Piano World Home Page Forums Our Most Popular Forums Piano Tuner-Technicians Forum Anybody Deliberately Tune Pure Twelfths?

Oh yes, I meant to say 19ths. But still, is that enough stretch for the brightest possible high treble? Or would wide 19ths be possibly called for? I don't know - I was just wondering if you have any tricks to doing that last octave.

Let me know how your different temperament areas work out when you get a chance to try them. I am curious to find out whether that will do the trick.

Thanks,
-Erich

Erich:

“Brightest possible stretch” would be just before you hear KERBANG! Reminds me of a riddle an Amishman once gave me. “How do you know when it is time to plant corn?” My best answer was to plant corn in Columbia County a week before they plant it in Sullivan County (about 90 minutes north.) The real answer is, now you have to understand that Amish humor is a little different, “You know it is time to plant corn when you get up in the morning and your wife’s bare butt is sticking out from under the covers and you put your hand on it and it isn’t cold. Ahgagagaga!” Believe it or not, it took me three days to get the answer out of him. See, I wasn’t married at the time so he figured I didn’t need to know...

Anyhoo, I was curious about the difference in stretch so I crunched some simple scenarios. If the F5/F6 octave is tuned as a 50/50 compromise between a 2:1 and 4:2 octave the difference between C8 being tuned as a pure 12th and a pure 19th is about 38 cents on a Baldwin Spinet and 30 cents on a Baldwin L with the 19th being the greater stretch.

What I do isn’t what I would call a trick, just comparing (as much as I can…) what is heard objectively with what is heard subjectively. As I ascend into the upper octaves, I will play the major tonic in the temperament and then the note I have tuned. Subjectively, I will listen and decide how it sounds. If I think it is definitely flat, I will objectively temper the 12ths a bit wide.

But the whole decision process does get “fuzzy”. I think somewhere in the top octave I am hearing the 12ths more melodically than harmonically (even though the notes are played simultaneously), which is where, I think, the extra stretch is needed. Listening to the chords and tuned notes a bit lower down is important, I think, to get a feel of how the treble is really sounding so as to keep the same quality going up. By melodically: I mean subjectively, separately pitch-wise. By harmonically: I mean objectively, beating interval-wise.

Maybe in the top octave I hear what I believe it should sound like if I actually could hear the beats when playing the 12th. When I am very honest with myself, it needs more stretch melodically than what I come up with harmonically.

Thanks Jeff for the info.. and the joke! Yes, you kinda have to be married to fully appreciate that one.

I had no idea that the 12ths and especially 19ths created that much stretch. See, that's why I'm glad to have forum fellows like you to keep me educated on such matters! I'm thinking now that the 19th would make a sufficiently bright treble.

One more question - if you deliberately widen the 12th a little, will the 19th be wide of pure automatically, or not quite yet, due to its increased size? Also, I would assume it would not be necessary to make the 19th wide of pure (probably at that point, the single octaves might beat too much). I guess I am wanting to know the limit, so that I can approach it, but not pass it (just for peace of mind). In other words, I want the top C to be a sharp C, but I don't want it to be a C#.

-Erich

Erich:

Thanks for the words of appreciation. You are very welcome!

This Forum has been great for me, too. There are many things about tuning theory that have been posted here that were contrary to what I was hearing. When I learned how to crunch the numbers I found that my ears were correct. But the value in all this was in learning what stretch is really about and different ways of controlling it. The latest factor is the 19ths, which I have only very recently looked at. I have to wonder just how valuable they are for aural tuning, though. They are ok for a check I suppose.

Anyway, to answer your question, it depends on where in the scale as to the difference between 12ths and 19ths. It should be apparent that with no iH 12ths would produce a tiny bit more stretch than 19th. But with iH 19ths produce more stretch. And the more iH the more stretch, such as in the high treble. So for much of the piano the 12ths would be tempered a little wide to produce pure 19th. But higher in the treble the 12th would have to be tuned very wide. More than what would be considered tempering.

Where the limit is, is of course subjective. The 12th is a totally unique interval because of the combination of its span and the partials involved. No they do not have to be pure, and I am learning that in some cases they should not be. If I was to give a rule of thumb, and that is what is seems that you are looking for, if the 12ths are obviously beating the stretch is wrong. Of course in the very high treble where you can’t hear beating I suppose almost anything goes.

A few years back, when I was expanding with pure sounding octaves, I found that if I played an ascending G arpeggio everything sounded fine. But if I played a descending G arpeggio and then repeated the top G7, it sounded very flat. In fact G#7 would sound much better. That does not happen with 12ths, but I think it is best to widen the 12ths a bit in the top octave. 19th would do the trick up there, too, I think. It just isn’t easy aurally.

There is an odd thing about the relationship of 12ths and 4:1 double octaves. If pure 12ths are tuned to beat wide at the same speed as the 4:1 is narrow (mindless octaves), they switch places in the last half octave in the treble. It is like the 12ths “run out of steam” or something. This is just another indication that in the last octave 12th may not produce quite enough stretch.

I probably won’t post again until Monday. I have some concrete work to do. They never found Jimmy Hoffa, and they won’t find the latest Amishman that told me a lousy joke either. Bwhahahaha.

Thank you for the advice - I will try widening he 12ths a bit (but not obviously beating) in the last octave and see how it sounds. That's interesting about the 12ths and double octaves trading places at the top. Ah, the wonders of mathematics and music (and the wonders of how we can't make music fit perfectly with the mathematics).

Something is wrong. The 3:1 is narrow and the 4:1 wide (and equal beating).

Kees

Yes, I was thinking the same thing - I think that's what Jeff meant to say.

Yep, I got it bass ackwards, sorry.

But what do you mean by they (3:1 and 4:1) "switch places"? Don't they just get further apart?

Kees

No. In the 7th octave they both become beatless and then the 12ths are wide while the 15th are narrow. Not that I can hear it. The interesting thing is that because this relationship is based on the inferred P4, it means in the 5th octave the P4s become narrow. But again, not that I can hear it. There are some graphs and discussions about it in the CHAS Topic.

Something is wrong somewhere. For example if I compute a tuning curve (using tunelab) based on pure 4:1 everywhere, the 3:1 are narrow, the higher up you go the narrower they are.

Why do you think if you tune pure 4:1 at some point the 3:1 becomes less narrow and eventually pure?

Kees

Two reasons, and I could be wrong. But what do you get when tuning 12ths or mindless octaves. There may be a difference.

First, that is what I calculated with my simulation. Now there is a difference between the calculation in my simulation and Tunelab. I use the original square of the partial numbers as in Young's paper while Tunelab uses an empirical table.

Second, when you really listen to the 4ths, the beatrate less than doubles each octave. It seems much more likely to me that they would reach a greatest value and then start decreasing and eventually becoming narrow than they would find a greatest value and stay there. It is even less likely to me that the beatrate progression would be a straight line. NOTHING in tuning is naturally a straight line! And if the 4ths become narrow the 12ths and 15ths must "switch places."

All the beatrates I have investigated show a curve that crosses the just intonation line when they less than double in each octave. M3s and M10s, have this character. A am sure the M17 would too if the piano had more notes. Ever wonder how the M17 is useful up to the 7th octave? If the M3 family doubled each octave, or even were on a 4:5 CM3 curve (which is less than double each octave…), they would be unusable.

Perhaps the iH curve has more to do with it than we think. Here are all the A's for the ideal curve that I used. It was based on based a Walter Console Verituner file that was kindly given to me and is similar to iH curves from Pscale for that size piano:

A0 0.27
A1 0.19
A2 0.13
A3 0.20
A4 0.52
A5 1.37
A6 3.60
A7 9.43

Enjoy!

With those particular IH constants 4:1 is wide when 3:1 is pure and 3:1 is narrow when 4:1 is pure. In an imaginary tuning with pure 3:1 everywhere and 4:1 everywhere for the second. And of course with mindless octaves 3:1 is narrow and 4:1 wide.

One of us is doing something wrong, could be me. The tunelab constants are not far from Young's ideal values. I guess it would be hard to show me your simulation.

Kees

Doel:

Here is a box.net link where you can look at the results:

http://www.box.net/shared/rxb631v2yz

And here is the box.net link for the simulator itself, but it does not include this particular simulation or the code. It requires Access 2003 or Access runtime to run it:

http://www.box.net/shared/ky02x9rneq

Or if you have Access 2003 I could send you the .mdb project for my simulator which includes this particular simulation. This version includes all the VBA code.

If interested, I will PM you an email address for further correspondence. But please, no gossip.

Thanks Jeff. Unfortunately I don't know how to use Access.

I did find out why we get different results. It is (as you suggested) because you used Young's constant for the 4th harmonic (16), whereas I use the Tunelab value which is 13.18. If I run my code with 16 instead of 13.18 indeed the 4th gets narrow.

Because the value 13.18 is obtained from measurements from actual piano's I would think it's closer to reality. Surprising this has such a large effect.

Kees

You wouldn't need to know how to run Access. It acts like an independent application. But why would you, now that we know how significant this error is! Thanks so much for pointing it out. I did what I could with what I had at the time. I will have to figure out how to remove the Box.net files.

(I continue to be disturbed that the inventor of Tunelab confirmed and did not correct me in a post on this Forum about the use of Young's original equations. I suppose that in the context of the exchange he may have been trying to avoid confusing the issue. But I sure would have appreciated the inclusion of the word "about" or "approximately" in the discussion!)

I wonder how big the variance of 13.18 (or any of the significant modified Young's values) is in the real world.
Or is there some specific physics behind this value?

Robert Scott, do you know?

Kees

Doel:

The difference depends on the iH.

16.00 (Young's) - 13.18 (Emperical) = 2.82 (Difference). So if the iH was 1.0, the difference in the 4th partials would be 2.82 cents.

Maybe you were asking about something else?

Yes. The partials are located at c(k)*iH with iH some single parameter. According to the ideal bar model c(k)=k^2 but empirically the constants c(k) are a bit different, in particular c(4) = 13.18 on average. I just wonder what the variance across piano's is of the best fit for c(4) (and the others).

Kees

Hey guys,

Just tried the 12th tuning on a 5'11 Wurlitzer. I am happy to report it worked out great! I was worried getting out of the temperament area because the D3 is where the two copper wound strings begin on this piano. But when I tuned it to the A4 above as a beatless 12th, it fit right in. D-F# M3rd was progressive, D-B M6th was, and even D-F m3rd was too.

I used the hybrid temperament-area method (Stebbins + Marpurg) that I posted on this forum a couple of weeks ago.

Jeff, I found that using the Stebbins method (that uses the C#3 which on most pianos is below the break) lets me ease out of the temperament area and into the 12th-method area more smoothly. This method gave a noticeably faster F3-A3 right from the get-go, so that by the time I got to the D3-F#3 check, it fit right in. The D3-F#3 wanted to beat faster than I would normally put it using other methods (like you encountered), but the slightly faster-than-normal F3-A3 above it(which initially uses a wound string to help arrive at its speed), allowed a smooth transition with no fussing. Maybe this is part of the answer to setting a temperament (or temperament area in my case) across the break that you had mentioned - I don't know. Just some thoughts.

Anyway, I love using this method! It is very easy to hear 12ths, even on poor pianos. Since I don't own a 12th-spanner however, I should add that while expanding, I tune an octave by ear with my left hand pinkie, while using the tuning hammer with my right. This gets the bass note to be tuned close enough in the ballpark. I then let go of the tuning hammer to play the 12th with both hands (maybe Rach had a 12th hand-span, but I sure don't ), and I check that it is beatless, and adjust the bass note if necessary. This applies to the treble as well, especially high treble where 19ths are used. Tune a quick octave, check 12th, tune the next quick octave, check 12th, and so forth. Once I find my rhythm, it goes pretty quickly.

Please let us know how your recent 12th tunings (Kimball console?) turned out.

-Erich