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.

Yes, I calculated that. MO (applied above F5) lies between 4:1 and 4:2 octaves. For low inharmonicity MO is closer to 4:2 than to 4:1, for high iH it's closer to 4:1.

4:2/6:3 mix gives even more stretch than 4:2.

Kees

Here's the iH tunelab fitted iH curve and the measured data for my Heintzman upright. The jump you mentioned is very visible!



Kees

Well what do you know - I've been away for a week or so, and this is turning into the NASA of tuning already. Should have known better than to hook the two of you up in a dedicated thread

This looks great, guys! Needless to say I need some time to catch up. Keep the good things coming, I will be back when I've gotten a grip of your earlier conversation.

No. First, if you match 2:1 partials in an octave, this means by definition the beat rate of those partials is zero -- beatless.

A flat deviation curve means that TuneLab is giving you beatless octaves all the way across, for whatever octave styles (4:2, 6:3) you have selected. The deviation is deviation from beatless.

--Cy--

Cy:

I don't quite follow you, but that's OK. If I get into ETD tuning I will learn it then.

Ahhh, glad to see that you have joined the rest of the world and are graphing iH on log paper.

From what I hear in the beatrates across the break, your iH curve is typical of big old uprights. I like to see how many semitones the beatrate of RBIs jump across the break. The largest I have seen was on a Keystone. I think it was 7 semitones. In other words B2-D#3 beat as fast as F#3-A#3. I am pretty sure the break was at B2-C3.

One of the most bizarre ones is the “K” console. The bottom two notes above the treble break are wound bichords and the next two are unwound bichords. I finally figured out that the unwound ones have lower iH and the wound ones have higher when I had expected the opposite. I am thinking that some “K&_” consoles have a similar jump. Just because a string is wound doesn’t mean it has lower iH.

Interestingly, one of the factors of higher iH in a wound string is the "bare length": the amount of core wire left unwrapped in the speaking length.

--Cy--

P.S. Pardon me for restating the obvious, OK? Let's say that you're tuning A3 to A4 (already at 440), and because of iH, the second partial of A3 is at 443 Hz when its fundamental is at 220. A pure, beatless 2:1 octave is obtained when A3 is flattened so that its second partial is at 440 exactly (maybe 218). So there is no "beat rate" in a pure octave. TuneLab would show the deviation curve for this note at zero.

Now, lets say that to smooth the overall tuning curve, either because of a jump in iH, or a change in octave styles between bass and treble, that TuneLab calculates a target frequency for A3's fundamental to be 217 Hz instead of 218, and (just to keep the numbers easy) this results in A3's second partial sounding at 439 Hz. There would then be a one-Hz difference between A3's partial 2 at 439 and A4's fundamental at 440, and you would hear one beat per second. TuneLab would plot this deviation as about four cents from beatless (you can also switch it to plot beats instead of cents).

OK... But you said, "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!" So... if the deviation curve is flat when 2:1 octaves are beatless, and octaves cannot be beatless at the 2:1 match when they are beatless at the 4:2 match, then how could be there be a flat deviation curve with 4:2 octaves??? That is why I was thinking that a flat deviation curve of the cents from 2:1 octaves could be, say, 3 cents deviation throughout much of the piano, be a flat curve (with a value of 3 cents), and beatless 4:2 octaves. The proof of this being true would be a 2:1 octave beatrate that doubles every octave. Pardon me for stating the unobvious.

Same here, break is 7 semitones if you tune according to tunelab with a smooth tuning curve. Bottom 4 notes (A#2-C#3) on the treble bridge are wound bichords and they have unusually low iH. The unwound tri's D4-E4 have unusually high iH. Almost as if someone expected the wound ones to have high iH and tried to smooth things out but overcompensated.

I wonder how valid it is to use just a single iH "constant" per string. In fact if I use the iH measurement of 2 notes to compute an exact 4:2 octave, then tune them with tunelab near the break the result is NOT a 4:2 octave. Of course in a real string the shift in partials is not exactly the iH constant times the square of the partial number (or some averaged correct to that that tunelab uses), and you should really work with 5 different iH curves for partials 2-6.

So to really tune accurately with an ETD you should first measure all the partials for every string before tuning, which probably takes longer than an accomplished aural tuner takes.

Kees

Kees:

Just got done replacing a combustion air actuator on a regenerative thermal oxidizer out in the sun so I am going to write briefly.

Yes, yes, yes.

I was given a nice selection of tuning files from a Verituner that showed the cents deviation for a wide number of partials. The wound strings are always pretty wacko. If certain partials were used, negative iH would be the result. And I do not ever remember tuning a piano where something did not add up somewhere. One test would show a bass note to be too low and another would show it to be too high. This would be after checking the notes that where involved in the tests.

Interesting that your piano has the same 7 semitone RBI beatrate jump. Imagine trying to tune CM3s across the break, or even just above the break.

And thanks for the verification on octave types for mindless octaves.

Have a good weekend. I am taking the four year old fishing!

Hi, Jeff,

There are two other factors thrown into the mix: the iH measurements of this particular piano, and Robert Scott's algorithm for taking those measurements and calculating a tuning curve.

6:3 octaves are wider than 4:2, and the higher the iH, the bigger the gap between the two. Since this was a high iH piano, smoothing the transition from 6:3 to 4:2 had to cover a big pitch change. That's why the deviation from beatless was so high in this case, and went to zero when I kept both bass and treble at 4:2.

--Cy--

P.S. Please help me understand what a 2:1 octave beatrate is, and how you calculate it. My 2:1 octaves don't beat at all (that rate does double every octave, though! Triples, too! :-)

Cy:
The 2:1 octave beat rate of, say, A3 A4 is the difference in the frequency of the 2nd partial of A3 and the 1st of A4. If A3A4 is not a 2:1 octave it will be nonzero.

In tunelab you can select different octaves for bass and treble. At the extremes what you choose is what you get, in between you get something in-between.

So if you select bass and treble octave types the same the deviation curve will always be 0, since you just get that.
If you select 6:3 for the bass and 4:2 for the treble (for example), A0 is a 6:3 octave and C8 a 4:2 octave, and the other notes are something in-between. So at for example C4 you will have roughly an equal beating 4:2 and 6:3 mix, and the deviation curve just shows this (in beats or cents).

If you select a tuning curve based on 6:3 bass 4:2 treble (for example), you can then also select 2:1 and 2:1 for both. Then, unless you press auto adjust, it will show the beat rates of the 2:1 octave for this particular tuning curve (based on 6:3/4:2).

Kees

Uhhh, since you understand the relationship between 6:3 and 4:2 octaves, you should understand that the same is true between 4:2 and 2:1 octaves (or any two different partial matches of any interval). When you say that your 2:1 octaves don’t beat at all, do you mean that you do not hear them beating, or the ETD shows that they are beatless?

There is an odd phenomenon that when an octave is tuned so that the 6:3 partial match beats narrow at the same speed as the 4:2 partial match beats wide, then the 2:1 partial match also beats wide at the same speed. Maybe that is why you don’t hear it?

As far as calculating the 2:1 beatrate, I am not sure where you want to start from. Do you mean “How do I determine the 2:1 octave beatrate when A3 has an iH of x and a fundamental frequency of y?” Or do you mean something else? And why do you want to know, so I can try to give a decent explanation. If it is an ETD display question I cannot help you.

Thanks for trying to clarify what you mean. When I tune a pure 2:1 octave, the second partial of the lower note is beatless with the fundamental of the upper.

Of course, other coincident partials may be beating, but when you describe the beat rate of a 2:1 octave doubling every octave, it sounds like you're not talking about a pure, beatless 2:1 octave.

Help!

--Cy--

He's not referring to a pure, beatless 2:1 octave. He is talking about the beat rate of the 2:1 partial match of the octave tuned in whatever way. 2:1, 4:2, and 6:3 are used here to identify the partials being referred to, rather than the partials to be tuned pure. So "the beat rate of the 2:1 partial match doubles every octave" is the meaning here.

Thanks, Phil; that clarifies it.

--Cy--

[exit lurkmode]

Kees referred to a 4:1 octave. Can one of you explain to me what this is, i.e. which partials of the two notes in the octave are being referred to? Does 4:1 not refer to a double octave? Why then does Kees call it a 4:1 octave?

[re-enter lurkmode]

The notation "4:1" means that the fourth partial of the lower note is tuned to match the first (fundamental) of the upper. You're right on that this is never, to my knowledge, a tuning style used for a single octave. The stretch would be enormous, since the higher the partial number, the greater the inharmonicity.

Rick Baldassin's book "On Pitch" is an excellent introduction to, and reference for, the bridging of ETD tuning and aural.

--Cy--

Cy:

What got me to wondering if your 2:1 octaves were doubling in beatrate every octave, is that is what would happen if they were all, say, 3 cents wide of beatless. I was just trying to figure out what you meant by a flat deviation curve. Not that the deviation curve was necessarily zero, just flat. Wish I had never started on this tangent. It seems to be an ETD thing.

Yes, tuning a single octave 4:1 would result in a stretch of about 1200 cent. Probably a bit too much.

When I say "4:1 octave" or even "3:1 octave" I just mean tune the upper note to the note 2 octaves below or octave and fifth below. I apologize for my sloppy terminology.

Kees

It's usually said "4:1 double octave" and "3:1 12th" (or octave fifth).

Thanks, all, for the clarifications. It's worth taking time to understand words, especially as we get into esoterics!

--Cy--

Please, post it. I find this very interesting.

DoelKees in Chas thread (May 25, 2010):..."The thing to optimize is the smoothness of the tuning curve and the constraint is whatever octaves you decide on. To understand the solution you need calculus and matrix algebra. If you have that background (I do this kinda stuff for a living, easy for me) it's straightforward. Perhaps this goes to far for this forum and certainly this thread, though it would be educational for Alfredo. You can email me for details if you want, or start another thread if you think there are other math heads besides us. More important that the details is I think the message that any such "theory" just formalized and makes precise the actual best tuning practices. There can be no such thing as a theory that tells you how to tune, unless you tell it exactly what you want."...

DoelKees in this thread (June 04, 2010):..."I wonder how valid it is to use just a single iH "constant" per string. In fact if I use the iH measurement of 2 notes to compute an exact 4:2 octave, then tune them with tunelab near the break the result is NOT a 4:2 octave. Of course in a real string the shift in partials is not exactly the iH constant times the square of the partial number (or some averaged correct to that that tunelab uses), and you should really work with 5 different iH curves for partials 2-6."...

Other educational conclusions?

Regards, a.c.

Welcome back Alfredo, hope you had a nice trip.

Jeff and I believe we have solved all mathematical tuning problems, hence no more activity in this thread.

If you have some ideas to contribute, or disagree with anything we wrote, feel free.

Cheers,
Kees

Hahahahahahaha!

If that were true then tackling the real problem of the world should be easy: Pride.

Jeff, this is more physics than math but maybe appropriate in this thread. I am thinking about using the 'smooth' versus the 'jerky' style of using a tuning hammer. The explanation I've heard for using the latter is that you don't twist the pin but move it 'at once'.

Now thinking about the physics this explanation makes no sense to me. If you would view the 'jerk' in slow motion would it not be just the same as a smooth move, hence the same amount of twist in the pin? Also the hammer is connected to the pin in the same way so why should it move 'all at once' when you jerk it, but twist when you pull it?

Have you ever thought this through?

Kees

Kees:

This would probably be a good subject for a new Topic. My short answer is the difference is due to the break-free friction and residual torque in the pin.

Once the pin starts turning throughout its length, the friction is reduced and it takes less torque to turn it. If the pinblock is very tight and a smooth pull is used, there is a lot of twist still in the pin when the pin breaks free, friction drops, the torque in the pin is sufficient to move the pin, and the pin moves too far. But when a jerk is used, often a series of more and more forceful jerks until one is made that just barely is forceful enough to break the pin free, the residual twist is released at the same time as the pin breaks free and it does not move as far.

For the best control with very tight pin blocks I prefer a heavy hammer. Mine weighs over 2 lb.

My long answer would include flagpoling, string rendering, the final state of the torque within the pin, and the phases of the moon.

My very shortest answer is: whatever works.

Jeff, a short answer is not the point, actually for learners can be misleading (a new Topic?). "Whatever works" does not relate to short time nor long time effects, and "tuning things" should work in the most advisable/convenient/correct way.

Regards, a.c.
.

Ok, my next to very shortest answer is: There is a time and a place for everthing.

I am interested in the details if you want to start a topic. As I've mentioned I have some ideas about the physics, but have only thought about them for a few days and would be happy to hear what you have come up with over the years.

I should perhaps mention I have a PhD in physics and have a hard time understanding this. It's quite complicated!

Cheers,
Kees

Here's my take:

I don't think the two are the same, especially if there is a large difference between the static and dynamic friction of the pin in the block - which I'm pretty sure there is. If the sudden movement is fast enough, the pin moves in dynamic friction, i.e. with less friction, hence less twist on the pin. If the movement is a slow pull, the pin moves on the cross-over point between static and dynamic friction, where friction is typically the highest. More twist on the pin. (In fact, close to maximum twist.)

I would think that the same goes for all contact points in the string.

Think of a tire on a road surface, when braking the car: the best braking happens when the tire just starts to skid, i.e. the cross-over from static to dynamic friction (ABS tries to keep the brakes on this point). Once the tire skids, the braking deteriorates because friction is lower.

Other than friction, one may also want to consider inertia, but I doubt that it would play a serious role, given the relatively small masses of tuning pins and strings?

Also, could a sudden movement cause a longitudinal pulse along the string, that helps to settle the string on the bearing points? By way of analogy, high speed crash videos reveal details that are clearly different from a simple "slow crush". One can see, for example, vibrations moving through the chassis of the car.

Just my 2 cents, from the vantage point of a Chemistry PhD with only two university years in Physics... (Hope it doesn't bother you that I'm posting here.)

Hello Mark.

..."If the sudden movement is fast enough, the pin moves in dynamic friction, i.e. with less friction, hence less twist on the pin."...

I agree, a sudden jerk leaves less time to the pin for twisting.

..."If the movement is a slow pull, the pin moves on the cross-over point between static and dynamic friction, where friction is typically the highest. More twist on the pin. (In fact, close to maximum twist.)"...

If it will be maximum twist or other twist degree will depend on how slow the pull is. In a way, this reminds me of Non-Newtonian fluid.

..."I would think that the same goes for all contact points in the string."...

Sudden jerks, like slow pullings, will also effect the distribution of tension all along the string. And that tension should be well distributed amongst the three lengths of the string.

Regards, a.c.
.

Physics aside, why would you want someone else to start a Topic that you are interested in?

If you think of some questions and post them, a much better discussion could result than if I posted some kind of article or lecture or sermon or something.

This reminds me of the scene in One Flew Over the Cuckoo’s Nest where the psychiatrist is interviewing the patient played by Jack Nicholson. It goes like this (No offense meant, just remembering something funny):

Young Psychiatrist: Have you ever heard of the old saying "a rolling stone gathers no moss?"

McMurphy: Yeah.

Young Psychiatrist: Does that mean something to you?

McMurphy: Uh...it's the same as "don't wash your dirty underwear in public."

Young Psychiatrist: I'm not sure I understand what you mean.

McMurphy: I'm smarter than him, ain't I?

Kees,

I saw that you would be willing to share your MATLAB code for calculating tuning curves? I would be interested in taking a look if you don't mind. I am trying to put together a Max/MSP patch for myself for use while tuning.

Hopefully I'll be able to make some final adjustments to the code with Bill next week, after which I'll clean up the code and release it. I'll post that event here. Probably end of July since I'm traveling till then.

How could a Max/MSP patch help with tuning?

Kees

Sounds great!

Oh, Max has all those great DSP objects. With a pitch tracker, spectroscope, fft's, and some inharmonicity external that i'll have to write, it could make a pretty good etd. i'd probably incorporate more constraints than just 4:2 octaves, ideally for all audible coincident partials..... but that may require more computing power than would be practical.

i look forward to your code.

That's a very cool idea. Computing power won't be a problem. I've been thinking we should have an open source free ETD. I guess it could be built on top of pd. Are max/msp and pd compatible at all these days? I haven't worked with those languages for over a decade.

I now realize you asked for the tuning curve code, not the EBVT code. That is ready. I'll post it in a few days.

Kees

I would love to see a good open source ETD, and I have been formulating a plan to create one. Max/msp is closed source so that would not be the way to go for a general release, it's just what I was using for myself. It should be pretty easy in Pd, though. They even have Pd for smartphones and such. It would be way cool.

Kees, let me double-check that I understood you right. A 4:2 temperament octave will give you MO's above F5 that falls narrow of 4:2's, right? (And wide of 4:1's/ narrow of 3:1's, by the very idea of mindless octaves.)



How stretched? wide of 4:2?

There is another thing that differs too... no matter if it looks the same in slow motion, the way you pull (or press) the hammer is very different.

When you jerk the hammer, there is most pressure on the hammer when it starts to move, right? When you pull slowly, there is (and this is only an empirical reporting) at least as much pressure just before you release the pull as there is when you start the motion, if not more.

So i guess that if you jerk the hammer, you kick-start the tuning pin in the direction you want it to go, then it finds its own place to settle, whereas when you pull/press slowly, it will most likely find its own place in a different position from where you left it.

Before jerking , I find a slow pull necessary in any case.

slow pull will allow to charge the pin more. and give a more precise control on the exact twist .

Jerking, dEspite the good sensations it provide, oblige you to jUmp the pin in place, consistency is lower. But it is a fastest technique indeed

O.T.

Jerking vs Steady Pull:

I like the analogy posted by Cy Shuster in a thread about impact hammers:



Can someone tell me what is this dynamic vs static friction difference?

When I studied physics there was only friction.

Anyway I think of it as a question of response time. In a steady pull you must release the pull once you feel the pin starts to move, if you are not fast enough in your reaction then the pin turns too much.

In the Jerking method when the pin starts to move you have already released the hammer, so the minimum possible movement is achieved.

Also you have to make less effort in the Jerking because you take advantage of the inertia of your body mass (hammer included).

Less control on the bottom of the pin position, while you put it where you want with the slow pull.
but the way most of you talk of pin setting makes me understand you see the things differently than what I ve been told.
No one dEal with the pin constrain for instance, hoping that the pin is not constrained is unrealistic.

and no one seem to settle the pin out of hard blows, which are not giving any control on the pin position.

To build the tone, one need the utmost control, most of those methods count on the coupling of the string to have the unison, once in the zone the unison build by itself, whatever the position of the pin is. a good unison should last +- forever.

There is probably no end to analogies regarding hammer technique. At some point they are all inadequate, but they can help give some idea what is happening.

Here’s one more. You want to put exactly $20 of gas in your car. If while you are filling you can slow down the rate to a crawl, you can stop at exactly $20.00. If you cannot, then you must stop before $20 and then give the handle quick squeezes. If each squeeze is more than one cent (this analogy may be better than I thought…) it may not be possible to stop at exactly $20.00.

Now this does not take into account overshoot, which can be necessary for either smooth pull or jerking, but is always less, in my experience, with jerking. I use both techniques depending on the piano. If the piano allows a smooth pull, I can leave a better, more stable tuning. If the piano does not allow a smooth pull I prefer a very tight block so that I can bury some pin twist in the block without breaking the pin completely free. I think this may be what Bill was mentioning when talking about “massaging” the pin on GPMs piano.

Let me differ slightly : in the slow pull the pin begin to move so slowly you can really turn it where you want, because when it begin to move it is really free and easy. What may be less is the string indeed, but in that case jerking does not help much.

Yes overpull is way larger with a slOw pull, but for some reason the precision of the tactile feedback is better, probably because of that larger rotation of the hammer.

Jerkin mean using the pin as a spring that jump from place to place (and crack audibly most often) the pin is left mo free, as you say. I noticed that often with the Verituner, the pitch can lower evenly by a very little amount when the piano is played. I dont recall the amount, was less than a cent. Id like to compare the 2 settings evolving with an accurate tool.

In any case after jerking, the last move is to charge the pin . then some downward pressure on the hammer is necessary to spring the pin and make the "knot" . Probably less good for the block than the straightforward motion of the slow pull.
an the



s

If the position differs that is only because the metal need time to stabilize. I see that as the reason why this tuning is slower, you enter in a different time "dimension" , but that control is so agreable it is worth the experiment.

I use jerking for corrections on a well loaded system jerkin mean braking the pin while adding stress until you release the brake, so you can keep the pin "charged" and limit the move.

Kamin,

You have several times explained what you mean by the pin being "charged".

But I've not undestood the idea. Can you please explain that, once again?

I was taught to try to let the pin in a stable state, with almost no torsion in it, only the amount necessary to resist the pull of the string. Is that little amount of torsion what you name "charged"?

Yes, and the charge can be more or less high depending on how deep the twist take place.
With a high charge, the pin is more active in transmitting strings vibes within the block ; Gives a definitive raise in power and a cleaning of the tone

I also have used minimal twisting for a long time, and noticed that the pin sometime get firmer after the piano have been played.

With that low grip the pin get stiff, very easy to compare the quality of the setting. it gives the sensation that the pin is in a harder metal, but also, it is difficult to move the pin, even to raise pitch, the pin does not want to turn in no direction (just forcing the point !)

Let us know if you experiment please !

Yes.

Yes. 4:2/6:3 is wider than 4:2 (barring extremely non-Youngian strings).

Kees

Yes, but my question was vague, and I'm not sure what you answered upon, so I'll rephrase: Would a 4:2/6:3 stretch of the temperament give octaves wide of 4:2 from the top note down in mindless octaves?

Aye, there's the rub!!!

I haven't computed exactly that, by I am pretty sure they will be narrow from 4:2.

Kees

Awww, I'll jump in then. It depends on the piano. I believe it would create narrow 4:2 octaves except perhaps on the largest pianos. But what do you hear Pat? Best to try it on a number of notes. No temperament is perfect and this is a fine distinction. Like, how do you make a 4:2 beat exactly the same as a 6:3? And how do you make a 12th beat exactly the same as a 15th? You can only get an idea (aurally) by listening to a number of examples.

Good question, Jeff, I will make the tests and get back with my findings!

But what you and Kees say is that a 4:2 temperament octave, and even a 4:2/6:3, would produce narrow 4:2's just one octave higher? Why then do we need a temperament octave of that wideness, if it all will be compressed just outside our temperament?

I did the calculation and MO produces essentially 4:2 octaves on a Steinway D, whether the midrange is tuned with 4:2, 4:2/6:3 or even 6:3.

On my Heintzmann upright MO is always narrow compared to 4:2.
For C8 the difference is stretch between 4:2 and MO is 12cent. Again it makes no noticeable difference how the midrange octaves are tuned.

These are of course just simulations, let me know what you observe aurally.

Kees

I have been busy with work and have not been on PW for awhile.

I can't offer any experience with the math, and tuning by ear, however, the posts here about hammer technique are something I would like to contribute to and explore further.

When Bill was here the last time, he had to spend a great deal of time taming those 5th-6th octaves. He had some suggestions for me regarding how I can get a better, more stable tuning, especially in those areas. Here are his suggestions along with some constructive and informative advice about why strings go slightly flat etc.

In the past, I have read differing opinions on whether one comes from the flat side or the sharp side when setting the pitch, depending on how the piano reacts. A few days ago, I tried coming from the flat side on a few of those pesky notes, opposite of what I had been doing It's the first time I had some stability! I pounded the string very hard from the flat side, and it was stable! Looking forward to trying this and Bill's suggestions on the whole piano!

Bill's advice below is for the RCT...also, this new Plateau tuning hammer from Joe Goss is great! www.mothergoosetools.com/ I have a much better feel for the pins and much better control.


1. Use the "Smart Tune" mode first in the low bass, low tenor and high tenor. Follow the first pass in smart mode by a another pass in "Fine Tune". The latter should reveal nearly every note already perfect.

2. In the treble, add +2% to the "Smart Mode". In the high treble, cancel the "+2%".

3. In both the treble and high treble, use the mute strip or tune just the left string through the section first with the device offset +0.5 cents in the fine tune mode. Then, as you tune the unisons, cancel the +0.5 cents. You will see each pitch be very slightly sharp at first but as you tune out each unison, you will see it "sink" to exactly on pitch in the fine tuning mode. If the entire unison still reads a tiny amount sharp, don't worry about that but it should never read as any amount flat.


That tiny amount of sharp tuning of the first string is necessary because of two important factors: 1. There is always a tendency to "pound" the strings a bit flat from where they were initially set. 2. There is a curious phenomenon about unisons in that area of the piano. A single string will read slightly sharper than three strings tuned as a beatless unison.

The latter has been discussed and written about extensively by Virgil Smith RPT who was strictly an aural tuner. He described his method of making the unison settle in properly as "cracking the unison". By this, he meant that the first string had to be tuned very slightly sharp of where it is intended to be in order for the unison to settle exactly to where it should be when the other two strings are tuned to it.

Long ago, I identified that "slightly sharp" amount as being 1/2 cent. I most often use a muting strip, as you know. I found that if I tuned the middle strings in the treble and high treble with a +0.5 offset first, I could get the whole unisons to settle nicely to the desired pitch. If I didn't do that, I could get perfect unisons but they would all sag just a tiny amount flat. If, upon seeing that a whole unison read flat, I put a mute in and played one string, it was right on pitch but all three strings together would read flat.

This is a very fine point in tuning but very important in producing a broadcast quality sound. Believe me, I struggled every inch of the way with that every time I tuned your treble and high treble. Offsetting the pitch first at +0.5 cents for the first string, then canceling the offset, tuning the other two strings to that initially slightly sharper pitch is the "smart" way to address that problem.

You will always on occasion find notes that slip down further as you try to tune the whole unison. That would be a stability issue and not related to the "coupled motion of strings effect" which was identified by Virgil Smith. On your piano, the G6 (I think it was) at the end of the treble section was always a problem. For that note, try setting the first string 1 or 2 whole cents sharp first. Otherwise, you may find that each time you try to tune that unison, it will always end up flat.

Pat:

I think the three of us are looking at this from different angles. I think you are looking at it as MOs always being tuned. I am looking at it as what is the result when 4:2 octaves are tuned, and Kees is looking at what happens when MOs are applied after tuning the midsection to a certain octave type or compromise.

So your first question is: “….. a 4:2 temperament octave, and even a 4:2/6:3, would produce narrow 4:2's just one octave higher?” I assume that you are referring to a case where the temperament is extended with the same octave type and then the first double octave is tuned wide of just and beat the same as the narrow 12th with a common note on top. Usually, yes. And it is more likely with 4:2/6:3 because this is a wider octave type. So when the first MO is tuned, it is tuned at a stretch less than what has been tuned so far resulting in the top octave (that have been tuned so far) being narrower than the prevailing stretch and is narrower than the prevailing octave type.

And your second question is: ” Why then do we need a temperament octave of that wideness, if it all will be compressed just outside our temperament? I do not think we need an octave type wider than 4:2 to tune MOs. Others think you do. Even others may desire a wider midsection, which is fine. And if they desire other sections to be a certain width, that is fine, too.

I suspect that some start with 4:2/6:3 octaves and when they reach the first 15th are tuning the resulting 12th wide and only think they are tuning MOs. The beat of the 12ths and 15ths are slow and they seem to beat about the same speed even though they are both wide. The RBI tests are a bit tedious and may not be given much attention. This is the only explanation I can come up with when I read posts about tuning octaves with a bit of a beat, MOs and the fifths become slower and eventually pure or wide.

When I tune 4:2 octaves, I invariably have at least MOs and on smaller pianos I have pure 12ths. But then very large pianos are rare for me. Perhaps 9 footers need a little more than 4:2 octaves for true MOs.

I had to read that last post about 6 times before I got it. Thanks Jeff, it does explain a lot.

My sticking point was the idea that a mindless octave could be narrower than a 4:2 octave, as in my head the use of the wide double octave below meant the 4:2 partials match on the octave would be wide. It takes a bit of thought to realise these two things are completely different.

I'd like to test this out, because it seems to explain what I've been finding - I tune MO with a 4:2/6:3 temperament, and when I get to the top I try and stretch the top octave out, but when I check it with the middle octaves I'm always going too far. If the octaves actually have been getting slightly narrower towards the top, then this makes my stretch curve rather wonky!

I should, and usually do, just end up tending towards pure 12ths as I go upwards.

Interesting discussion!

larger than pure 12th have the same quality than a too large 5th, it can be confusing But not as much once you recognize that quality.

the 12th seem to have much "plastiicity" and sound just with different sizes, as the octaves.

Yes, I agree, most interesting!

Phil, are you leaning towards pure 12ths because you can handle that stretch comfortably, or because you like the sound? No offense whatsoever, I'm just curious!

Kees, that clever MATLAB code of yours, is it fully compatible with an open source alternative such as Octave (or any other programme)?

I'd be really interested in using something like that, I get a kick out of every chart you post here (and in other threads). They are crystal clear to me, and visualize what I believe I hear...

It should be octave compatible except perhaps the fitting of the iH data which I read from a tunelab file and fit to a double exponential using the optimization toolbox.

Kees

Grandpianoman kindly provided me with some notes of his piano to measure. Look at this partials structure of his C2. This does not fit the Young model, or the tunelab modification thereof at all!



Kees

I'm staying out of this thread, but I hope I can ask:
What program does this readout come from?
Are the numbers in the Cents column the number of cents flat or sharp from the ideal Fourier multiples?

Here you can not ignore that the predominant partial, the 4th partial, has a negative iH, while the upper partials behave as expected.

The tuning of C2 will be averaged by all ETD's to fit in a smooth progression with its neighbor notes.

All intervalls in which the 4th partial of C2 is involved will sound out of tune:

C1-C2 (P8 8:4)
F1-C2 (P5 6:4)
G#1-C2 (M3 5:4)
C2-F2 (P4 4:3)
C2-C3 (P5 4:2)
C2-C4 (P15 4:1)

If you tune these intervals to sound good then the 3rd and 6th partials, which are very audible, will sound bad:

C1-C2 (P8 12:6 and 6:3)
D#1-C2 (M6 10:6 and 5:3)
F1-C2 (P5 9:6)
G1-C2 (P4 8:6 and 4:3)
A1-C2 (m3 7:6)
C2-D#2 (m3 6:5)
C2-G2 (P5 6:4 and 3:2)
C2-C3 (P8 6:3)
C2-G3 (P12 6:2 and 3:1)
C2-G4 (P19 6:1)


A difficult choice.

But do you know how it was actually tuned?

I do. And it illustrates and reinforces your point that ETD's that assume Young's model are inadequate. I was not sure until I saw this data and now agree wholeheartedly with you on this point.

Kees

I am curious, I suspect that Bill Bremmer tuned the piano. Can you confirm that?

What are the beat rates for the critical intervals mentioned above?

Do they stand out from its chromatical neighbors?

Thanks for posting this, Kees!

Is it that we have a bad model or a bad string? I have found the [Edit:] unwound strings to be very close to Young’s model. Regardless, it confirms in my mind that wound strings are best tuned by playing more than one note at a time. An octave and a twelfth above the lower note is my choice.

I tuned the wound strings of that piano aurally and noticed nothing unusual at all. Indeed, it is a fine set of would strings made by Ari Isaacs. They are smoother sounding and more easily tunable than average. It seems to me that the more mathematics try to get involved in tuning predictability, the more they predict impossibility when in fact, there is no problem at all.

You should have seen the comments one "scientist/engineer" had about my original tuning chart for this piano. It was a predictable example of "It wouldn't work, it couldn't work and shouldn't be tried". Since all of the upper octaves that I tuned were determined by direct interval (the piano itself telling me what it had to offer), the numbers generated did not come out of thin air nor were they a calculation based on some "predictable" constant or any other imagined factor.

The Mason & Hamlin pianos are known to have lower inharmonicity than most. Therefore, even though on the second tuning, I used 8:1, 12:1 and 16:1 (for C8) octaves for the high treble, the numbers were far lower than I would get on a Steinway.

The wound strings would be somewhat different than the originals but they were still confined to the scale configuration that the piano has. They are just higher quality and more carefully made than many sets of wound strings that I see.

One thing I would say is that I never, ever use a calculated program to tune the wound strings of any piano. The SAT does make its calculation based on readings of the difference between the 4th & 8th partials of F3, the 2nd & 4th partials of A4 and the 1st and 2nd partials of C6. How would it know what to do with the wound strings? Answer: it would have no idea what to do; it could only make an assumption. I would never count on that and I just shake my head in disbelief at how many tuners use that program and start tuning the piano on A0.

Well, there is no mathematical problem, but it seems an ETD would need to use each partial and not assume Young's model.
The negative offset of the 4th partial is perhaps due to bridge/soundboard coupling effects. The C2 sounds beautiful, there is no problem at all.

The ETD program I made myself (which assumes Youngs model) mostly gets pretty close to reproducing Bill's EBVT3 tuning charts, but not as close as I would like.

Bill, thanks for telling us how you tuned the high treble. I will program that in and see how close I can get to your numbers there. If you could describe how you tuned the wound strings I would like to simulate that mathematically and see how close I can get.

When I saw you tuning charts my reaction was the opposite of the guy you described: I think it is a challenge for ETD design to produce anything as intricate as that!

Kees

Kees:

I am confused. I thought you were using the look up table from Tunelab, not Young's model.

And my experience is the partials of bass strings can be wacky and sound good until an interval such as an octave is played with another bass string, especially if it is also wacky. Then there are clashes that just cannot be tuned away.

I use a "modified Young's model" which I sometimes lazily confuse with Young's model, sorry. It still tries to capture all the partial deviations with one number.

So then perhaps tunelab's ignoring of these wacky partials and just fitting them into a modified Young's model captures more or less what an aural tuner does too?

Kees

I really don't know much about what is being discussed here but it did occur to me that what may have happened is a reading anomaly or error. I know, for example that with some of this software, inharmonicity samples seem to change slightly every time a reading is taken. It is obviously not the piano that is changing inharmonicity from one moment to the next but the interpretation of it is very sensitive to a number of factors.

I don't have access to the chart I created for this tuning because I left it with GP and I did it on a borrowed SAT IV. I don't think you'll find however that the C2 is out of line somehow with where it would be expected. I also had no trouble reconciling any intervals with it. What Jeff is saying does happen, yes but it did not on this piano.

I have found such inconsistences in my readings many times in many pianos. I think of it as being the rule more than an isolated incident.

And also I've found what Bill says, be it electronically or be it aurally, there is no noticeable problem with the tuning of the piano.

When tuning electronically, I don't know how the ETD calculs its targets but I suspect it averages the tuning of all notes to fit into a smooth curve.

When tuning aurally we tune to the best sounding spot without noticing iH uneveness. We can only run into problems if we are tuning based on a unique single interval, which we usually don't do.

Bill:

Here's the note with the unusual inharmonicity:
C2

I'm not sure how I can check if it's a reading error as I can't play GPM's piano from here so this note is all I have.

I have your tuning charts from GPM. Do I have your permission to post a link to them for this discussion?

I do notice something unusual at C2 in both charts, but perhaps it is normal. I think only you can tell.

Kees

Kees, post anything you want. If there is an anomaly in C2, then the aural tuning result should reflect it. As I said, I never noticed it is there is one. The sound file does not produce anything conclusive to me.

I'm still tweaking my EBVT program to get the correct octave stretches, but a comparison with Bill's master tuning chart in the midtrange is very good.

On the temperament octave F3-F4 my program, compared to Bill's June tuning of grandpianoman's piano, had a total of 3 error points (as computed per PTG exam standard). By comparison, using just the theoretical EBVT offsets from Bill's website (which are computed disregarding inharmonicity)and entering them as offsets in an ETD, produces 8 errors.

For the rest of the midrange (C3-B4) my program had one more error point. A total score of 92.5 for the temperament octave and 94 for the midrange.

An interesting point is that to get these good results a "pure" fifth has to be treated as an equal beating 3:2/6:4 interval.

Kees

Here are the tuning charts from Feb. and June. Esp. in the Feb. tuning I notice C is sharper than C# in terms of offset in octaves 3-6, but this reverses at C2.

Chart

Kees

Kees,

Those two readings may well point to the inharmonicity anomaly that was noted. Whatever it may be, it did not create an aural dilemma. The question is, however, what would a calculated ETD program do with it? If only that string is different from the others, I would expect that a calculated program which is always based upon assumptions to tune that note somehow "off" from the way an aural tuner would tune it. The SAT FAC doesn't even sample any wound strings. RCT samples A's and so presumably would only tune that note incorrectly. Tunelab samples C's and this is where I see that all of you found the problem. Would Tunelab provide information for a substantial range of notes that is slightly "off" based upon that one anomaly?

The facts you noted in in post #149854 are interesting. Ron Koval has often been perplexed in how my direct interval/program tunings "sound great but they don't match the numbers". I don't have quite the time right now to get into all of that but what is always interesting to me is that when I use that method, many of my "numbers" are the same or very similar from one piano to the next. Far from needing 1/100th of a cent resolution, 0.5 is quite sufficient.

I never have, for example, found any piano where A3 entered at 0.0 and read on the 4th partial calls for F3 to be anything other than 1.0. So, I enter 1.0 and listen to it and it always sounds like 6 beats per second to me. If i were to tune 0.9 or 1.1, I wouldn't be able to hear the difference, so I always simply enter 1.0. Since F3 is 1.0 (read on the 4th partial), F4 is also 1.0 (read on the 2nd partial) and that cannot create anything but a perfect 4:2 octave which it always tests out perfectly to be.

Similarly, C4 always ends up being 2.0, C#4 is always -2.0 as are E4 and F#3. I have gotten to the point when I construct these programs that I already know what many of the numbers will be, so I enter them. A few of the in between numbers vary slightly but they are also fairly predictable. Some of the outer octave numbers also end up always being the same. F5 is always 2.0. E5 is always 0.0. C6 is almost always 6.0 and so is F6.

That is why I could use that one Steinway model D tuning as a "generic" tuning for nearly any moderately high inharmonicity piano and have to do very few aural corrections. I would simply tune the high treble and the wound strings aurally. Any Kimball, Baldwin Hamilton or Acrosonic, any A.B. Chase grand, any Baldwin Grand, just about any crummy piano or good piano you can imagine could be at least pitch corrected using that program and simply be tweaked aurally here and there to produce a tuning with that signature EBVT III sound.

That flies completely in the face of the notion that new samples need to be taken every time a piano is tuned. that just has not been my experience. The first time I tuned GP's piano, I used the Steinway program and after about one hour of tuning, we got a first impression of what the EBVT III would sound like. He liked it, so I went on the next day to create a custom program for it, having already pitch corrected it using the Steinway program.

Those figures are posted somewhere here on PWF. If you look at them and compare them to either the February or the June tuning of GP's piano, you will find more similarity than difference.

Bill,

Tunelab can sample any notes you want, the C's is just the default. It basically just ignores the anomaly and fits as best as it can. Interestingly the anomaly in C2 disappears if I use just the first 2 seconds of the note; you can also hear a tiny self-beat later on in the envelope, so it's probably an effect of coupling to the rest of the piano.

On my Heintzmann upright with A3 0.0 and F3 +1.0 both at the 4th partial, F3A3 beats at about 4.7 bps. I measured this by tuning the partials at the offsets with my ETD, then record the F3A3 third and timing the beats with a spectrogram. The prediction from my inharmonicity model is 5.1 bps which is not bad.

In this case I think your offsets are still good, because the ET F3A3 beats about 6.5bps on my upright so to get the EBVT effect you probably want to go below 6bps to hear a clear effect?

In your Feb. and June tunings F#3 is -1.5 and -0.5, F6 is 8 and 9.

Also, the difference between the Feb. and June is quite big even in the midrange:

C#3(-2.5/1.0), D#3(0.0/2.0), F#3(-1.5,-0.5), G3(2.5/1.5), G#3(2.0/3.0), A#3(3.0/4.0), B3(0.0/2.5), D4(2.0/1.0), D#4(1.0/2.5)

Was there a difference in the EBVT variant you used?


Kees

Bill, Keys,

this is interesting reading! I'm not using my RCT taking samples either, just reading directly on the partials.

One thing that was confusing to me in my rookie ETD week (last week) is that if you read with RCT set on A3, and you read the on the 4th partial, the offset value will be for the fundamental. That was not of much use to me.

So now i set my ETD to listen to A5, use the advantage of two different note recording windows in the device display, and make the 5th partial of F3 be -11.84 cents lower than the 4th partial of A3. THAT will indeed produce a F3-A3 beating at 6bps, if anything

I have heard that direct interval tuning is not possible yet with the RCT but soon will be. Any of the Sanderson devices work well that way, however.

Well, if you read say on the 4th partial of A4 and set it for an offset of +4, that means the 4th partial of A4 is 4 cent above the 4th partial of a 2^1/2 ET without inharmonicity. So if your 4th partial happens to be 8cent sharp because of inharmonicity, the fundamental of A4 will end up -4 cent from 440.

I agree it is confusing, having been confused by this myself.

Kees

The real difference from Feb to June was that in Feb, I had used the Steinway chart as a pitch correction, then tweaked it by ear. In June, I started from scratch.

Compare any two exam master tunings on the same piano, sometimes even by the same master tuning committee. You will find similar differences. Was one then wrong and the other right? . No, they were both aural tunings and each has its on variances based upon perception at the time.

So, even though I started the June tuning with A3 at 0.0 and F3 at 1.0, etc., the rest of the differences were due to perception at that time. I recall, for instance, changing the F#3 I had preset to -2.0 to -1.5. That is a small difference but it did make the F#3-A#3 beat a little more mildly.

At Jerry Groot's, I first tuned the EBVT III aurally and it sounded good. Then we programmed the RCT to produce a calculated version. I expected some differences and there were but many notes seemed to agree perfectly. I expected the F3-F4 octave to be a little wide but it really wasn't, same with G#3-G#4. The D5 was flat as expected. There were a couple of 5ths from F4 to F5 that proved a little narrow but it took lowering the bottom note rather than raising the top to fix them.

Where I really noticed a difference was in F5 to F6. The RCT tuned too sharp. Then after F6, it wasn't sharp enough. So, I did all of that aurally as well as the wound strings. It made for a super clean sounding example of the EBVT III.

Would I be correct in stating that there is a bit more to tuning EBVT3 than blindly following the tuning instructions from your website? For example on a very inharmonic piano the ET F3A3 will already beat around 6bps, in which case 5 bps would probably be more appropriate to get a well-tempering?


I find the same in tunelab: the low bass and high treble stretch "leaks" too much to the rest of the range.

Kees

Kees,
I try to understand you,
1. t(i) = [1200/log(2)] * log(p4(i)/p2(i+12)), i = 1,…,76. ? why log(2)?
2. why n(48) = 4700 measn 440hz? what is n?
3. why we don't use curve fitting?
thanks in advance,

1. Definition of cent.
2. Per definition. n is defined in the post.
3. Because there is no data to fit a curve to.

Kees

HOOOOOOOO! Setting an f temperament need not be so complex.
Here is simple aural:

set middle c. (wherever...)
F below, a fifth, set to middle C and narrow by 3 beats in the time sound dies....
F above middle c, a fourth, raise by 4 beats same as above. Should match the first f. Wait & see....

Set B flat, in above ratios to the F's. EZ

So from there set above ratios relative to middle c:
down to G, widen
up to D, decrease (you have a 3rd check with B flat and a 6th check with lower F)
down to A , widen, (3rd check with lower F)
up to E, decrease, (3rd check with C, 6th with G)
down to B, widen, (3rd check with G, 6th with E)... check flow at this point...
down to F #, widen, (3rd check with B flat)
up from B flat to D#, widen, (3rd check with B, 6th with F#)
down to A flat, decrease, (3rd check with C, 6th with higher F)
up to C#, widen, (3rd check with high F,6th check with A flat)

should be pretty.

beats in fifth = too sharp
beats in forths = too flat

that would be bass, then for treble, beats in forths = too sharp, beat in fifths = too flat. It works well . Use your ear to balance, very nice....

SM:

The original purpose of this Topic was for calculations, but like a good Topic, (not saying this is one) other constructive things come up.

Nice to see another C-forker using fourths and fifths and including F4 and A#3 at the beginning of the sequence.

Something I want to mention is the G3-A#3 minor third should always beat a hair faster than the C4-E4 major third. Theoretically they should beat at the same rate, but due to inharmonicity, no. This is a great way to limit the pitch of E4 for those that have a tendency to make the A3-E4 fifth too pure, and may not find out until G#3 (if A#3 is not tuned at the beginning and used for checks and tests.)

agree, I posted to my "aural" question, will send you private

Well, my math thinks the sound never dies out completely and will tune a perfect 5th. (3 beats per infinity = 0 bps.)

If you would introduce an arbitrary cutoff dB level pianos with shorter sustain time would then get narrower fifths.

Your recipe may work in practice, but in theory I'm not so sure.

Kees

Hi Kees, happy B-day!, mailed you 2 boxes today. Can't wait to actually read all of the above. As you know I absolutely love the physics of sound, so I love your approach!

At this point I'm not into investment in devices. Sometimes my fifths come out fairly pure, depends on piano and string scale. I set f first, then work with it...

Life/tuning in practice v theory, always open for discussion... love this topic! SM