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Tuning Stretch. Is there a better way?

  • 1.  Tuning Stretch. Is there a better way?

    Registered Piano Technician
    Posted 06-19-2020 17:14
    There are two types of technicians and both can do high level work. It's just that the ones who know why their tunings work, can teach others.

    Case in point. Tuning the stretch. The accepted method is to "stretch the octaves". Just what does that mean anyway? Guess!?

    I get it. After 10 or 20 years of guessing, you get good at guessing.

    But what I have found is that if the mid-range octaves are tuned as pure 6:3 which is the best size for a piano where the 4:2 and 6:3 are close to each other (I call this a Small Scale), then we can tune pure P12 AND pure P19, at the same time.

    AND, if the mid-range 8:4 works out to be as narrow as the P4 is wide, you can tune pure P12 AND pure P22, all at the same time.

    Pure P12, pure P19, AND pure P22? Now that sounds like a very accurate and precise stretch. No guessing and I can explain how it works.

    Join me for a FREE Zoom meeting this Monday at 12:00pm Eastern.

    https://zoom.us/meeting/register/tJwtcuiuqD0tGtVYYEwBEmZytJFPQo-c5EHd


    ------------------------------
    Mark Cerisano, RPT
    B.Sc.(Mech.Eng.), Dip.Ed.
    https://howtotunepianos.com
    http://mrtuner.com
    1-866-678-8637
    ------------------------------


  • 2.  RE: Tuning Stretch. Is there a better way?

    Posted 06-19-2020 17:53
    You've peaked my curiosity! So have signed up, if I can manage it UK time. 

    But signing up you require a choice about what one's greatest challenge is at the moment . . . . I haven't got one so I had to choose one of your categories at random.

    HAHA - the greatest challenge at the moment is being able to go out and tune. And the other day, I did go out, for a friend, and the weather's been dry . . . The instrument's been treated with CA before and on this occasion . . . tuning took ages. I had to tune, hold the lever in position, apply CA and wait for it to set . . . on nearly every pin!

    Yes - greatest challenge - getting notes to hold! HAHA

    Best wishes

    David P

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





  • 3.  RE: Tuning Stretch. Is there a better way?

    Registered Piano Technician
    Posted 06-19-2020 21:24
    See you then David,

    I hope you find it interesting.

    ------------------------------
    Mark Cerisano, RPT
    B.Sc.(Mech.Eng.), Dip.Ed.
    https://howtotunepianos.com
    http://mrtuner.com
    1-866-678-8637
    ------------------------------



  • 4.  RE: Tuning Stretch. Is there a better way?

    Registered Piano Technician
    Posted 06-23-2020 09:01
      |   view attached
    Greetings, Mark,


    Missed the zoom explanation,but thought I'd add a comment.
    "Pure P12, pure P19, AND pure P22? Now that sounds like a very accurate and precise stretch. No guessing and I can explain how it works."

    The P12 is a 3:1, the P19 in a 6:1, and a P22 is an 8:1. In his book "The Craft of Tuning", Dan Levitan has shown that any of these coinciding harmonic relationships inherently produce their own stretch, depending on how far away each harmonic is from from its 1st partial. The actual stretch of these intervals on any given piano is the product of the inherent stretch at each of the notes in the interval and that note's inharmonicity. 

    Yes the above intervals can be tuned, but the choice of a "default" interval for any region of the piano has its implication for the width (or narrowness) of the intervals above (or below) it.

    ------------------------------
    William Ballard RPT
    WBPS
    Saxtons River VT
    802-869-9107

    "Our lives contain a thousand springs
    and dies if one be gone
    Strange that a harp of a thousand strings
    should keep in tune so long."
    ...........Dr. Watts, "The Continental Harmony,1774
    +++++++++++++++++++++
    ------------------------------



  • 5.  RE: Tuning Stretch. Is there a better way?

    Posted 06-23-2020 10:06
    It was a very interesting seminar, although I came in only in the last half hour with computer not equipped with video or microphone.

    Thanks so much for organising it.

    I tune pure 12ths in the bass, but find that no one solution fits all instruments. 

    Best wishes

    David P

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





  • 6.  RE: Tuning Stretch. Is there a better way?

    Registered Piano Technician
    Posted 06-23-2020 16:58
    Thank you David and William,

    It's true that each piano is different but what I have done is identify the "ideal" piano, for lack of a better term, and with that comes some choices.

    I would like to discuss this from the point of view of how many intervals we can get to be pure.

    But to do that, we need a definition of what pure is.

    For this discussion, we will say that "pure" (with quotes) is defined as an interval whose check intervals "sound" the same speed. I have shown that all humans, regardless of tuning or musical experience, can identify non-equal beat rates only after they are more than about 3% different. Each person may be higher or lower, but every person has a limit! "Pure" intervals are tuned within that limit.

    Try it yourself: https://howtotunepianos.com/ear-training-for-piano-tuners-beat-speed-difference-sensitivity-test/

    How to Get Pure Intervals in the Piano
    The pianos inharmonicity will allow some intervals to be tuned "pure", but after that, we must make choices if we want more "pure" intervals.

    For example, on many well-scaled pianos, it is possible to tune a mid-range "pure" 4:2 and a "pure" 6:3, at the same time, where M3=M10 and m3=M6, as far as we can tell. (i.e. "pure" with quotes) if the piano will let us.

    I call these Small Scale Octaves since the distance between the 4:2 and 6:3 is small.


    But as we can see here, the m3 is actually narrow. We just can't hear it. (The 6:3 is always to the left of the 4:2 on almost all pianos)

    This means that the m3 is actually faster than the M6 for a "pure" 4:2, "pure" 6:3.

    m3 M6
    In Figure 13 we see that this "pure" 6:3 is actually narrow but it has the m3 and the M6 existing in each other's 3% bubble. That's why we can't tell it's narrow; why it "sounds" pure to us.

    In this way the piano allows us to tune a "pure" 6:3 and a "pure" 4:2 at the same time.

    But it is what we do with the M17 that allows us to have a "pure" P12 and a "pure" P19 at the same time.


    What is interesting to note in figure 14 is that since the m3 and M6 exist in each other's bubble, the area where the M17 can be in both bubbes is relatively large, and the distance between the P12 and the P19 is small.

    When I started creating these figures for my book, I noticed something else very interesting.

    For mid-range octaves where "pure" 4:2 and "pure" 6:3 are not possible, I used to say that the "pure" P12 and "pure" P19 were not possible either, but that is not entirely true, as shown in figure 19.

    medium 6:3
    There are also procedures to identify when the piano's inharmonicity will allow "pure" 11, "pure" P4 at the 8:6, and "pure" P22. This depends on the P4/8:4 assumption. It is possible when the P4 is as wide as the 8:4 is narrow. These tests are for specific intervals called the "window".
    Using the same check note, we tune M3=M10<M6=M17=m3down=m6down. In there you can see the P4 (M3<M6) and the 8:4 (m6>M3)

    This is all based on actual inharmonicity. Intervals have been recorded, filtered, and measured to show that beat rates in real pianos prove the existence of these intervals in a piano aurally tuned to this high level.

    Personally, for me, tuning a piano any other way is just guessing. But some people are pretty good at guessing. πŸ™‚

    Star Trek IV - The Voyage Home

    Kirk Mr. Spock, have you accounted for the variable mass of whales and water in your time re-entry program?

    Spock Mr. Scott cannot give me exact figures, Admiral, so... I will make a guess.

    Kirk A guess? You, Spock? That's extraordinary.

    Spock [to Dr. McCoy]  I don't think he understands.

    McCoy No, Spock. He means that he feels safer about your guesses than most other people's facts!

    (love that movie)





    ------------------------------
    Mark Cerisano, RPT
    B.Sc.(Mech.Eng.), Dip.Ed.
    https://howtotunepianos.com
    http://mrtuner.com
    1-866-678-8637
    ------------------------------



  • 7.  RE: Tuning Stretch. Is there a better way?

    Member
    Posted 06-23-2020 16:51
    Here's a graph showing all the Verituner data for a recent tuning of a nice Yamaha grand. I think this visualization should help with the current discussion. 


    ------------------------------
    Jason Kanter
    Lynnwood WA
    425-830-1561
    ------------------------------



  • 8.  RE: Tuning Stretch. Is there a better way?

    Member
    Posted 06-23-2020 17:07
    And here's a graph of a Wurly. A poorly scaled, compromised bass.


    ------------------------------
    Jason Kanter
    Lynnwood WA
    425-830-1561
    ------------------------------



  • 9.  RE: Tuning Stretch. Is there a better way?

    Registered Piano Technician
    Posted 06-23-2020 17:30

    Thank you Jason,

    It is my opinion that one problem with ETD's is the danger of having "too much" information. One example is the ETD that weighs partials based on volume. Humans hear partials subjectively as we all know from days when we couldn't hear them at all. πŸ™‚  Assuming the ETD can guess how loud we hear any partial is a bit arrogant in my opinion.

    It is also my opinion that ETD's do not tune intervals to the sizes they advertise. I recently posted a video on Youtube that proves this. See https://youtu.be/uKOFTsH6dD4

    The argument that too much information can be counter-productive is illustrated in the development of the Goldman Cardiac Risk Indices. 
    https://en.wikipedia.org/wiki/Revised_Cardiac_Risk_Index 

    Prior to the use of these questions, patients underwent extensive tests and interviews to determine if they were at risk of complications from surgery. The tests took so long that many patients died before the operation.

    Goldman (and Lee) came up with a simpler, easier set of questions to ask, that looked at less data.

    This is a clear example of "less is more"

    "Compared with the Original Cardiac Risk Index, the RCRI was easier to use and more accurate." - https://en.wikipedia.org/wiki/Revised_Cardiac_Risk_Index#:~:text=In%201977%20Goldman%2C%20et%20al,In%201999%2C%20Lee%20et%20al.

    However, the proof is in the pudding.

    If anyone believes they have an ETD that can produce as many pure intervals as can be done aurally with this system, just send me a recording of the following intervals and we will know for sure, once and for all.

    C#3F3
    C#3F4
    F3G#3
    G#3F4

    F#3A#3
    F#3A#4
    A#3C#4
    C#4A#4

    C#4F4
    C#4F5
    C#4A#4
    C#4F6
    A#3C#4
    F3C#4

    Feel free to transpose these intervals anywhere on the piano.

    Mark

    P.S. I am well aware that many technicians "feel" the pitch and tune to what sounds good and feels good. This is not that kind of system, and even though these technicians can do great work, not everybody is attracted to this type of system. What I present is a more objective and empirical system that puts the power back into the tuner's ear instead of the ETD. If we can find an aural system that claims to get better results than an ETD, why not look into it I say. Anytime a human can beat a computer is a great day for humanity - but that's just my opinion. πŸ™‚



    ------------------------------
    Mark Cerisano, RPT
    B.Sc.(Mech.Eng.), Dip.Ed.
    https://howtotunepianos.com
    http://mrtuner.com
    1-866-678-8637
    ------------------------------



  • 10.  RE: Tuning Stretch. Is there a better way?

    Registered Piano Technician
    Posted 06-24-2020 18:59
      |   view attached
    @Mark Cerisano, OK, I'll bite. Here's a recording of the intervals you requested, taken on an old 1970 Yamaha U1, single strings only. For completeness I also recorded some progressions of slow and fast-beating intervals (octaves, fifths, fourths, twelfths, thirds, tenths, and sixths) between F2 and A5 (the length of my temperament strip plus a few mutes). ​​​​With the exception of a couple notes (G#4 and maybe F5) that apparently drifted between the time I tuned and recorded, all the notes should be within 0.5 cents of their ETD targets. I guess I should take your stability class.

    I can't claim that any specific intervals are being tuned pure here. This particular ETD doesn't do that, but instead seeks a compromise that minimizes the beat rates of all slow beating intervals. The 12ths end up being fairly pure, but even that's a compromise between 3:1 and 6:2. You can see an example of that in the attached images of target interval widths (in cents). The algorithm ignores fast-beating intervals, so I'd be interested to see where the thirds fall if you score those.

    You didn't specify a file format so I uploaded .WAV. Sorry for kiddo noises in the background. I've got a 6-month-old who loves to rattle a set of big plastic keys :-)


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



  • 11.  RE: Tuning Stretch. Is there a better way?

    Registered Piano Technician
    Posted 06-24-2020 20:02
    Nice. Let's see what we got here.

    ------------------------------
    Mark Cerisano, RPT
    B.Sc.(Mech.Eng.), Dip.Ed.
    https://howtotunepianos.com
    http://mrtuner.com
    1-866-678-8637
    ------------------------------



  • 12.  RE: Tuning Stretch. Is there a better way?

    Registered Piano Technician
    Posted 06-24-2020 20:19
    Can you send C#4A#4? I think you played C#4G#4 instead. πŸ™‚

    ------------------------------
    Mark Cerisano, RPT
    B.Sc.(Mech.Eng.), Dip.Ed.
    https://howtotunepianos.com
    http://mrtuner.com
    1-866-678-8637
    ------------------------------



  • 13.  RE: Tuning Stretch. Is there a better way?

    Registered Piano Technician
    Posted 06-24-2020 20:36
      |   view attached
    Whoops! You're right. Here you go. (I just grabbed a snip from 5:53 in the recording where I was running the M6ths.)

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



  • 14.  RE: Tuning Stretch. Is there a better way?

    Registered Piano Technician
    Posted 06-25-2020 10:34
    Thanks.

    I'm trying to make a file that other people can listen to. The C#4A#4 you sent was a little short. The other recordings were about 3 seconds each. Can you send another. Sorry to have to ask again.

    ------------------------------
    Mark Cerisano, RPT
    B.Sc.(Mech.Eng.), Dip.Ed.
    https://howtotunepianos.com
    http://mrtuner.com
    1-866-678-8637
    ------------------------------



  • 15.  RE: Tuning Stretch. Is there a better way?

    Registered Piano Technician
    Posted 06-26-2020 09:06
      |   view attached
    No problem. Here's a new recording.
    (Sorry for the delay, I thought I sent this yesterday, but it looks like I forgot to hit "send" after the attachment finished uploading.)

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



  • 16.  RE: Tuning Stretch. Is there a better way?

    Registered Piano Technician
    Posted 06-26-2020 13:55
    I don't know precisely where this puts me in this picture (e.g. contrarian, guesser, don't care, just thinks differently...), but to be truly honest about it, when I'm tuning equal temperament I DON'T want any "pure" intervals. I want my slow intervals to roll slowly but nicely, I want all my octaves to roll a bit (some more than others), i even want my unisons to START to roll...but not complete the roll (a "micro-beat"  as someone coined the term...I forget who). 

    I'm sure someone somewhere will label me a nut job but i can produce a very very nice tuning with these parameters and I don't need a machine to help me. 

    Pwg

    ------------------------------
    Peter Grey
    Stratham NH
    603-686-2395
    pianodoctor57@gmail.com
    ------------------------------



  • 17.  RE: Tuning Stretch. Is there a better way?

    Registered Piano Technician
    Posted 06-26-2020 20:08

    We need to understand and recognize that there are two different approaches. The holistic and the analytical. 

    The benefit of the analytical is that everyone gets a score; ETD and human. 

    The real power comes when the analytical suggests an improvement that the holistic tuner likes. πŸ™‚



    ------------------------------
    Mark Cerisano, RPT
    B.Sc.(Mech.Eng.), Dip.Ed.
    https://howtotunepianos.com
    http://mrtuner.com
    1-866-678-8637
    ------------------------------



  • 18.  RE: Tuning Stretch. Is there a better way?

    Registered Piano Technician
    Posted 06-26-2020 15:46

    Here are the marks.

    The challenge was determining the ideal size. Read what I have come up with at the end of this post:

    ETD Mark
    OCTAVES
    F3F4(Medium): 30%
    A#3A#4(Small): 100%
    Average: 65%

    STRETCH
    Ideal Stretch defined as Wide
    Mark: 100%

    Total Mark
    83%

    As a comparison, this was a mark I got tuning the same intervals by ear.

    Aural Mark
    OCTAVES
    F3F4(Medium): 100%
    A#3A#4(Small): na
    Average: 100%

    STRETCH
    Ideal Stretch defined as Wide
    Mark: 80%

    Total Mark
    90%

    It's important to note that these are just snapshots of one tuner/device tuning one series of intervals.
    To get a really accurate picture, we need to measure many interval sets tuned many times.

    One application for this procedure is to test tunings objectively without having to do a master tuning.

    Notes:
    "Pure" means that the check intervals are so close that we can't tell the difference. According to my data, this is 3%. For this marking system, I am using 5% as the limit

    Octave Definition and Goals:
    If tuneable as "pure" 4:2, "pure" 6:3, (4:2 and 6:3 are less than 10% apart) they are called Small Scale. (The distance between 4:2 and 6:3 is small)
    If not tuneable as "pure" 4:2, "pure" 6:3 (4:2 and 6:3 are between 10% and 20% apart), then octave is called Medium Scale.
    More than 20% apart, ideal size tuned as "pure" 4:2, resulting ion very narrow 6:3. (Large Scale)

    Stretch Definition and Goals:
    If A#3A#4 is "pure" 6:3 (Small Scale, m3/M6 < 5%) then P12 and P19 can be tuned "pure". (Pure P12/P19 stretch)
    If A#3A#4 is a narrow 6:3 (Medium Scale, m3/M6 between 5 and 10%) then P12 and P19 can still be tuned "pure". (Pure P12/P19 stretch) 
    If A#3A#4 is a very narrow 6:3 (Large Scale, m3/M6 more than 10%) then P12 is tuned wide and P19 is tuned narrow by half the span (difference between m3 and M6) (Spilt P12/P19 stretch) 

    If P11 is less than 10% different than P12/P19 average, then I define the stretch as "Pure". ("Pure" P12/P19/P22 are possible and the goal) 
    If P11 is more than 10% different than P12/P19 average, then I define the stretch as "Wide". ("pure" P12/P19/P22 not possible. The goal is wide P12/P19 and narrow P11 by the amount calculated as half the span of the P11 and P12/P19 average.)

    Octave Marking Criteria (Mark one interval, 4:2 or 6:3, not both.)
    Small Scale. Lose marks if more than 5% wide/narrow. (2% per 0.1% over 5%)
    Medium Scale. Span = difference between 4:2 and 6:3 (Should be between 10% and 20%) Lose marks if size is more than 5% away from half the span. 2% for every 0.1% over 5%)
    Large Scale. (Only measure 4:2)  Lose marks if more than 5% wide/narrow. (2% per 0.1% over 5%)

    Stretch Marking (Just measure the P12/P19 average))
    Pure P12/P19 Stretch. Lose marks if more than 5% wide/narrow. (2% per 0.1% over 5%)
    Split P12/P19 Stretch. Lose marks if size is more than 5% away from half the span. 2% for every 0.1% over 5%)
    Pure P22 Stretch. Lose marks if more than 5% wide/narrow. (2% per 0.1% over 5%)
    Wide P22 Stretch. Lose marks if size is more than 5% away from half the span. 2% for every 0.1% over 5%)



    ------------------------------
    Mark Cerisano, RPT
    B.Sc.(Mech.Eng.), Dip.Ed.
    https://howtotunepianos.com
    http://mrtuner.com
    1-866-678-8637
    ------------------------------



  • 19.  RE: Tuning Stretch. Is there a better way?

    Posted 06-26-2020 16:22
    The intervals at 0:42 and 0:50 in that recording don't sound very nice . . . 

    Best wishes

    David P






  • 20.  RE: Tuning Stretch. Is there a better way?

    Registered Piano Technician
    Posted 06-26-2020 21:06
    Mark, I'm a bit unclear on what the percentages mean or what they're calculated from. Assume that for my F3-F4 octave the 4:2 is 0.5 beats per second wide of pure and the 6:3 is 0.75 BPS narrow. If I understand correctly you would want the one to be the same amount wide as the other one is narrow. But what's the math for scoring that? I'm wondering how wide and narrow you found those octaves and where you would have preferred them to be. 

    On this piano I see a wider spread between the 4:2 and 6:3 octaves for the plain wire strings at the tenor break (F3 and F#3 in particular). I assume it's from high inharmonicity from foreshortening of the bridge. 

    Anthony Willey
    (206) 307-4533

    (Sent from my mobile device)






  • 21.  RE: Tuning Stretch. Is there a better way?

    Registered Piano Technician
    Posted 06-27-2020 13:04

    Hi Anthony,

    Thank you for sending in the recording and updating them as well.

    To answer your questions:

    I measure the beat rates by filtering the recordings. See my free lesson on my website where I show you how to do that yourself using a free audio program called Ocenaudio (no typo)
    https://howtotunepianos.com/ocenaudio/

    The percentages are calculated from the criteria I listed at the end of the post. The criteria are developed from the research I have been doing with beat rates and pure intervals.

    As far as your example of bps for your F3F4 octave, I don't use bps. I use percentage differences in the M3/M10 and m3/M6 tests.

    Given clean beats, humans can't hear differences less than 3%. Try it yourself: https://howtotunepianos.com/ear-training-for-piano-tuners-beat-speed-difference-sensitivity-test/

    For the marking, I defined "pure" as within +/- 5%. I also used that as a window of tolerance so that any mathematically determined relationship was given 100% if it was within 5%, and then we lose 2% for every 0.1% after a 5% error.

    As far as Pianometer's graph, I'm afraid I don't understand it completely. Are the lines indicating the calculated octaves sizes based on how the piano was tuned? If so, it shows some pretty wild variations from the standard 6:3 in the bass instructions often suggested.

    Also, how can the graph for 12ths go from A0 to C8? Not possible.

    Finally, so you can see exactly how I did the calculations, I have included a link where you can get your own copy of the spreadsheet I used. This will be your own copy. Don't worry about changing things on it. It will not affect the original. 

    Here it is: https://docs.google.com/spreadsheets/d/1wVAQCpykHYURSRWrytKXOecOwhrKYlbsQVdybTyOC5Q/copy

    Also, take a look at this infographic:

    Aural Tuning Loop - How to Get BetterI have found this to be a very powerful way to become the best aural tuner I can be. Basically it means, "Don't take what technology says as accurate unless you can hear it for yourself." This process - guess, test, confirm - results in a fast-tracking of aural skills needed for aural tuning.

    For that reason, I will post the beat speeds with a recording of the filtered beats so we can confirm be ear that the technology is correct. Very powerful. 😎


    ------------------------------
    Mark Cerisano, RPT
    B.Sc.(Mech.Eng.), Dip.Ed.
    https://howtotunepianos.com
    http://mrtuner.com
    1-866-678-8637
    ------------------------------



  • 22.  RE: Tuning Stretch. Is there a better way?

    Registered Piano Technician
    Posted 06-27-2020 19:50
    Here are some pages that people may find useful.

    Treble Stretch Reports

    Anthony's PianoMeter Submission (With filtered recordings and graphs 😎)

    Instructions to submit a treble to be scored

    Volunteers are needed to submit tunings so we can get an idea of what different ETD's can produce.


    ------------------------------
    Mark Cerisano, RPT
    B.Sc.(Mech.Eng.), Dip.Ed.
    https://howtotunepianos.com
    http://mrtuner.com
    1-866-678-8637
    ------------------------------



  • 23.  RE: Tuning Stretch. Is there a better way?

    Registered Piano Technician
    Posted 06-28-2020 00:42
      |   view attached
    Cool, thanks. I'll check it out. From a glance at the numbers in the spreadsheet it looks like the F3-F4 6:3 octave was pure and the 4:2 was wide (0.8 BPS if I'm calculating correctly). Right?

    Re your question about the graphs, those are the interval widths calculated from the app's internal frequency targets, not any actual measurements. You're correct that I'm not tuning pure 6:3 octaves in the bass. If you look at the original octaves graph it looks like it's a compromise between 6:3 and 8:4 octaves...the 6:3 octaves are about as wide as the 8:4 octaves are narrow. But keep in mind that this is in cents, so when you convert to beats that still translates to slow octaves in the bass even though it looks a bit crazy in the graph.

    Here's another graph to give you a more in-depth picture of what's going on. There's a lot of lines (sorry) but they are color coded... octaves are all blue, twelfths are black, fifths are red, fourths are green, double and triple octaves are orange and yellow respectively, and the 19th is magenta.
    The transparency of the color of each line is related to how "loud" the individual partials are, and by extension, how strong of an effect that interval will have in the tuning calculation at any given note. So when you look at the low bass you can see that the "important" intervals include the 8:4 octave, the 8:1 triple octave, and the 6:4 fifth. The 6:2 twelfth is moderately strong as well. Note that while the 10:5 octave and 9:3 are both very strong they're not having a big effect on the tuning because they have a lower "weight" in the app's settings. (Tuning those more pure would add a ton of stretch.)

    I've attached another recording if you're interested in another data point. This was a large Kawai grand I tuned today in a client's home. (I think it was something like KG-7D). It's full unisons, not single strings this time. I've downloaded oceanaudio and I'll mess around with it tomorrow if I have time.


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