I think we're talking across each other.

A4= 440 Hz fundamental pitch
A4 2nd partial mathematical without inharmonicity = 880 Hz. (Half string length / doubles the Hz. value)

The numbers that are added or subtracted in this discussion are cents, which have a logaritmic relation to Hertz. Every half step in the mathematical model is made of 100 cents. Therefore, an octave = 1200 cents.

So, A4 set to 440 (measured second partial = 1.08) Means that the second partial of A4 measures 880 Hertz, plus 1.08 cents, or about 1% of a half step. (Which really isn't much at all!)

This crossing back and forth between beats and Hertz - which have a direct relationship, to cents, which is something all together different, often causes confusion.

Think of the doubling... Between A4 and A5, there's 440 hertz, yet between A5 and A6, there;s 880 hertz. That's why we hear increasing beat rates, the scale that relates to beats is changing throughout the scale of the piano.

Helpful, or no?

So (something which has some sort of relationship with A4 approximating) A5 would be 1201.08 cents more than 440. An inharmonicity of .09%

Then we get to:

This does not make sense. This is an inharmonicity of .59%. Why is the inharmonicity of the second partial of A5 so much greater than that of A4? Are you sure that you are not measuring the inharmonicity of the fourth partial of A4?

I'm not sure about the inharmonicity percentages that you got - how'd you get them?

But yes, if you wanted to match the second partial of A4 with the fundamental of A5, (2:1 octave type) the machine would be set to on A5=1.08. While the machine is set to A5=1.08, you can play A5, see that the display is stopped, and then play A4 and also see that the display is stopped.

Looking at A5, the second partial measures 1207.1 cents above the fundamental - that we just set to plus 1.08. That's why to get a 2:1 octave above A5, we've got to add the number we started with, plus the inharmonicity measured from the string.

As to why the inharmonicity changes, think of a slinky toy in wave motion. Each segment adds up to equal the whole length IF the wire is perfectly flexible and the endpoints vibrate without any wire stiffness. Anyone that's had to tie off a string knows that piano wire has significant stiffness, which makes each segment "sound" shorter than it would be expected using the slinky model. As the segments of the wire get shorter and the diameter of the wire vs. the segment is greater, the stiffness of the wire effectivly shortens the individual vibrating segment, causing the partial to measure and sound higher.

Wow, that's hard to describe! Anyway, what just happens naturally to an "ear" tuner is all part and parcel to this topic. What sounds beatless and unstretched is in fact following the inharmonicity of the strings in the piano. If you tune a piano and have nice single, double and triple octaves, you are ending up with tunings as high as we are talking about.

I think Ron's getting his readings from Verituner, and I earlier posted a bit from TuneLab. Accu-Tuner III comes with a bunch of pre-programmed tunings included. This is the same mathematical formula applied to different pianos, according to preliminary inharmonicity readings someone apparently took on each piano. Accu-Tuner’s calculated tuning isn’t intended to produce pure octaves/intervals of any given type in the treble. They're wide of pure 4:1 double octaves, but that’s about all I can say about it. Accu-Tuner III produces slightly more stretch than earlier models.

It's interesting to me that this is the style chosen by the Sandersons as the default for their unit. I assume they increased Accu-Tuner's stretch in response to feedback from people they respected.

Here's how many cents “sharp” C8 ends up on Baldwin and Steinway grand pianos. (I put quotation marks around the word sharp, since it just depends on what part of the piano's sound we're relating C8 to.) For brevity's sake I'm just posting numbers for these two brands, because they're the most dramatic.

Baldwin grands:

M 5'2" 44.1
R 5'8" 40.6
L 6'3" 45.8
F 7' 48.8
D 9' 33.5

Steinway grands:

S 5'1" 38.0
M 5'7" 41.5
L 5'11" 41.7
O 5'11" 44.0

A 6'1" (85 note) 43.2
A 6'1" (88 note) 44.9
A III 6'4" 45.6
B 6'11"(85 note) 40.9
B 6'11"(88 note) 36.3
C 7'2" (85 note) 57.0[/b]
C 7'5" (1886 up) 34.2
D 9' 37.1

I have run into a few people who prefer the lesser stretch of their earlier Accu-Tuners. I wouldn't say the tunings produced by the 80's vintage Accu-Tuner I recently traded in were that conservative, though.

Jeff

What does not add up for me is that when I compare with a tone generator on my computer, notes in the top octave of the piano are reasonably close to the theoretical pitch from the tone generator. You may object that the tone generator may not be all that accurate, but it is the same electronics used on computer-based tuning devices. Also, pianos play with other instruments that have less inharmonicity.

If you have a laptop or pocket PC, maybe you could download the free trial version of TuneLab and take a few numerical readings from one of your tunings. Or maybe just borrow a friend's ETD for a day. It would be interesting to see what you come up with. Without numbers it's hard to say much.

Jeff

Most tone generators have only an octave or two range. All other pitches are generated from combinations causing overtones or resultant tones. It is unlikely that the standard computer has a greater range than that.

I'd be interested in hearing what your readings are on an ETD as well. Stretch is pretty much well documented. Not even S&S argue about the concept of stretch.

I just made an funny test.

I calibrate Tune Lab Pro to a Seiko SQ70 Quartz Metronome, to A4=440Hz, on a Laptop and on a Pocket PC. Then let the laptop to produce a A4=440 tone, and locking it on PPC, it is 11.74cent higher than 440. The laptop is a HP nx6325.

But the PPC in windows mobile 5.0 can generate an accurate tone, and then I put the seiko SQ70 aside it to make a compair, it can be trustful.
PPC model Mio P350

There are two separate calibrations in TuneLab Pro - one for input and the other for output. If you calibrate for input, that does not automatically calibrate for output. The reason is that the soundcard in many laptops has two separate timebase crystals - one for generating sound and the other for listening to sound. If you want to trust the output pitch, you must calibrate for it specifically. TuneLab Pro offers separate means to calibrate for both functions.

Robert Scott
TuneLab

"Reasonably close" Care to put some numbers to that? At A7, the difference from a zero offset to a +30 cents offset translates to A=3520 Hz. to A=3581 Hz.. For comparison, at that range, A#=3729 at a zero offset. Could you try resetting your tone generator to 3581 and see if it is still reasonably close?

Any discussion of other instruments involves dealing with adjustable pitch. Only in fixed pitch instruments like piano, organ, xylophone do we have to deal with pre-setting a tuning, everything else makes adjustments during playing. (voice, embouchure, fingers, etc.) That's why the temperament discussion doesn't apply to choirs, bands and orchestras the same way it applies to pianos.

I am aware of the theory of inharmonicity, as well as Fourier analysis, and difference tones, and things like that. All of those factor affect the proper pitch.

I am also aware that inharmonicity will also affect other transducers such as microphones and speakers. They too are stiff enough to distort the frequency of sound that gets transduced.

I have compared to other frequencies. Unfortunately, I need to do it reasonably close to when I have tuned the pianos, and I am not in the mood to tune them now. However, 30 cents sharp sounds sharp compared to pitch, and 50 cents sharp is unbearable.

Actually, I think that the best test is to compare to the sense of pitch of the best musicians in the world who are sensitive to pitch. There are not many of them, but I am fortunate enough to tune for them from time to time.

"I am aware of the theory of inharmonicity, as well as Fourier analysis, and difference tones, and things like that. All of those factor affect the proper pitch."

Cool! You're ahead of me on that, but there still seems to be some confusion between us..

"However, 30 cents sharp sounds sharp compared to pitch, and 50 cents sharp is unbearable."

30 cents sharp of what? Where? How are you measuring that?

When Sanderson (as well as the other develpers, I'm sure) came up with his machine approach to tuning, he started with the best aural tunings he could find. Remember, this was in response to those first electronic "scope" tunings that sounded so dead to aural tuners - because the natural stretch the piano required to "sing" wasn't taken into consideration. These approaches, though using a scientific single-partial method, grew from the aural tradition.

Do you have gear that can measure tones? Even if you are limited to Hertz, I'd be interested in seeing where your A7 (or C8) ends up for any particular instrument. It's hard to discuss specifics without any real data.

"Actually, I think that the best test is to compare to the sense of pitch of the best musicians in the world who are sensitive to pitch."

I'm confused. Are you talking about sensitivity to pitch around A4? Pitch recognition in the upper range will change depending on whether one listens to a melodic, or harmonic interval.

Could anybody give me some idea in how to tuning an upright kawai model KS-3F for a best fit in playing Mozart by tune lab pro? I am playing Mozart sonatas and concerto no.20 all these months.

I do uses 6:3 single octave for bass and 4:1 double octaves for treble.

My piano tech, also my best friend used to tune it in A4=443Hz, It really created a much Mozart tone than 440Hz. Since I am practise a concerto and followed by CD player, I had to retone it to 440Hz, now my piano fit the CD, but while play the sonatas, the piano sounds too morden for a Mozart sonata...

Your best bet may be to overlay an alternate temperament on the piano. I would recommend the Bach tuning as interpreted by Bradley Lehman:
****************************************
"I use it almost "all the time," too -- on my harpsichords and clavichord
here, and on our church's piano. (The only time when I don't use it is
when I'm playing earlier harpsichord music that was written for meantone.)"

I'm giving a lecture and two full-length harpsichord recitals in Chico
CA next month, featuring all of this. Details here:
http://www-personal.umich.edu/~bpl/2007mar13.htm

For any needing the offsets to do this electronically (I always do it by
ear, no device), they're here:
http://www-personal.umich.edu/~bpl/larips/math.html

A (0), Bb (+3.9), B (0), C (+5.9), C# (+3.9), D (+2),
Eb (+3.9), E (-2), F (+7.8), F# (+2), G (+3.9), G# (+3.9).

Recordings are listed here:
http://www-personal.umich.edu/~bpl/larips/recordings.html
Brautigam's disc of Beethoven sonatas 4-7 there, on fortepiano, makes a
good demo for any pianists who might want to hear what it's like on
piano.... Robert Hill's disc of WF Bach's fortepiano music is
forthcoming sometime this year, I believe: he sent me one sample track
so far.

Also, this page by Ross Duffin
http://music.cwru.edu/duffin/Norton/Letter.html
offers Gershwin's "Rhapsody in Blue" plus some Brahms, Debussy, and
Liszt further showing its flexibility, even in extremely chromatic music.


Bradley Lehman
http://www.larips.com
********************************
You will calculate a tuning in ET, as normal, then insert the offsets, (just once in the temperament section - Robert Scott could help you.) which add or subtract a certain number of cents from the tuning file. The idea of alternate temperaments is to bring back the concept of "Key Color" - where keys nearer the top of the circle of fifths will sound "more calm", while keys nearer the bottom of the circle offer composers a "more agitated" sound... Each temperament differs in how to move from one extreme to another. Feel free to start a new thread if you'd like to discuss temperaments more. It's a safe bet that Mozart didn't compose for equal temperament...
I also have modeled a few of my own if you'd like a more even machine-based progression that optimizes even-ratio beating in a tuning.

www.rollingball.com has graphs which help visualize different temperaments.

Ron Koval
Chicagoland

Kawai pianos often suggest to me a more conservative approach to octave stretching than I would normally use. For me this brings out their best. Generally Kawais are more deep and dark-sounding than the average piano, and their partial signature in the midrange doesn't seem to provide much support for a lot of stretch in the treble.

If it's this basic Kawai tonal character that isn't fitting with Mozart, you might try going for a bit more stretch in the treble octaves. The brightening effect this would have could sound similar to tuning to A443. Maybe the tone would also sound a little thinner, not as full and round; more like an instrument of Mozart's day.

In a way you would be trying to get the Kawai to sound less like itself, but it might help satisfy your desire to get a sound more appropriate to Mozart's period.

You could first try 3:1 instead of 4:1 and see what you think. I've not tried historical temperaments like Ron is suggesting with TuneLab, but I think you would still have the choice of what octave types to use.

Jeff

It is so moist in Shanghai and I think a Kawai is a best fit in such kind of climate due to the ABS action. Although everybody come to my home and played my piano and saying it is a wonderful Chopin piano. I do seldem play Chopin.

I have asked my friend to find a second hand kawai RX-1 for me this year, with two piano, I then can start all these interesting experiments. Thanks everybody for the help.

Good luck. Kawais are wonderful pianos and I always look forward to tuning them, because they're unique and suggest a different approach.

My thoughts on adding a bit more stretch to get a different tone mainly apply to equal temperament. If you use an historical temperament you may find the offsets sharpen the sound enough to give you the desired effect. For example, following a link Ron provided above, here are the offsets for Bradley Lehman's Bach Temperament:

http://www-personal.umich.edu/~bpl/larips/math.html

With those offsets you'd already be raising the general pitch level, so I'd first try sticking with your usual amount of stretch in the treble before increasing it.

Thanks for your perspectives on different composers and the interesting conversation.

Jeff

I remembered when first time I came to this BBS, the 1st post I read mentioned somebody need to have a steinway, a yamaha, a kawai, and so on, and adjust them to different temperament to fit different pieces form various composers.

I think this had already happened to me, I have set a "new year resolution" for a new or second kawai RX-1, and delayed my orignal plan in replace my 7-year-old corsa. What a BBS here... Glad to know all of you.

Sorry for my poor English, since I do only spend a postgraduate study for less than 2 years in UK, and without daily uses of English since early 2005.

Hi Folks,

I'm back again after losing my computer for about a month but just got a new one. I've seen the argument put forth here before about how the note C8, tuned at +50 cents would "sound" 1/4 sharp. Plain and simple, it doesn't. Many skilled concert tuners will tune much sharper than that. I have often given this simple demonstration with my Accu-Tuner:

Tune the note C4 (of most any piano) on its fundamental and zero it out so it reads 0.0. Now move the note selection to C5 and play the C4 again. The cents will now read between 1.5 and 2 cents. Move the selector to C6 and it will read between 5 and 6 cents. Move it to C7 and it will read from 18-20 cents. Move it to C8 and it will read a whopping 75-80 cents! All from one note, C4, middle C.

In my view, this gives a compelling reason to tune these registers to some approximation of this spread. I tuned aurally for well over 20 years before ever using an ETD and found that indeed, my aural tunings roughly matched this kind of stretch.

At the PTG Annual Convention last year, I demonstrated what I now do routinely with my Accu-Tuner III. It's normal curve does a fine job up to about F6 but then does not stretch the last octave and a half enough to suit my taste. It does have a handy tool on it called the Double Octave Beat (DOB) control which can quickly approximate what I want but I showed that with just a couple of minutes of overriding the program, I can very precisely tune the range from F6-C8 by simply playing the note which lies 2 octaves and a 5th below the target note. For F6, this starts on Bb3. I play the previously tuned Bb3 but have the note indicator set on F6. I stop the lights and enter whatever value is shown and continue that way to the top. C8 inevitably ends up at approximately +50 cents.

To me, this is conservative and the way I tune most pianos. If I really want a concert hall "super stretch", I start at F6 as described but at C7, I tune to the note 3 octaves below and when I reach F7 (sometimes I wait until G7), I tune to the note, 3 octaves and a 5th below. This puts the very top notes of the piano precisely in tune with the notes in the Temperament octave (F3-F4) and the note C8 ends up from +75-80 cents. I have observed many a Steinway D concert tech tune at least this high, if not higher.

Not only is the piano actually in tune with itself by doing this but it actually begins to match the enigmatic desire of the ear to actually hear the highest notes of the piano that high in pitch. There is a concert hall with a Steinway D which I service regularly and have since 1990. When the newly renovated hall opened with a newly engaged sound technician about 3 years ago, he listened carefully as I tuned the highest octaves. He said to me afterwards, "I listened as you were tuning the high treble and this is the very first time I ever heard a tuner tune those notes where my ear really wanted to hear them". I tuned it that way for George Winston about a year and half ago and he was quite pleased. Just coincidentally, I tuned that piano today but for a far more pedestrian ensemble and chose instead to give it my usual amount of stretch with C8 ending up about +50 cents.

Does stretching the high treble either of these ways produce beats between the single octaves? You bet it does but you have to remember that the sustain is short in this part of the piano, so even in the context of someone playing single octaves in that range in a large concert hall, what a tuner might think would be objectionable at close range literally disappears from perception out in the audience. The same kinds of concerns are noted with voicing such an instrument. The brilliant tones will "tear up" your ear if you don't wear hearing protection while tuning but they sound crisp and clear in the audience. Similarly, you wouldn't want to stand right in front of a trumpet or trombone playing fortissimo or an opera soprano belting out a crystal shattering high pitch. (Incidentally, would anyone care to guess how high violins and opera sopranos push upwards their highest pitches? You'd be quite amazed).

On the other hand, a piano in more intimate circumstances is quite another matter but I still tune most of my home pianos with the C8 ending up at least +40 cents. The clientele I serve calls for and recommends my work, year after year, decade after decade because I give (in the opinion of those who engage me) the piano the most musical sound of anyone in my area.

I should also note that I have been involved with setting up many a "Master Tuning" for the PTG exams. By the rules, the 7th octave is to be tuned at 2:1 octaves with the 6th. The examiners always shake their heads and mutter about how "flat" that sounds. Having seen scores of Master Tuning records, I can attest that the note B7 (the C8 is not used), usually ends up about +28-30 cents. Most tuners and musicians agree that this is not a musically satisfying sound and will comment that they "only do it for the exam".

In the exam "prescreening" process, the applicant is instructed on this matter and by the rules, is reminded of it just prior to commencing Part 2 of the exam. It should be noted however that the tolerance for error in that range is much larger than for the temperament and midrange. If the examinee tunes the rest of the piano well but cannot restrain him or herself from tuning the 7th octave in the habitual way, there may be points scored off in that section but it does not usually lead to a failing score. On the other hand, tuning such a wide octave in the midrange would count heavily against the examinee. This reflects what anyone would consider objectionable.

Simply stated, there are different standards and expectations for octave width in different parts of the piano. Despite what all the old tuning books say, tuning what would sound like "pure" octaves throughout the piano's 7 1/2 octave range is a rather poor way to tune. All of today's modern Electronic Tuning Devices (ETD) (or software) designed for tuning a piano work toward avoiding this kind of "flat" and contracted kind of sound.

For the Bass (from A0 to B2), I agree that 6:3 type octaves are generally ideal. Food for thought based on 38 years of full time professional experience, in depth study and understanding.

Bill, are you using the SAT III's FAC program quite a bit now, for the tenor and treble below F6? Do you ever use the DOBA?

Thanks,

Jeff

I actually have gone back to using several of the old but good programs I worked up on my SAT II and then transferred manually to my SAT III. I only use the FAC program on really good pianos. The FAC program works fairly well on good pianos but often seems to throw me a curve in the 5th octave and uses a partial selection that is sometimes hard to read. My custom programs read the 5th octave clearly on the fundamental and I have the amount of stretch I need in that area worked out very well. That's why I think people get "different numbers" virtually every time they measure the inharmonicity. It's not the piano changing, it's a weak signal that produces a wobbly pattern. From my experience, I haven't needed to use the DOB until the High Treble but different people use it in different ways. For example, I still tune my wound strings exclusively by ear, wouldn't waste my time with a calculated program. It may be useful in the low bass but I have never even tried it there.

Incidentally, I tuned two Baldwin Acrosonics today and know that they are one of the models some techs love to hate, I measured the inharmonicity of C4 as I described in my previous post. Here are the results:

C4 read on C4: 0.0
C4 read on C5: 2.2
C4 read on C6: 6.0
C4 read on C7: 26.6
C4 read on C8: 105.2

Wow! When tuning such a piano with one of my custom programs, I usually tune the 7th octave by ear too (faster, better and cheaper than struggling with the ETD) but I always notice that I tune considerably shaper when tuning an Acrosonic but don't when tuning a fine grand or well scaled vertical. Now I think I know why.
I also note that the spread between the 1st and 2nd partial and between the 1st and 4th partial is fairly typical, so a program worked out on a Steinway D tunes an Acrosonic just fine up to the 7th octave, so when I reach that point, I just turn the device off. I usually tune at least 20 but often 25 or more pianos a week and my SAT III battery easily lasts two weeks before needing a recharge.
Also, please note that PTG has again engaged me to provide tuning tutoring at the Convention this year in Kansas City, June 20-24, 2007. I am most interested in helping people who normally use an ETD to be able to get through Part 1 of the Exam where they are required to match the A4 within 3 cents of A-440 (whether their pitch source is accurate or not) and to tune the span from C4-B4 with no more than 8 cents worth of error between F3 and F4 and no more than 6 more cents worth of error in the rest of that span on a piano which is detuned in a way which you will otherwise virtually never find.

It stumps many an ETD tuner but I have a plan which they all seem to understand very well and produce either perfect or nearly perfect results. After getting through that much, for Part 2, you can just fire up your ETD and tune the way you usually do and "sail through" the rest of the exam (which nearly all ETD users do). If anyone has passed part 2 of the exam but needs to retake Part 1, you need to come to me first, then try again! I will fill all available spots. Last year, I tutored 9 people but can do as many as 15. I can also teach how to tune either the EBVT or EBVT-II aurally.

I have a good tuning fork that when measured at body temperature, (by carrying it in a pocket for a while) produces a perfect A-440 but out of curiosity, I measured it on the zero degree days (-17.8 C) that we have had in Southern Wisconsin for the past several weeks. It measure +5 cents when cold. During the few sub-zero days we had, as low as -20 degrees F (-28.9 degrees C), it went up another cent to 6 cents. Getting with an instructor for tutoring or exam prescreening will get you around this problem.

I do not understand this. If there is higher inharmonicity on a spinet, it will be at the lower notes, not the higher ones, where both the string lengths and thicknesses are similar to those in larger pianos. Yet these figures seem to show much more inharmonicity in the higher notes than figures previously given, unless you are measuring something entirely different than what was measured before.

Is this tuning fork steel or aluminum? I think my fork was calibrated at 68° F, so body temperature would be too warm, just as freezing would be too cold. But it is steel and does not change as much as an aluminum fork. In any case, this is not rocket science, so deviations of a fraction of a hertz from A 440 does not matter.

Read the chart. There is only one string being measured: C4. I don't doubt that the bass strings have extremely high inharmonicity, I can hear that when I tune them.

My fork is a high quality steel one. At room temperature, it reads about 2 cents sharp. at about 85 or so by putting it in my pocket, it reads perfectly. I could alter it if I wanted to but I rarely use it since I have an ETD.

The chart means nothing to me. Does the last line mean that your ETD is measuring something it thinks is C8 when you are playing C4?

The machine would be measuring the 16th partial of the vibrating C4 - which corresponds to C8

How would it distinguish between the 16th and 17th partial? I am not convinced that it could read either.

Yes. Playing C4 the machine measuring is set to C5 or C6 or C7 or C8. The machine begins with the theoretical as the starting point C5=523hz C6=1046hz C7=2052hz C8=4104hz. By measuring how many cents it takes to stop the lights/strobe whatever one can determine where the partial is sitting.

Your math is wrong (440*2*1.08 = 950.4). You are actually mixing cents and hertz and oversimplifying. The correct math would be

fn = f0n(1+Bn^2)^0.5
f(2nd partial of A4)=(440*2)(1+B*2^2)^.5
f(A5)=880(1+B*4)^.5

where fn is the frequency of the n-th stretched partial, f0 is the fundamental frequency of the ideal string, and B is the inharmonicity coefficient. At this range of the piano B is likely to range between 0.001-0.005. To explain B gets into Young's modulus etc which I haven't studied enough to be able to simplify.

Check out http://lib.tkk.fi/Diss/2003/isbn9512263149/article4.pdf
http://tinyurl.com/2u2byq

Your ear automatically matches the partials when you do your aural checks. The ear can not hear partials past the 6th partial individually. Higher partials are all blended together.

Your tone generator is just not the proper device for comparison because it is not actually generating a tone at the required frequency. If you went to an audiology lab to have your hearing tested they use a completely different type of tone generator which would in fact be adjustable to any hz setting required.

My math was not wrong, his explanation was not clear. I just used that equation to demonstrate.

As for my tone generator, it is as accurate as an electronic tuning application on a laptop computer. It is the same equipment.

It is a much different thing from a mechanical perspective to measure a frequency with a small transducer than to produce a frequency with a single transducer. Your computer can measure the wave much easier than it can produce a sinusoidal replica.

That means that you cannot trust the pitch of computer-generated MIDI. I doubt that.