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There seems to be enough interest and talent on this Forum to try a Topic on Advanced Tuning Math.

This is a subject that I find fascinating because I believe it is within my reach, but not my grasp. I am hoping that those that do have the subject within their grasp will join in, so I will start this Topic at the level of understanding that I am at, rather than try to make this a primer for those that are new to the subject. Where this Topic may end up is anybody’s guess. For a basic understanding of the subject, I suggest the Wikipedia pages on Piano Tuning, Cents and Inharmonicity, Young’s paper on the inharmonicity of plain wire strings and the Pscale web page that shows the iH curve of many pianos.

Until iH rears its ugly head things are straight forward. The frequency of each note is the 12th root of 2 times the frequency of the note a semitone lower if beatless octaves are desired. If beatless 5ths or 12ths or equal beating 12 and 15ths are desired, then the multiplier is the 12th root of some other number than 2. And a set of frequencies using a fixed semitone ratio will produce a straight line on a Railsback diagram. But that is not how actual pianos are tuned.

Because of iH there really isn’t such a thing as a beatless interval. Whenever any pair of coincident partials are at the same frequency the other pairs will not be. Also, because the values of iH form a curve, the octave ratio changes from note to note even when the octave type (such as 4:2) is kept the same.

So given the iH value of the notes and a given octave type (I will use the 4:2 octave type for this post) all the A’s of a tuning can be precisely calculated. This will produce eight points on a Railsback diagram. The slope of the lines connecting the eight points will be the average slope of the semitone ratios of the notes between the points, but does not give us either the frequency of the individual notes or the individual semitone ratios.

It would seem that some type of nonlinear interpolation would give us the frequencies, but neither logarithmic nor polynomial interpolation give satisfactory results. The Railsbacks diagram does not show a smooth curve. Using the spline function may work, but is beyond my abilities.

So I am not sure what to try next. If a very theoretical piano is used where the iH curve can be written as a function, there should be a way to directly calculate the frequency of each note. But if tabular values for iH are used (and if available, should be used) it may only be possible to estimate the frequencies and then refine the estimates until a desired accuracy is achieved.

One estimate that is available is the octave ratio (or 5th ratio or 12th ratio or…) Given the theoretical frequency of one note and the iH values of both notes a very close estimate of the octave ratio can be calculated that can then later be refined when a better estimate of the note’s frequency is available. And the twelfth root of this octave ratio will be the average semitone ratio of the notes within the octave (or 5th or 12th or…)

But I don’t know if this information is what is needed to produce a tuning curve. For each semitone (at least in the middle of the piano) a very close estimate of the average semitone ratio for all the octaves that spans the semitone is available. Would an average of these twelve averages produce a more refined estimate for each semitone ratio? Of course these semitone ratios would have to be applied starting with one of the A’s. I do not have much confidence that these averages of averages would produce a chromatic scale that would result in the chosen 4:2 octave type. It seems that there would probably be a discrepancy with the last note.

So folks, that is where I am in my understanding. I hope those that are more knowledgeable can help me along.

Thanks for starting this topic. To simplify an already complicated subject let's follow your suggestion and tune 4:2 octaves across the keyboard. Once can understand that it will be easy to make necessary modifications for other octave choices. I also propose for analysis purposes to make the simplifying assumption we start from an F3 tuning fork, so F3 is our starting point.

I will notate partial numbers of a given note X by p2(X), p3(X), etc. So p4(X) will be slightly larger than 4, depending on the iH of X. This means that to tune a note Y an octave up from a given note X it will have frequency p4(X)/p2(Y) * X. If Y was on octave down from X, Y would have frequency p2(X)/p4(Y)*X. (Here I use note names to stand for the frequency of the note.)

Now if iH was constant, p2,3,4 would not depend on X and the semitone ratio would simply be (p4/p2)^(1/12) and there is no problem. If it is not constant we can still tune all octaves and the problem remains how to define the ET notes F3,...,E4.

In reality iH plotted versus note number looks something like a parabola, with a minimum around the tuning octave. This means the iH curve is approximately constant in the temperament octave so let’s tune our second note F4 as F4=p4(F3)/p2(F4) * F3, and fill in the temperament octave with semitones of size (p4(F3)/p2(F4) )^(1/12). After that we tune all the other notes in octaves, now using the known (not constant) values of p2,4 in the temperament octave. This will get something that looks like a Railsbacks curve, except it is a straight line in the temperament octave, whereas theoretically it should have some curvature there too. This is the point Jeff brought up earlier as objection to what I just wrote.

Now let’s look at an actual tuning curve obtained from tunelab (for my piano):

As you can see it is a straight line by any practical measure in the temperament octave, though a very close inspection reveals that it has actually a small curvature. This could be obtained by computing the curve as I state above, i.e., a straight line in the temperament octave and the rest computed with our iH values, and then fitting the whole thing to some spline or other function as you suggested.

While I think this solves the problem in a practical sense I share your discomfort about what happens to the semitones in the temperament octave. When using the spline smooth fit they will not be identical, though off by much less than is audible (or within practical tuning accuracy). Also there are many choices of splines and each will give a slightly different temperament, theoretically. It would be nicer if we could derive the infinitesimal variation in semitones within the temperament octave directly from the iH curve. Also, one might run into pianos where the iH curve is very nasty and not flat in the temperament octave. I’m not sure if that is a realistic possibility.

Before going on, let’s see if you have any objections to what I just wrote.

Kees

Kees:

You posted:

”I will notate partial numbers of a given note X by p2(X), p3(X), etc. So p4(X) will be slightly larger than 4, depending on the iH of X. This means that to tune a note Y an octave up from a given note X it will have frequency p4(X)/p2(Y) * X.”

I do not follow you here. A 4:2 octave means that the frequency of the 4th partial of the lower note [p4(X)] is at the same frequency as the 2nd partial of the upper note [p2(Y)]. When these two terms are equal then p4(X)/p2(Y) * X = X. So then you have tuned Y to the same frequency as X and have tuned a unison to a note that should be over twice the frequency.

”In reality iH plotted versus note number looks something like a parabola, with a minimum around the tuning octave. This means the iH curve is approximately constant in the temperament octave so let’s tune our second note F4 as F4=p4(F3)/p2(F4) * F3, and fill in the temperament octave with semitones of size (p4(F3)/p2(F4) )^(1/12).”

No, the low point in the iH curve is below the temperament octave, and even there is nothing near “approximately constant”. Here is a link to a number of iH curves: http://www.goptools.com/gallery.htm And, again, (p4(F3)/p2(F4) )^(1/12) equals 1 for a semitone ratio. We now have 13 notes tuned as unisons.

” Before going on, let’s see if you have any objections to what I just wrote.”

I think you underestimate just how small of a difference can be noted in tuning. An M3 need only be 0.8 cent wider than the one chromatically higher to be unprogressive. So to guarantee that RBIs are progressive, all notes would have to be tuned within 0.2 cents of ideal. SBIs require 7 times the accuracy, but I have yet to be able to set a pin that reliably.

I suggest that we keep this theoretical and expect an accuracy of +/- 0.01 cent.

If we take p2(x) to be the partial given as a multiple of the fundamental p1(x) then this works.
How about [p4(X)/X]/[p2(Y)/Y]* X = Y?

Not sure how this equation could be solved though... god my maths is rusty

You are confused by my notation, which I didn't explain clearly enough, sorry. p4(X) is the partial number (multiplier), not the partial. The 4th partial of X has frequency p4(X)*X. So Y=p4(X)/p2(Y) * X which is about 4/2 * X.


OK, but just look at the actual tuning curve I posted. If you believe the straight line approximation on F3-F4 produces audible differences I think the ball is on your court to show this. I don't believe it, but I could of course be wrong. By visual inspection of an enlargement I eyeball the differences are less than 0.1 cent.

Agreed. If my notation is now clear I will move on to the more interesting stuff.

Kees

EDIT: PS. A more correct (but clumsy) notation would be p4(s(X)) where s(X) denotes the string associated with note X.

PPS. You can of course define the temperament octave to be exactly at the minimum of the iH curve, I suspect ETD's do that. So you'd have to look at the inflection point of the tuning curve and see how close it is to a straight line there.

Kees

Here I will show how to compute the tuning curve directly from the inharmonicity curve.

Let's index the notes from 1 to 88. Let n be a vector (array) of dimension 88 which contains the pitches of all keys in cents measured (arbitrarily) from A0. So without inharmonicity we’d have n =(0 100 200 …8700).

Next assume we have another vector iH which has the 88 inharmoniticy constants or, even better, the actual partials (expressed as ratio w.r.t the first partial).

Next we define t as a 76 vector, which will be the octave sizes in cents measured from the bottom note. (It has only 76 elements as C#7 has no octave above it.) The t vector can be computed, its values are

t(i) = 1200/log(2) * log(p4(i)/p2(i+12)), i = 1,…,76.

(As before p4(i) is the ratio of the 4th and the 1st partial of note i.)

Now we are ready to write down equations that, when solved, yield n.
First of all we must have

n(i+12) = n(i) + t(i), i=1,...,76. (Eq 1)

This is a system of 76 equations in 88 unknowns, so left are 12 unknowns. We also must have

n(48) = 4700, (Eq 2)

which expresses that A4 = 440. That eliminates one more variable, and we are left with 11 remaining variables.

The 87 semitone sizes are given by

s(i) = n(i+1)-n(i), i = 1,…,87

and their change (induced by inharmonicity) is

ds(i) = s(i+1)-s(i), i=1,…,86.

We know we can’t make the change in semitone size arbitrarily small, but we can try to make the size change as smooth as possible. For this we define the change of the change, which is defined by

q(i) = ds(i+1)-ds(i), i = 1,…85.

Since we want the total change of change to be as small as possible, we define a penalty function

L = q’*q/2 (Eq 3),

where q’ is the transpose of vector q. This simply is the sum of the squares of all the q’s and can be zero only if all the q’s were zero. Note that q depends (linearly) on n, the variables to solve. L is called a penalty function because it is a function of the 88-vector n, and the bigger it is the unhappier we are.

Instead of zero-ing L, which is not possible, we minimize it subject to the octave constraint (eq 1), and the position of A4 (eq 2). To do this in practice requires some matrix algebra and results in a system of 88 linear equations for the unknowns n(1),…,n(88) which is easily solved using standard numerical linear algebra in a microsecond.

To try it out I made up an inharmonicity curve which looks like this:

Applying what I wrote above results in a tuning curve like this:

Here is the size of the semitones:


This method is not limited to a 12 tone equal division, we can divide the octave more finely. In the limit of an infinitely fine division we end up with a 6th order differential-integral equation for the tuning curve and the current procedure can be viewed as a discrete numerical approximation to solve it.

To use different octaves (4:1, mindless, mixes) we just have to change the constraint equation (1) and everything else remains the same.

To apply this to non-equal temperaments is possible but not so trivial, and very interesting. You first have to formulate the temperament as a particular constraint. E.g for WMIII the constraint would be all P5's are pure except CG GD DA BF# which are equally off.

This method would then find the optimal tuning that would most closely implement the goals of the temperament, which generally can't be met completely. Perhaps you'd want a more faithful implementation in the midrange, so you may want to weigh the penalty function to pay more attention to that.

I can post the MATLAB code if anyone is interested. MATLAB is expensive but there is a free open source version called, believe it or not, OCTAVE!

Kees

Jeff, I calculated the notes in the range F3-A4 with the trivial method which you criticized, then computed the actual beat ratios. I used the inharmonicity values I measured on my piano.
F3-A4 are numbered 1 to 17. The M3's and P5's (not shown) are fine, but you are right the P4's are not. I am surprised and was wrong about that. However I don't think anyone would notice. Only if I artificially boost the inharmonicity to go from 0.42 to 2.25 the M3 beat rates are no longer progressive.

oct stretch = 2.987400
iH(F3)=0.430000 iH(F4)=0.750000
M3 beats
n=1 fb= 6.697633
n=2 fb= 7.004596
n=3 fb= 7.324309
n=4 fb= 7.657197
n=5 fb= 8.003692
n=6 fb= 8.364230
n=7 fb= 8.739252
n=8 fb= 9.129198
n=9 fb= 9.534511
n=10 fb= 10.166462
n=11 fb= 10.839916
n=12 fb= 11.557578
n=13 fb= 12.322327

P4 beats
n=1 fb= 0.874295
n=2 fb= 0.872430
n=3 fb= 0.867218
n=4 fb= 0.858265
n=5 fb= 0.845143
n=6 fb= 0.827386
n=7 fb= 0.804487
n=8 fb= 0.775893 <------ Look at that!!!
n=9 fb= 0.898792
n=10 fb= 1.033622
n=11 fb= 1.181372
n=12 fb= 1.343105


Kees

Kees:

It is a wonderfully busy weekend for me. I do not want you to think you are being ignored. Nor do I want to give a quick reply. I may not get back to you for a few days. Your approach is refreshingly different than mine.

Looking forward to hear your approach Jeff. It really surprised me that even after you fix the octaves there is still room for choices within ET. So really ET is not defined precisely when there is iH. Looking again at the 4ths in my previous post, maybe it is audible after all like you said before. The 4th on E4 beats about twice as fast as the 4th on C4 in my "approximate but I think good enough" method.

Kees

Here is the result (for 4:2 octaves everywhere) of my method versus what tunelab computes. I use precisely the same inharmonicity model for my piano in both cases.


Kees

And here the tunelab tuning curve for 6:3 bass 4:1 treble octaves versus a custom offset (red) I computed as above and imported in tunelab, which uses mindless octaves from A3 on. (Normally this is not possible in tunelab.) It gets a nice boost in the upper octave.



Kees

Kees:

I was looking for someone that knows more than I do and that is what I got – Thanks!

To mathematically stretch ET is not as straight forward as it might seem. Take a stretch defined as A0-A1 as a beatless 6:3 octave and C7-C8 as a beatless 2:1 octave. Six different possibilities come to my mind, but there are many, many more. The beat rate for the resulting C7-C8 6:3 octave could be determined and then each octave could be defined by an interpolated 6:3 beatrate. But the interpolation could be linear or logarithmic. Or these two variants could be used with the 2:1 octave beatrate or the 4:2 octave beatrate. These would not result in the same tuning curve.

And I am not sure how to define ET with inharmonic tones. So I don’t know how to judge the +/- 0.01 cent accuracy I mentioned earlier. An obvious goal would be that the beat rate of all intervals be progressive. But what about a piano with jumps in iH? Only some intervals can be progressive. And can you call the beatrate of an interval that beats faster, then slower, then faster again but then as a narrow interval instead of a wide one as progressive like some SBIs do depending on the stretch? And why base the calculations of a tuning on the first partial? With jumps in iH, the 3rd or 4th partial may be a better choice. Or perhaps the theoretical fundamental rather than the first partial would be more mathematically elegant. I even wonder about using fractional and negative partials with all notes having a deviation of 0 cents, but the partial number defines the curve.

But really these are muses. My limited mathematical abilities force me to be more practical. My background is in marine navigation and cargo operations. I understand the concept of solving multiple equations, but I am more used to making estimates and then refining them. I cannot say that this method is better, just that is what I am able to do.

Here is something that I just tried that resulted in progressive 4ths and 5ths. This should give you an idea of how I look at the problem. I used a simulator that I programmed using VBA with Microsoft’s Access database. By entering either cents deviation or the frequency of the first partial the beatrates of the intervals are calculated. The iH values can be changed and there are various graphs that can be displayed. Some beatrate ratios such as CM3s and M6-M3 test are also available.

I had been thinking about the octave ratio and how it can also be defined as the algebraic difference in cents. And how the tuning curve is 87 segments and wondered if the octave ratios could be used to estimate and refine the semitone ratios, which can also be defined as the algebraic difference in cents.

Rather than spend the time to write the code, I just used a calculator and the simulator. Staying with beatless 4:2 octaves, I determined the algebraic difference in cent deviation of the octaves encompassed by A2 and A4, using an artificially ideal iH curve for a studio upright with the lowest value at E3. As a first estimate for A3-A#3 the deviation difference for the octaves A3-A4 and A#2-A#3 were added together and divided by 24. The idea being that the semitone deviation would be very close to the average octave deviations. This was done for each semitone from A3 thru G#4. These differences were added together giving a total of 1.70 cents, but the A3-A4 octave had a difference of 1.43 cents. This is a ratio of 0.84. So each individual first estimated semitone ratio was multiplied by 0.84 to give a refined second estimate. Starting with A440, the second estimated semitone ratios were applied to the notes from A#3 thru G#4. Then A2 thru G#3 were calculated as beatless 4:2 octaves to the temperament octave of A3-A4. As I said before, the 4ths and 5ths were progressive and I am not sure how else to judge the accuracy but by the slowest beating intervals. Oh, the 6 twelfths beat slower than the fifths and were also progressive.

As I said in a previous post your method is refreshingly different. It will take me a while longer to understand it more. And there is also the question of how to estimate and then refine a stretch involving not just octaves, but double octaves and 12ths and whatnot also.

[EDIT:] I checked the beat rate ratios and they wre good, but not perfect. The M6-M3 test showed 1.01/1 for all but two which were 1/1. (At least one regular poster believes this test is not trustworthy.) The various 4ths and 5ths and octave ratios had some waver to them also. But then everything was being rounded to the nearest 0.01 hz, cents or bps and I expect that there would be high and low points in the SBI ratios. There were not enough notes to see if there was an actual wave in the curve. I will have to write some code sometime to see if two estimates are enough. I wonder what all could be used for further refinement?

Jeff:

I like the method I explained because it is flexible in what you put in as absolute (or very accurate) constraints, namely the octaves, and then you find the tuning curve that satisfies the constraints and which maximizes smoothness. To use other octave types is exactly the same, I already did that (see the plot with mindless (equal beating double octave and octave and fifths). Regarding the math all it involves is some linear matrix algebra.

Another approach I came up with (perhaps closer to how you are thinking) goes like this.

First tune F3F4 as 4:2 octave. So their frequency ratio is

F4/F3= p4(F3)/p2(F4)

where p4(F3) is the 4th partial multiplier (4 + something) of F3 and p2(F4) the 2nd partial multiplier of F4 (2 + something).

To tune the other 11 notes between F3 and F4 proceed as follows:

First temporarily tune F#3 as a 4:2 octave to F3 (you probably don't want to do this in a real piano). You end up with F#3 = p4(F#3)/p2(F4) * F3. Then take 1/12th of this interval and end up with F#3 = (p4(F#3)/p2(F4)^(1/12) * F3.
Do the same for G3, but then take 2/12'th of the resulting octave, so you get
G3=(p4(G3)/p2(F4)^(2/12) *F3. And similar up to
E4 = (p4(E4)/p2(F4)^(11/12) *F3.

This will stretch the semitones progressively over F3-F4. A similar construction over 2 octaves starting from a 4:1 octave is possible.

Kees

To put calculations up to 0.01 cent in perspective, here are the iH values on the range F3-A4 (numbered 1-17) on my piano.
One may question the validity of fitting this with a smooth iH curve and computing a tuning from that. On the other hand maybe that is precisely what one should do to hide this imperfect scaling.

For tunings like EBVT (actually I don't know of any other tuning that is like it), where specific intervals are supposed to be equal beating no ETD will achieve this which such a scaling unless you really measure every note and don't fit the measured iH values to a smooth curve. I have tried using these iH values to compute the theoretical pitches of EBVT and all the beat rates come out perfect when I then tune with tunelab with custom offsets for every note. When I just use the cent offsets from Bill's page computed ignoring iH and machine tune EBVT the result is very poor in the sense that supposedly equal beating intervals are nowhere near equal beating.



Kees

Kees:

Yes, solving multiple equations would be more accurate, but still beyond me right now. You posted:

”Another approach I came up with (perhaps closer to how you are thinking) goes like this.

First tune F3F4 as 4:2 octave. So their frequency ratio is

F4/F3= p4(F3)/p2(F4)

where p4(F3) is the 4th partial multiplier (4 + something) of F3 and p2(F4) the 2nd partial multiplier of F4 (2 + something).

To tune the other 11 notes between F3 and F4 proceed as follows:

First temporarily tune F#3 as a 4:2 octave to F3 (you probably don't want to do this in a real piano). You end up with F#3 = p4(F#3)/p2(F4) * F3. Then take 1/12th of this interval and end up with F#3 = (p4(F#3)/p2(F4)^(1/12) * F3.
Do the same for G3, but then take 2/12'th of the resulting octave, so you get
G3=(p4(G3)/p2(F4)^(2/12) *F3. And similar up to
E4 = (p4(E4)/p2(F4)^(11/12) *F3.”

It seems that this method substitutes the iH of the note you are tuning for the iH of the lower note in the reference octave in order to have the semitone ratios describe a curve. But why would this be the correct curve? Have you tried it to determine the accuracy?

I tried another variant of my latest method with even better results. Rather than average just the end-most octave ratios that encompass a semitone, I averaged all of them. The result was a correction multiplier of 0.91 instead of 0.84. This time all the M6-M3 tests resulted in a 1.01/1 ratio. But then again, the additional accuracy may have been due to an additional decimal place. There was still some waver in the SBI ratios, though. Again there may be a need for additional decimal places.

I am afraid that I have little interest in UTs. They do make me wonder about octave stretch, though. If an UT is to have a certain character, what is the best way to keep that character with the effects of iH and stretch? Everything seems to point in the direction of ET.

Your example of the problem of using a smooth iH curve and trying to tune an UT makes me think that the ETD manufacturers are very correct in stating that they are a tuning aid. The ears must be the final arbiter.

Jeff:

I finally understand your method. I believe you that it produces very accurate results. A theoretical weakness is I think that the equal adjustments of all semitones in the correction sweep is somewhat arbitrary, but from a practical point of view it is probably much more accurate than can be tuned. And it makes total sense to me to use the octaves around a semitone as reference.

Kees

Kess:

Thanks for the reply.

I have had the same concern about accuracy. I expect that there would be a ripple or seam when comparing the semi-tone ratios of G#3-A3 and A3-A#3. But since the result is progressive SBIs, it is indeed more accurate than I can tune.

Heck, now what can we talk about?

Here's an idea: What is the typical octave type that is required for mindless octaves? I have found that 4:2 works well, though some say that a compromise between 4:2 and 6:3 is needed. Do you have an opinion?

It's interesting to discuss what the goals are for a tuning curve: what makes a good one? Rick Baldassin's "On Pitch" is an excellent resource for this discussion. He takes into account not only which partials are present in various parts of the compass, but also the relative volume of each one.

The TuneLab documentation refers to a goal of having the deviation curve (the bottom part of the tuning curve editor) as flat as possible. This curve shows how TL "feathers" the pitch to smooth the transition from the bass octave style to the treble octave style, by plotting the difference in pitch from a beatless octave (BTW, what do you mean by a "mindless" octave?).

In The Real World the other day, I measured iH for a new customer's upright and looked at the tuning curve. My normal 6:3 bass, 4:2 treble had deviations of more than five cents in the middle. This was a console, so I tried 4:2 in the bass. Sure enough, the deviation curve went flat! I tuned up a few octaves from the bass, and did some aural tests along the way. By the time I got to C3, the C2:C3 octave sounded horrible! Sure enough, the 4:2 was pure, but the 6:3 partials were much louder, and were beating furiously. I redid the curve to 6:3 and retuned what I had done so far. So "flat deviation curve" is not always a good goal (but a maximum deviation of five cents in the middle is almost always better than a deviation bigger than that; I've never had, say, ten cents deviation).

So, back to goals: what makes a good tuning curve?

As an aural tuner, my first goal is to make the whole piano sound musical. By this I mean simply that the pianist (not my ears) should not find any unison, octave, or any other interval that sticks out obnoxiously. Admittedly this is a modest goal, much like setting a goal for washing a car so that there are no obvious clumps of bird dirt, but you'd be surprised how hard this is on a poorly-scaled spinet. Fortunately, you have the relative volume of the various partials to work with. It's impossible to have an octave where both 4:2 and 6:3 are beatless, but you can pick the one that sounds the best, or even fudge a bit between the two.

BTW, you might be as amazed as I am that people tune pianos every day with the SAT by measuring iH on just three notes: F2, A4, and C6. I measure A1, A2, A3, A4, A5, and A6 when I use TL, an approach that intentionally skips the highest wound strings where iH is often way off from the rest. It's instructive to measure iH on the highest wound string and the plain string next to it, to see how big a jump there is on many common pianos (U1, P22, spinets). This is why Robert created the "split-scale" mode.

--Cy--

Hi Cy:

“Mindless octaves” is a Bill Bremmer term for tuning double octaves to beat wide at the same speed as 12ths beat narrow.

You posted:

”As an aural tuner, my first goal is to make the whole piano sound musical. By this I mean simply that the pianist (not my ears) should not find any unison, octave, or any other interval that sticks out obnoxiously. Admittedly this is a modest goal, much like setting a goal for washing a car so that there are no obvious clumps of bird dirt, but you'd be surprised how hard this is on a poorly-scaled spinet. Fortunately, you have the relative volume of the various partials to work with. It's impossible to have an octave where both 4:2 and 6:3 are beatless, but you can pick the one that sounds the best, or even fudge a bit between the two.”

My latest solution is to tune to two other notes instead of just one. For most of the piano it means using a 12ths spanner and including the 5th down from the top. It works especially well on the bass of spinets.

So a flat deviation curve in Tunelab would mean that the beatrate of 2:1 octave partial match would double each octave? I am not saying that they shouldn’t, just trying to understand what it would mean.