Is there anything similar to a Railsback curve for the iH of partials on a single string? Or for each string, as I guess it would have to be. (Has there been an extended comparison of the data about the pattern of iH for each partial on single strings on different pianos?)

In other words, as we all know, the lower partials are often sharp from an idealized Rameau\Fourier partial structure, and the amount of iH rises as one goes up the partial ladder on a given string. However, the rate of deviation--how low on the ladder the partials start becoming very sharp--and the way in which some lower partials fall back closer to being exact multiples of the fundamental, varies from piano to piano.(And from string to string, of course.)

Is there any pattern to be expected at all? I don't mean the obvious patterns--that the iH will increase as the partial number rises, and that partials on strings with high iH will, by definition, have higher iH on lower partials. I mean instead the patterns of deviation on the lower partials as one goes up the ladder. The deviation doesn't just rise in a linear manner. It's slightly erratic, but is it erratic in a similar way on similar strings on most pianos? (I'm not sure: I think I'm finding that some piano strings that seem to have high iH--a strong deviation on the 2nd partial, say--may have a 3rd partial that is almost just. Others don't.)

Or are there too many variables to make a comparison or a curve valid--metal content, string thickness, and age etc? But those would affect a Railsback curve, too?

I only have been told that the iH of srtings vary in the same unison, (from one string to another, there are too much parameter sthat influence the string ihj to have them all matched)
Then the partials cant be coupled well, (while tuning the unison) if we do so (if we fight to couple the partials to the most) they take advantage on the fundamental and give a bad (too clear, metal toning) tone. that is the reason why unisons are tuned "open" , to help the fundamental to couple (plus whatever partial you are able to couple with)

ALso, listening to the whole sustain is misleading, because the partials finally couple with time, the tone of pianos is all but "fixed" despite the official appellation.

The first thing a tuner may learn is to recognize the coupling at the fundamental level, it is heard as a thickening of the tone, and a better projection (the apprentice tuner can even work with earplugs to filter the high spectra and learn toà recognize that basic process.

The formulas for computing iH are all approximative, the data shown by different EDT also differ. I belive it will be difficult to obtain what you are looking for (but all the formulas are available there in English and German http://www.sengpielaudio.com/index.html

Jake:

The iH in cents for each string is the difference in cents between what the frequency of the fundamental should be ignoring stiffness and what it actually is. To determine the difference between what the other partials should be and what they actually are you apply the iH times the square of the partial number.

Example:

A string has an iH of 2.0 cents and for its diameter, density, length and tension it should vibrate at 1000 hz. It will actually vibrate 2.0 cents higher than 1000 hz. The 2nd partial will vibrate 2^2 * 2.0 or 8.0 cents higher than 2000 hz. The third partial will vibrate 3^2 * 2.0 or 18.0 cents higher than 3000 hz.

…..

This is accurate for unwound strings, but not as accurate for wound strings.

Here is a link where you can view some iH curves. Notice that the scale is logarithmic. The values for a straight line on a logarithmic scale double at regular intervals: Pscale iH curves

What affects the Railsback curve is both the iH and the octave type. A short piano tuned with 4:2 octaves in the midsection may have a similar Railsback curve as a longer piano tuned with 6:3 octaves in the midsection.

Yes, basically, but you are over simplifying the thing, Jeff. What you call octaves type is a concept, not a reality.

And soundboard stiffness, bridge thickness, and the pins on the bridge (plus the voicing) are modifing the iH of any single string.

SO the formulas gives you ballpark figures, and in reality the output is slightly different (even from one string to the next in the same unisons)
How much, I cant say, but it is.

Kamin:

I have analyzed the measured cent deviation of partials for a number of pianos and I stand by what I have said.

Hmm. Working from a recording, I can't determine the diameter, density, or exact length of the string.

Let me at least show you why I was asking the question. Maybe you guys can tell me something by looking at the chart below. I seem to be finding strings in which the partials don't vary at a regular or logically irregular rate from the fundamental. Bear with me. I'm just working with the raw pitch information, taking a recorded note and using Audacity's spectral analysis window to view the partials. Here's the partial structure, in hz, for the first 15 partials of a recorded G#2:

Partial Freq Deviation from multiple of P1 in hz

P1 102
P2 208 4
P3 310 4
P4 414 6
P5 519 9
P6 621 9
P7 727 13
P8 832 16
P9 936 18
P10 1041 21
P11 1148 26
P12 1254 30
P13 1362 36
P14 1468 40
P15 1576 46

They're sometimes the same distance from the idealized multiple of the fundamental. At other times, there's a sudden bump: see partial 11, where there's a leap to a deviation of 5, followed by partial 12, where the deviation drops down again to 4 hz. That's why I was wondering if there was an expected curve of any kind--which partials would remain about the same distance from the multiple of the fundamental, where bumps and dips might occur, the rate at which the deviation rose, etc. I expected to find a deviation that rose in a predictable, albeit not linear, way with the partial number.

Jake:

G#2 would have to be from a good sized piano to be an unwound string. If the G#2 is a wound string, you cannot expect the partials to be completely predictable. Also, the unison should be muted to allow only one string to sing. And I have no idea if the program you are using is very accurate or not. It bothers me that there are no decimal places.

The math that I described can be used to calculate the iH from any two frequencies that you have listed. That is the sort of thing that ETDs do, and they work. But as you noted the frequencies do not follow a completely predictable pattern. Different pairs of these frequencies will produce different results when calculating iH. I don't use an ETD, but this is a good reason to tune the bass aurally.

I think a lot of what are be seeing here is the effect of rounding errors on the calculations made by Audacity. Is this how the information is presented - all the partials on exact Hz values? Because rounding errors could easily give you misleading information here. For example, P2 and P3 have identical offsets, but if P2 is at 207.51 Hz, and P3 is at 310.49Hz, the offsets will be 3.51 and 4.49. If the P4 offset is similarly reduced, the curve becomes smoother.

To really analyse this you really need to take the average offsets for a number of different strings, or at least for all the strings on a trichord.

But at this level there are so many interacting effects that it's going to be very difficult to separate real from anomalous results.

By the way, I should mention that this piano sounds very good. I analyzed the note because I liked it, not because I heard anything as being off. I'll confess: It's from grandpianoman's piano. It's the very first note in the recording of "Reflets dans L"eau linked here for us at http://www.box.net/shared/pi155b7yq7 .

NA Mongoose: I did the rounding, actually. I'll go back and get the exact freqs. They are the peak freqs using a Hanning window for the analysis. (I chose a soft\medium strike to minimize transient noise.)

Audacity is usually good. I've compared its results to those in Spear and other similar programs with high resolution, and they are usually in agreement. But let me go back and check.

Jeff--about the wound strings. I didn't understand that the winding would make a difference. (I thought that the intention of winding was to prevent the need for very long strings. Didn't see that a possible side effect was to change the partial structure.)

In the meantime, if anyone else cared to do a quick look at that same note on another piano and wrote down the exact partials and their offsets from their multiple of the fundamental...

Jake:

Well, if the note you are analyzing is from a recording, then at least two strings are sounding. Also, if the sustain pedal is pressed, the strings for other notes may also be vibrating.

Here are the cent deviations of the measured partials from G#2 on a M&H BB that a Verituner measured (Thanks, Joe!):

P3 -4.45
P4 -3.72
P5 -2.56
P6 -1.47
P7 0.12
P8 1.86
P9 3.88
P10 6.42

These figures are very handy for calculating iH. Simply (?) take the difference between two deviations and divide by the difference of the squares of the partials.

Examples:

P4 is -3.72 and P3 is -4.45. (-3.72 - -4.45) / (4^2 – 3^2) = 0.10 iH.

P5 is -2.56 and P8 is 1.86. (1.86 - -2.56) / (8^2 – 5^2) = 0.11 iH.

And for comparison here are the deviations from G#2 on an Acrosonic Spinet. (I’ll let anyone interested do their own math.)

P3 -4.81
P4 -5.21
P5 -4.83
P6 -2.59
P7 -1.12
P8 2.27
P9 6.23
P10 9.52

Jeff--

Thanks for posting those figures. Could you post the figures for the fundamental and the 2nd partial, too? (Not the perfect figure for the note, but the actual freq, in cents if you prefer.) Without that data, we can't see how much the other partials vary from a perfect multiple of the actual fundamental... Or does the etd read on the 2nd partial, so there's no way to see the first?

Sorry, the figures for the Verituner files included only the third thru tenth partials for this note. (Other parts of the scale had other partials.)

But there is something you don't seem to realize. iH is applied to the multiples of the theoretical fundamental, not the first partial. So you would not want to compare the other partials to multiples of the actual fundamental (1st partial), anyway. But you can still see if the partials differ from what iH predicts by calculating the iH for two different pairs of partials.

As I said before, the frequencies of the partials of unwound strings are much truer than wound strings.

I could post the deviations for an entire scale, such as a Yamaha U1 if you want.

We may be talking about slightly different things--I'm not trying to examine the variations from the ideal pitch, or test iH correction figures, etc. I'm just trying to do the more limited, tedious thing of looking at how the partials vary from the actual fundamental. (When I mentioned the deviation of the partials, I meant the deviation from the multiple of the actual fundamental freq, not the deviation from the multiple of the ideal fundamental freq.)

Which means that I do need to move to the right of the decimal point. I'll look at the note in more than one program and post the more precise figures tonight, in case anyone is interested.

But, yes, I do understand what you said about the wound strings. Thanks for looking into this and for the explanations about calculating iH. Cheers.

Jake:

You have picked a tough row to hoe. Regardless of the number of decimal points, the numbers will not quite work out with what you are attempting, but give it a try.

Here is a perfect series of partial frequencies with an iH of 0.10:

100
200.0347
300.1387
400.3467
500.6936
601.2142

See what you can do with them.

Hm..The row is both hard and long. I found that Audacity lets you save a list of all of the freqs it registers. (Analyze\Plot spectrum\Export). Valuable. Easy to put into a spreadsheet...But it means that, with a Hanning window of 2048 samples, I have a 10 page list of freqs for that note. Stop laughing, Jeff. (Let's see. Do this for 88 notes and I get 880 pages or a large spreadsheet. How many pianos would constitute a valid data set…?)

And of course, I'm often getting two freqs that are very close to each partial, which may represent the separate strings (but this is a bass note--I'm not sure if an M&H has two strings here)or may represent a phasing and\or resonances with the bridge and\or soundboard and\or sympathetic resonance with nearby strings. If anyone is curious to see the entire list, I’ll be happy to share my misery. In any case, Audacity’s ability to quickly generate a text file of all the freqs it registers may be useful.

But here are the figures:

---------Freq----------Ideal mult.----Dev from ideal mult.

P1----102.282715
P2----204.56543-----204.56543------0
P3----309.539795----306.848145----2.69165
P4----414.51416-----409.13086------5.3833
P5----516.796875----511.413575----5.3833
P6----621.77124-----613.69629------8.07495
P7----726.745605----715.979005---10.7666
P8----831.719971----818.26172-----13.458251
P9----936.694336----920.544435----16.149901
P10---1041.668701--1022.82715----18.841551
P11---1146.643066--1125.109865---21.533201
P12---1254.309082--1227.39258-----26.916502
P13---1359.283447--1329.675295---29.608152
P14---1466.949463--1431.95801-----34.991453
P15---1574.615479--1534.240725----40.374754


So the deviation is often less than the rounded figures suggested. On the other hand, there are still dips in the deviations. Since this is a bass note, all I’ve really demonstrated, if I’ve demonstrated anything, is why people often tune the bass strings by ear: bass strings have high iH (and this bass string has erratic iH?). But I’m still curious about whether a similar pattern will come up on other pianos on that same note, or if there are so many variables with wound strings, soundboard resonances, etc, that it's a mud-bath.

Surprised me that partial two was perfect, and that partials 4 and 5 came as having exactly the same deviation. (I wonder if Audacity read on partial two, dividing it in half for the fundamental and then doubling that to get itself as a perfect 2nd partial. That would explain why the fundamental is read as 102 instead of the usual pitch for this note, which 103 hz. The 2nd partial would then actually be sharp by around a hrz.)

But thanks, all, for suffering through this so far. Sorry if I'm just reinventing the wheel, here, but I haven't been able to find much raw data on the deviations of actual strings.

Jake:

I am not laughing. Smiling, yes, but not laughing, mostly because you posted at 3 and then edited at 6 in the morning. Unless you are not actually on the east coast...

Bass strings do not have nearly the iH of high treble strings. The highest wound strings usually have the least iH of all.

In the bass of very short pianos I hear so many contradictory things and plain garbage I often use a 12th spanner and match the 3rd partial of the lower note to the 1st partial of the higher note and "be done with it." It is a lot faster and just as good, if not better than searching, searching, searching for the place that sounds least bad. Bass strings have a somewhat random partial structure; shorter ones are worse than longer ones. I have wondered if there is a general pattern to the randomness, too, but it doesn't keep me up all night.

Correction: The usual ET freq for that note is 103.826. Doesn't affect the chart, though, since it measures the iH pattern based on the real fundamental.

Yes, up most of the night. Coffee around 9:00. Funny thing about coffee--they must put something in it. And it took an hour to get the columns to more or less line up.

Something I've often seen, and saw again here, but I'm not sure that I understand: Whenever I record many piano notes, I often see 2-3 "near freqs," WITHOUT there being an exact pitch match for the partial. As I said earlier, I understand that the near freqs may be unison strings, etc, but I'm confused by the absence, at times, of the center freq. Does a single piano string (or rather the combination of the string, the bridge, impedance mismatch with the soundboard, the soundboard resonances, etc) sometimes actually create a wide band of freqs near a given partial's freq? The human ear then averages these, is a sense, to create the partial? Or do they beat, so that the ear hears the intermediate pitch?

Mr. Deutschle,

it would be nice to give a comment about the source of this useful tool and to which tuning it relates originally.

Jake, Jake, Jake:

It does not work to base an iH pattern on the real fundamental, but maybe you have to find out for yourself.

Not sure if I follow you...

Two strings at nearly the same pitch will "couple" and beat at the same frequency, at least for a while. And there have been measurements taken that show when a unison is tuned, at least the upper partials are at a lower pitch when two or more strings are vibrating together than a single string vibrating alone. Whether this is due to the "effective length" of the string changing or the "effective iH" changing I am not sure.

And beats are the “difference tone” between the partials of two different strings, usually different notes.

But since you are measuring from recordings of multiple strings with other things going on, the analyzer may be just giving the best answers it can. It probably does not display a probability factor or something.

If you really want to stay awake, try chocolate covered coffee beans!

Sure, be glad to.

Mr. Stopper sells this tool and advocates its use in aurally tuning his OnlyPure method that is similar to perfect twelfths. I made mine, but it isn't as nice.

Slight correction:
I don´t sell it, but invented it for tuning aurally pure duodecimes (twelfths).

Jeff--You say that "It does not work to base an iH pattern on the real fundamental..." But, but, I'm not trying to make anything work, if you mean tune a piano based on these figures or calculate an iH factor. I'm just trying to see the raw data, the actual iH pattern of this specific string or pair of strings.

Do you remember where you saw the research about unison strings lowering the upper partial freqs? You're speaking of unisons tuned "perfect" to the center string, yes? I understand that the coupling of the strings causes freq phasing, but I don't think I'd understood that the upper partials become\are heard as slightly lower with a perfect unison.

I can see that the software may be guessing, or that I'm hearing unisons, or phasing or little voices in my head...I obviously have to look at upper notes more.

(Coffee with chocolate? Get thee behind me, Satan. But the next time you're eating these while at a piano that has a good sounding G#2, if you want to record the note and send it along, I'll do another chart...)

Bernhard: Since you do scale design, I would imagine that you've examined the literature closely and have done similar calculations. Or am I caffeinating away time looking at this at all? Don't string manufacturers look at this, so they can try to design a string that has partials with a predictable\manufacturable degree of deviation from the multiple of the fundamental?

Is there anyone here who has large enough hands to play that interval without the aid of the spanner? I can almost do it, but not quite...

"Is there anyone here who has large enough hands to play that interval without the aid of the spanner? I can almost do it, but not quite..."

Even a child has hands large enough, but few people have a hand large enough. I can get a 10th, but it isn't honest except for some minor tenths.

Jake:

Well, I am probably splitting hairs. You say you are looking for an iH pattern, but I would say you are looking at a frequency pattern. If you want to look at the actual iH pattern, you have to determine the iH. This can be calculated from any two measured partials, but may be slightly different depending on which partials are measured, due to either the accuracy of measurement or due to the string being wacko. To me, a comparison of these different calculated iH values would be an “iH pattern”.

Here is a link to a Topic on this Forum about The Virgil Smith Effect