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Originally Posted by pyropaul
Originally Posted by prout
Zero is an integer, and an important integer. It defines unity.

1^0=1
2^0=1, 2^1=2, 2^2=4
3^0=1
4^0=1
...

This implies that the frequency of the first harmonic, if you will, is the fundamental frequency.

Great stuff people. Continue.

Let's also not forget subharmonics! These also contribute to the perception of the (missing) fundamental. For example, the difference tone of the 2nd and 3rd partials is a subharmonic one octave below the fundamental. These are produced in the ear by the non-linear way the eardrum works (if I remember correctly) and the difference frequencies are easier to hear than the summation ones. Fun to listen out for!
Excellent point and you've reminded me that difference tones played a large part in that telephone experiment I mentioned earlier.


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Originally Posted by N W
This thread reminds me of an experiment that was done when I was young. I'm searching my old IMIT records for details but basically it was that in those days the telephone system had a bandwith to only cope with speech. Charcoal microphones..anyone else remember them? Anyway this scientist musician wrote and recorded a piece of music to be sent over the telephone wires, constructed in such a way that bass notes could be heard even though they could not possibly have been transmitted and carried by the phone system. I just can't remember whether it was in UK or USA.

Glenn Gould used the telephone system to work with his sound engineer remotely. They bypassed the Microphone and loudspeaker of the phone system and used the plain wires, which AFAIR didn't have frequency range limitations that would distort music.

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The ear, due to its non-linear response, easily fills in missing sub-harmonics, as well as the sum and differences (the products in reality) of two or more frequencies. We hear this in the organ with an ersatz 32 foot pedal note using the technique as pyropaul mentions. We hear this also playing, for example, C5 and E5 simultaneously, which causes our brain to hear G4.

Just a point here. The organ pipes produce true harmonics, so octaves do not need to be stretched.

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Old reply that I wrote a couple days ago but apparently never pressed submit and just found in a stale browser tab (so a bit off topic with recent comments):

Nomenclature: I have to think about it every time. Harmonic above fundamental = 2nd harmonic = 2nd partial = 2nd mode = 1st overtone. (I think that's right).

Fundamental decaying faster: I usually think about it in terms of the string having just a little energy in its first mode to begin with, and that energy is dissipated really quickly. The energy in the 2nd harmonic is siphoned off more slowly, so it stays stronger for longer. Maybe it's more complicated than that, but it's enough to satisfy me. (When you get down into the lower bass the first couple of harmonics pretty much disappear entirely.)


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Old Square,

You are right on the money! Astin-Weight. Negative IH. How did they do that?

Peter Grey Piano Doctor

Last edited by P W Grey; 11/22/21 09:55 PM.

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Originally Posted by P W Grey
Old Square,

You are right on the money! Astin-Weight. Negative IH. How did they do that?

Peter Grey Piano Doctor

Some time ago I wrote a software that calculates inharmonicity for bass strings with non uniform mass distribution, turns out that if you add additional mass just close to the ends (thicker wrap on ends) inharmonicity of first 10-15 partials will decrease, in extreme case it can get negative, but inharmonicity of higher partials will shot up always.

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One can lower the inharmonicity of a bass strings measurably by making the end waste lengths shorter - extending the winding closer to the agraffe at one end and the front bridge pin at the other end. That is a much simpler and more controllable way to lower inharmonicity. Many bass string makers use 1" waste lengths, a safe waste length would be around 12 mm. or 1/2". Closer than that, and you run the risk of manufacturing errors allowing the winding to crowd the agraffe or bridge pin. We strive for uniformity in the making of bass strings. In practice, making the ends of the winding thicker may introduce overtones that we do not want to have. Clean bass strings are a tough nut to crack even for the better bass string makers.

Decreasing the core diameter will lower the inharmonicity also. This works particularly well in the lower monochords. For short pianos, decreasing the core size and also the outer wrap can also yield surprising results.


Does your software calculate inharmonicity for uniform mass distribution? It would be interesting to compare your designed bass string with one with longer windings.


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Originally Posted by AWilley
Nomenclature: I have to think about it every time. Harmonic above fundamental = 2nd harmonic = 2nd partial = 2nd mode = 1st overtone. (I think that's right).

Yes, that sounds right. Thanks, Anthony.

I think I wrote something different earlier in the thread (used overtone numbers for harmonics) - will check, and correct if needs be.

[Edit: yes, I treated harmonics and overtones as one and the same. Apologies for that. Can't edit that post anymore, though.]

Last edited by Mark R.; 11/23/21 09:22 AM. Reason: given in post

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Interesting diversion from my OP so keep it up.

As an aside perhaps I will repeat the etd test and instead of the mid range notes I mentioned I will use A0 and A1. I have constructed 3ft high transmission-line speakers that are tuned to have their lower frequency at 29Hz. Not as far as the 27.5Hz of A0 but very close.

The fundamental of A0 is not heard by all human ears but a partial thereof is heard. It will be interesting for me to find out how the ETD shows the decay of this fundamental and first partial.

I'm impressed by those who claim to hear the 16Hz fundamental of a Bosendorfer Imperial grand!


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Originally Posted by Beemer
Interesting diversion from my OP so keep it up.

I have constructed 3ft high transmission-line speakers that are tuned to have their lower frequency at 29Hz. Not as far as the 27.5Hz of A0 but very close.

Not the Capellmeister transmission line speaker design?

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Originally Posted by WilliamTruitt
One can lower the inharmonicity of a bass strings measurably by making the end waste lengths shorter - extending the winding closer to the agraffe at one end and the front bridge pin at the other end. That is a much simpler and more controllable way to lower inharmonicity. Many bass string makers use 1" waste lengths, a safe waste length would be around 12 mm. or 1/2". Closer than that, and you run the risk of manufacturing errors allowing the winding to crowd the agraffe or bridge pin. We strive for uniformity in the making of bass strings. In practice, making the ends of the winding thicker may introduce overtones that we do not want to have. Clean bass strings are a tough nut to crack even for the better bass string makers.

Decreasing the core diameter will lower the inharmonicity also. This works particularly well in the lower monochords. For short pianos, decreasing the core size and also the outer wrap can also yield surprising results.


Does your software calculate inharmonicity for uniform mass distribution? It would be interesting to compare your designed bass string with one with longer windings.

Of course compensating long unwound ends with that doesn't make much sense, they have to be short to begin with, 10mm is usually enough, below that there isn't much difference from fully wounded string.


With my software I designed and made a string that have negative inharmonicity, then tested it with very crude setup, string was mounted on thick wooden board with something resembling a bridge (with very special bridge pins) here are the result:


[Linked Image]

Core is roslau 0.9mm and wound is 0.55mm enamelled copper wire (I was experimenting with such wire, not important in inharmonicity context)

From "bridge" to "agraffe":
8mm unwound, 30mm double wound (3.2mm diameter), 1000mm single wound (2mm diameter), 5mm unwound

Double wound section doesn't look nice, I haven't figured out a good way to do it yet.

There is a sound file: https://gofile.io/d/YomaXm

Calculated by my software and measured inharmonicity of that string for each partial:

[Linked Image]

(only 12 partials measured because higher ones were very short and weak, there is something funny with 4th partial going on but crudeness of this setup may play a role here)


And here are measured frequencies of the first 12 partials:

79,62
159,17
238,65
317,32
397,16
476,01
554,44
633,05
710,70
788,53
865,61
944,32

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Ok I've added all partials that are above noise floor and technically can be measured:

[Linked Image]

I'm surprised myself how well it matches the prediction.

Here are frequencies:

79.62
159.17
238.65
317.32
397.16
476.01
554.44
633.05
710.70
788.53
865.61
944.32
1021.0
1105.0
1185.67
1266.72
1351.0
1434.9
1519.05

Last edited by ambrozy; 11/25/21 12:17 AM.
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[quote=Beemer]

As an aside perhaps I will repeat the etd test and instead of the mid range notes I mentioned I will use A0 and A1. I have constructed 3ft high transmission-line speakers that are tuned to have their lower frequency at 29Hz. Not as far as the 27.5Hz of A0 but very close.
/quote]

Which design did you use? It's been years since I looked at TL designs (17 years or so since I went down the route of buying a set of PMC IB1S actually). There were some companies offering DIY cabinets which allowed you to use drivers of your choice but also would build them for you if you didn't want the hassle of DIY. I don't recall seeing any designs that went down that low. The PMCs I have roll off quite sharply below 40Hz. I'd love to know which design you have.. the urge to buy new speakers hit me recently :P.

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Originally Posted by David Boyce
Originally Posted by Beemer
Interesting diversion from my OP so keep it up.

I have constructed 3ft high transmission-line speakers that are tuned to have their lower frequency at 29Hz. Not as far as the 27.5Hz of A0 but very close.

Not the Capellmeister transmission line speaker design?
As per:
http://quarter-wave.com/TLs/TL_Anatomy.pdf


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These ones:

[Linked Image]


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