals on a piano are out of tune insofar as it’s impossible to match all coincidental partials at the same time. Some intervals, e.g., 3rds, are quite badly out of tune, irrespective of whether you “stretch the framework” or not. Octaves can be pretty well in tune usually, even if all their coincidental partials can’t match at the same time. Higher coincidental partials are much more audible lower in the scale. There is no “solution” in the sense of finding the solution to a mathematical problem (like the area of a rectangle of 2 x 3 units). This is where piano tuners tend to get a little impatient or glassy eyed with theoretical speculation. We just need to get the piano tuned to sound as good as possible. There is “stretch” in the piano whether we choose to ignore it or not. If we ignore it, it is like painting by the numbers. The art/craft of piano tuning comes into play when we have to deal with the inconsistencies in a piano scale, like when inharmonicity suddenly changes between adjoining plain strings and wrapped strings.

    I hope this helps.

    Bob Anderson, RPT
    Tucson, AZ




  • 5.  RE: Fundamental Frequencies vs Coincidental Partials

    Registered Piano Technician
    Posted 6 days ago
    If one were to try and tune without stretch, what key signature should the use? Remember, depending upon the key signature F# is not the same as G flat.

    ------------------------------
    Larry Messerly, RPT
    Bringing Harmony to Homes
    www.lacrossepianotuning.com
    ljmesserly@gmail.com
    928-899-7292
    ------------------------------



  • 6.  RE: Fundamental Frequencies vs Coincidental Partials

    Registered Piano Technician
    Posted 4 days ago
    @Richard, PianoMeter does not currently have a way to do this.  ​The way I calculate the tuning I'm not able to tune anything narrower than 2:1 octaves. I've got some custom tuning style controls in development that I was talking about with David, but still not anything that would specifically support this.

    Based on what David has said, I would suggest that the best way to experiment with his "flat" tuning style would be to just download a generic chromatic tuning app. The app called "Pano Tuner" (P-A-N-O...not a typo) is decent, and with a $1.99 upgrade to the "Full" version it will support unequal tempermanets. Kirnberger III is pre-loaded. You'll probably want to tune the top and bottom octaves by ear.

    ------------------------------
    Anthony Willey, RPT
    http://willeypianotuning.com
    http://pianometer.com
    ------------------------------



  • 7.  RE: Fundamental Frequencies vs Coincidental Partials

    Posted 4 days ago
    David and Anthony,

    Could the following method be useful for tuning the central octaves of a piano's keyboard without stretch?

    Each note within the central octaves of a piano's keyboard should be tuned in accordance with its theoretical fundamental frequency.


    Example:

    If the frequency ratio of A4-E5 is 3/2 and the fundamental frequency of A4 is 440 Hz, the fundamental frequency of E5 can be calculated.

    E5 = 440 Hz × 3/2 = 660 Hz.

    If the frequency ratio of [insert interval] is [insert frequency ratio of interval] and the fundamental frequency of [insert 1st note of interval] is [insert fundamental frequency of 1st note of interval], the fundamental frequency of [insert 2nd note of interval] can be calculated.

    [Insert 2nd note of interval] = [insert fundamental frequency of 1st note of interval] × [insert frequency ratio of interval] = [insert fundamental frequency of 2nd note of interval].


    This method ignores inharmonicity and can be used to directly apply the theoretical model of each temperament without stretch to the central octaves of a piano's keyboard.

    Unequal temperaments are more mathematically complicated than equal temperaments due to the inequality of their intervals. It is easier to calculate the scale of a temperament if there are not many different types of fifths. The system of cents can be incorporated into the frequency ratios to calculate the frequencies of the scale of a temperament.

    ------------------------------
    Roshan Kakiya
    ------------------------------



  • 8.  RE: Fundamental Frequencies vs Coincidental Partials

    Posted 4 days ago
    We're not talking about calculating the frequency - we're talking about tuning it. 

    For years I have tuned both aurally and using Tunelab with standard stretch. Most of the time the results were superb.

    But since adopting no stretch in the central three octaves the results have been reliable. A reality is that in this section inharmoncity isn't generally very heavy so tuning to fundamentals doesn't give a lot of difference to tuning harmonics. There's a further thought here in so far as when tuning fundamentals it's as tuning organ pipes or sine waves from two laboratory oscillators. The harmonics of piano strings are at a lower level than the fundamentals. Therefore tuning to fundamentals gives a more solid sound than listening to beats from harmonics, which are musically irrelevant compared to the fundamentals. Tuning getting harmonics beatless may well have been an adjunct to Equal Temperament practise so as to give the piano an edge with which to get the sound to glisten and shimmer. Essentially I'm tuning a compromise between the two systems, and arguably or potentially accessing the best of both. 

    In the treble top two octaves, bearing in mind that the instruments with duplex stringing sound out a range of frequencies for every note as the Duplex is rarely nicely tuned, we're looking there more at the concept of tuned percussion so tuning bright, to the harmonics below is fine. Below the central three octaves for the bass two and a bit octaves were the harmonics not to align with the centre then it would jangle more which is why people take objection rightly to pure machine tuned fundamentals down there. Those notes musically set the harmony for above and so getting their harmonics to coincide with harmonic chords above is the priority above any relevance of their fundamental pitch.

    Incidentally I'm aware of a tuner in the UK who has record of a tuner in the 1920s bemoaning the loss of discussions when he was young in the 1870s or so as to where to put "The Wolf". This sets fire to Fred Sturm's insistence that after the 1830s when Montal laid out the textbook for tuning Equal Temperament the new tuning became universal. Far from it. In due course I'll be meeting up with him and will be able to find out more about the source.

    Best wishes

    David P






  • 9.  RE: Fundamental Frequencies vs Coincidental Partials

    Posted 4 days ago
    David,

    "We're not talking about calculating the frequency - we're talking about tuning it."

    If a piano is tuned with the aid of a device/machine, it would be beneficial to make use of its capabilities by calculating and using fundamental frequencies. This is because your method directly involves dealing with fundamental frequency ratios such as 2/1 for the perfect octave and 3/2 for the perfect fifth.


    "Therefore tuning to fundamentals gives a more solid sound than listening to beats from harmonics, which are musically irrelevant compared to the fundamentals."

    The method that I have described on my previous post facilitates this approach because it only involves tuning each note within the central octaves of a piano's keyboard in accordance with its theoretical fundamental frequency. It ignores inharmonicity by not considering any of the partials beyond the 1st partial/fundamental frequency of each note within the central octaves of a piano's keyboard.

    ------------------------------
    Roshan Kakiya
    ------------------------------



  • 10.  RE: Fundamental Frequencies vs Coincidental Partials

    Posted 4 days ago
    To put it simply, theory = practice, in practice, if the central octaves of a piano's keyboard and the intervals within them are being tuned without stretch.

    ------------------------------
    Roshan Kakiya
    ------------------------------



  • 11.  RE: Fundamental Frequencies vs Coincidental Partials

    Registered Piano Technician
    Posted 3 days ago
    @Roshan,  ​​What you're describing sounds more like a Pythagorean/Just tuning system. David is talking about applying the calculated (cent) offsets of unequal temperaments on top of regular old unstretched 12-tone equal temperament. So E5 would be 659.2551 Hz, not 660 Hz as you calculated above. The fundamental frequency doubles exactly each octave, and the frequency ratio of each minor second in the underlying temperament (before the unequal temperament is applied) is exactly 2^(1/12). I think I've seen you calculate the frequencies yourself in an earlier post, but they're also listed at https://en.wikipedia.org/wiki/Piano_key_frequencies#List

    ------------------------------
    Anthony Willey, RPT
    http://willeypianotuning.com
    http://pianometer.com
    ------------------------------



  • 12.  RE: Fundamental Frequencies vs Coincidental Partials

    Posted 3 days ago
    Anthony,

    The method that I have described can be used for any tuning system such as equal temperament, unequal temperament and just/rational intonation. I know that unequal temperaments can have a mixture of intervals such as perfect fifths and tempered fifths.

    I can use the Circle of Fifths and Fourths to construct the scale of an unequal temperament. I can also incorporate the system of cents into the frequency ratios so that cents can be converted into frequencies.

    I can calculate the frequencies of Kirnberger III for C3-C4-C5-C6.

    ------------------------------
    Roshan Kakiya
    ------------------------------



  • 13.  RE: Fundamental Frequencies vs Coincidental Partials

    Posted 3 days ago
    The frequencies for Kellner are from Tenor C up three octaves:
    131.4
    138.5
    147.0
    155.8
    164.6
    175.2
    184.6
    196.6
    207.7
    220.0
    233.6
    246.8
    262.8
    276.9
    294.1
    311.6
    329.2
    350.4
    369.2
    393.2
    415.4
    440.0
    467.2
    493.6
    525.7
    553.9
    588.2
    623.2
    658.3
    700.9
    738.5
    786.5
    830.8
    880.0
    934.5
    987.2
    1051.3
    1107.8
    1176.4
    1246.3
    1316.6
    1401.8
    1477.0
    1573.0
    1661.7
    1760.0
    1869.0
    1974.4
    2102.7
    Calculating the frequencies is not the difficult part.

    It's predicting how they will line up with harmonics of the lower strings when in practice inharmonicity differs between instruments. This is a matter that varies in the field and lends itself to analog rather than digital solution. The analogue solution to the problem is the task of the piano tuner. Why we are obsessed by digital solutions I don't know. But digital solutions such as Anthony's based on the inharmonicities measured in the field are particularly helpful.

    Best wishes

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





  • 14.  RE: Fundamental Frequencies vs Coincidental Partials

    Posted 3 days ago
    David,

    Have you tried using a device/machine to tune each note within the central octaves in accordance with its theoretical frequency?

    For example, according to your data, the theoretical frequency of C4 is 262.8 Hz. You could set 262.8 Hz as the tuning frequency on a device/machine and tune C4 until its frequency is 262.8 Hz.

    This procedure can be repeated to manually tune each note within the central octaves.

    A device/machine that does not account for inharmonicity would need to be used so that the intervals within the central octaves are not stretched.

    ------------------------------
    Roshan Kakiya
    ------------------------------



  • 15.  RE: Fundamental Frequencies vs Coincidental Partials

    Posted 3 days ago
      |   view attached
    Kirnberger III Circle of Fifths and Fourths



    Kirnberger III Frequencies (C3 to C6)

    C3 131.70 Hz
    C#3 138.75 Hz
    D3 147.16 Hz
    D#3 156.09 Hz
    E3 164.44 Hz
    F3 175.60 Hz
    F#3 185.00 Hz
    G3 196.89 Hz
    G#3 208.12 Hz
    A3 220.00 Hz
    A#3 234.14 Hz
    B3 246.66 Hz
    C4 263.40 Hz
    C#4 277.50 Hz
    D4 294.33 Hz
    D#4 312.18 Hz
    E4 328.88 Hz
    F4 351.21 Hz
    F#4 369.99 Hz
    G4 393.77 Hz
    G#4 416.24 Hz
    A4 440.00 Hz
    A#4 468.27 Hz
    B4 493.33 Hz
    C5 526.81 Hz
    C#5 554.99 Hz
    D5 588.66 Hz
    D#5 624.37 Hz
    E5 657.77 Hz
    F5 702.41 Hz
    F#5 739.99 Hz
    G5 787.54 Hz
    G#5 832.49 Hz
    A5 880.00 Hz
    A#5 936.55 Hz
    B5 986.65 Hz
    C6 1053.62 Hz

    ------------------------------
    Roshan Kakiya
    ------------------------------

    Attachment(s)



  • 16.  RE: Fundamental Frequencies vs Coincidental Partials

    Posted 3 days ago
    Yes - I use a Vogel CTS5 tuner, as recommended by Michael Gamble also here.

    One can set 0 inharmonicity as it's used for organ tuning and harpsichords as well as pianos. In addition it offers some 5 or so other stretch regimes. So in the treble I set it to C6 whilst playing C5 and then choose the inharmonicity setting that gives the closest match to the 2nd harmonic of C5. When I get to C7 I'll check the setting again against the 2nd harmonic of C6.

    This was the sort of technique used by an early tuning device available in the UK in the 1970s which had settings for the lower instrument tuned without stretch and then three fine tuning knobs for the top three octaves which one tuned to the successive harmonics. Anthony Willey's software has the advantage of being able to do this essentially on the fly.

    Best wishes

    David P

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





  • 17.  RE: Fundamental Frequencies vs Coincidental Partials

    Registered Piano Technician
    Posted 3 days ago
    @Roshan, Those are the right frequencies if you calculate the Kirnberger III using the Pythagorean Comma. I calculate it using the Syntonic Comma, placing the Schisma between C# and F#.

    For that case, the cent offset values would be:
    A3 0
    A#3 6.35
    B3 -1.47
    C4 10.26
    C#4 0.49
    D4 3.42
    D#4 4.4
    E4 -3.42
    F4 8.31
    F#4 0.49
    G4 6.84
    G#4 2.44
    A4 0


    ------------------------------
    Anthony Willey, RPT
    http://willeypianotuning.com
    http://pianometer.com
    ------------------------------



  • 18.  RE: Fundamental Frequencies vs Coincidental Partials

    Posted 2 days ago
    Anthony,

    I prefer to divide the Pythagorean comma because It is easier to deal with one comma (Pythagorean comma) rather than two commas (Syntonic comma and Schisma).

    Syntonic comma (21.51 cents) + Schisma (1.95 cents) = Pythagorean comma (23.46 cents).

    ------------------------------
    Roshan Kakiya
    ------------------------------



  • 19.  RE: Fundamental Frequencies vs Coincidental Partials

    Posted 2 days ago
    Roshan,
     Thanks for clarifying .... so where do you place your "schizma"? And what are the cents there...? The third question is, should
    one be setting Kirnberger III from C3 or from A3 (or even A2)?...that's pretty low. Doesn't that change the perfect thirds and schizma?
    I guess I don't understand the fine points of using historical temperaments. I usually set them using the Cs, rather than the As;
    I think as long as the perfect intervals are the same, it shouldn't matter where you begin. It is confusing though is someone
    is talking about starting from As and someone else talks about starting from Cs, at least it is to me.

    Richard

    ------------------------------
    Richard Adkins
    Piano Technician
    Coe College
    Cedar Rapids, IA
    ------------------------------



  • 20.  RE: Fundamental Frequencies vs Coincidental Partials

    Posted 2 days ago
    Although I tune Kirnberger III occasionally it might be too strong for general taste, and get unequal temperament a bad name as a result. 

    Kellner unstretched is capable of being inoffensive on modern pianos with modern repertoire.

    Kirnberger III is the next best thing to 1/4 comma Meantone if one wants to veer in that direction for earlier music. Again zero stretching is crucial to avoid C# to F being unacceptable particularly in the tenor octave. Ab major was the key of putrefaction and death . . . and for our ears it's best only to hint at it rather than let it scream.

    Best wishes

    David P



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





  • 21.  RE: Fundamental Frequencies vs Coincidental Partials

    Posted 2 days ago
    Richard,

    "Thanks for clarifying .... so where do you place your 'schizma'?"


    Pythagorean comma (23.46 cents) = Syntonic comma (21.51 cents) + Schisma (1.95 cents).

    The Syntonic comma and the Schisma are part of the Pythagorean comma. The Syntonic comma and the Schisma can be eliminated at the same time by eliminating the Pythagorean comma. Therefore, one can kill two birds with one stone by eliminating the Pythagorean comma. However, this is only one side of the story.


    The other side of the story must also be considered. Anthony's diagram provides a clear insight into the theoretical compromises that need to be made:

    If the Syntonic comma (21.51 cents) is equally spread across C-G, G-D, D-A and A-E, C-E becomes a Pure Major Third. However, this is achieved by reducing F#-C# by a Schisma (1.95 cents). Therefore, this method produces 7 Pure Fifths.

    On the other hand, if the Pythagorean comma is equally spread across C-G, G-D, D-A and A-E, F#-C# becomes a Pure Fifth. Therefore, this method produces 8 Pure Fifths. However, this is achieved by reducing C-E by a Schisma (1.95 cents). I can illustrate this by comparing the value of the Tempered C-E in cents with the value of the Pure C-E in cents.


    Kirnberger III Cents

    C 0.00 cents
    C# 90.22 cents
    D 192.18 cents
    D# 294.13 cents
    E 384.36 cents
    F 498.04 cents
    F# 588.27 cents
    G 696.09 cents
    G# 792.18 cents
    A 888.27 cents
    A# 996.09 cents
    B 1086.31 cents
    C 1200.00 cents


    Offsets in Cents from Pure Octave Equal Temperament

    C 0.00 cents
    C# 9.78 cents
    D 7.82 cents
    D# 5.87 cents
    E 15.64 cents
    F 1.96 cents
    F# 11.73 cents
    G 3.91 cents
    G# 7.82 cents
    A 11.73 cents
    A# 3.91 cents
    B 13.69 cents
    C 0.00 cents


    Difference between Tempered C-E and Pure C-E in cents = 384.36 cents − 1200 × log2(5/4) cents = 384.36 cents − 386.31 cents = −1.95 cents = Schisma.

    The Tempered C-E will be a Pure Major Third reduced by a Schisma if Kirnberger III has been constructed by equally spreading the Pythagorean comma across C-G, G-D, D-A and A-E.


    ------------------------------
    Roshan Kakiya
    ------------------------------

    Attachment(s)



  • 22.  RE: Fundamental Frequencies vs Coincidental Partials

    Posted 2 days ago
    The resultant schisma for CE seems to go against what David Pinnegar advicates a "pure" CE interval
    to get his "better" effect than ET, no?

    R

    ------------------------------
    Richard Adkins
    Piano Technician
    Coe College
    Cedar Rapids, IA
    ------------------------------



  • 23.  RE: Fundamental Frequencies vs Coincidental Partials

    Posted 2 days ago
    A tempered C-E in Kirnberger III (1/4 Pythagorean comma) that has a value of 384.36 cents is significantly "better" than a tempered C-E in Pure Octave Equal Temperament that has a value of 400.00 cents.


    Pure C-E = 386.31 cents.

    Tempered C-E in Kirnberger III (1/4 Pythagorean comma) = 384.36 cents.

    Deviation = 384.36 cents − 386.31 cents = −1.95 cents.


    Pure C-E = 386.31 cents.

    Tempered C-E in Pure Octave Equal Temperament = 400.00 cents.

    Deviation = 400.00 cents − 386.31 cents = +13.69 cents.


    ------------------------------
    Roshan Kakiya
    ------------------------------



  • 24.  RE: Fundamental Frequencies vs Coincidental Partials

    Posted 2 days ago
    To be clear I'm meaning that the two ways of getting a Kirnberger III ....Anthony Willey's way has a perfectd CE third... and Roshan's way....has
    a schisma CE third...which is 1.95cents....so there'll be a beat of almost 2 per second...very audible and slightly "wangy" sounding....

    That's what I was getting at.... yes, either one would be better than the tempered third, but only if you want to do a Kirnberger temperament
    in order to get David Pinnegar's effect.

    But Anthony's comes out pure and Roshan's doesn't. Roshan give you one more perfect 5th...but that to me is not as
    critical as still thirds....

    Maybe I just don't understand the point to do a Kirnberber III if you're going to compromise a third like that for the sake of 5th.

    I'll have to try both ways, I guess.

    Thanks Roshan!



    ------------------------------
    Richard Adkins
    Piano Technician
    Coe College
    Cedar Rapids, IA
    ------------------------------



  • 25.  RE: Fundamental Frequencies vs Coincidental Partials

    Registered Piano Technician
    Posted 2 days ago
    I'll be honest, I get a little muddled when it comes to the thirds in unequal temperaments. I usually just think of it in terms of spreading the comma between the 5ths.

    • Equal temperament gives 12 equal fifths, all 1.96 cents narrow.
    • Kirnberger calculated with the Pythagorean comma gives 8 pure fifths and 4 that are 5.87 narrow (yuck)
    • Kirnberger calculated with the Syntonic comma gives 7 pure fifths, 4 that are 5.38 cents narrow (slightly less yuck), and one that is 1.95 cents narrow (same as an equal-tempered fifth).

    I don't have an opinion on which way is better. I used the Syntonic comma because that's the way I understood that it was done historically, and why use a historical temperament if you're not going to try to be historically accurate? I'm fine with people arguing that one way sounds better than another (though I doubt 99% of people could recognize the difference listening to played music). But I would push back on arguments that one is better than the other because it is easier to express mathematically in terms of interval frequency ratios.

    ------------------------------
    Anthony Willey, RPT
    http://willeypianotuning.com
    http://pianometer.com
    ------------------------------



  • 26.  RE: Fundamental Frequencies vs Coincidental Partials

    Posted 2 days ago
    Anthony,

    There are always theoretical compromises that need to be made. The approach that I have suggested is one of the ways in which the Schisma can be distributed. The Schisma can also be distributed in other ways.

    The following is another way of distributing the Schisma:

    The Syntonic comma (21.51 cents) can be equally spread across C-G, G-D, D-A and A-E. This would cause C-E to be a Pure Major Third.

    1/4 Syntonic comma = 21.506 cents / 4 = 5.377 cents.

    Pure Fifth reduced by 1/4 Syntonic comma = 701.955 cents − 5.377 cents = 696.578 cents = 696.58 cents.


    The Schisma can be equally spread across the 8 remaining fifths.

    1/8 Schisma = 1.954 cents / 8 = 0.244 cents.

    Pure Fifth reduced by 1/8 Schisma = 701.955 cents − 0.244 cents = 701.711 cents = 701.71 cents.


    Verification:

    7 Octaves = 1200 cents × 7 = 8400 cents.

    12 Tempered Fifths = 696.58 cents × 4 + 701.71 cents × 8 = 8400 cents.


    Modified Kirnberger III (1/4 Syntonic comma and 1/8 Schisma):

    https://my.ptg.org/communities/community-home/digestviewer/viewthread?GroupId=43&MessageKey=72771916-275d-4dc7-bcca-d5d481656d7c&CommunityKey=6265a40b-9fd2-4152-a628-bd7c7d770cbf

    ------------------------------
    Roshan Kakiya
    ------------------------------