# Re: Pitch evoked by spectral regularity (inharmonic components) (Chen-gia TSAI )

```Subject: Re: Pitch evoked by spectral regularity (inharmonic components)
From:    Chen-gia TSAI  <gia(at)SNAFU.DE>
Date:    Wed, 19 Feb 2003 20:29:55 +0800

-----Original Message-----
From: beauchamp james w [mailto:j-beauch(at)ux1.cso.uiuc.edu]
Sent: Thursday, February 20, 2003 12:53 AM
To: gia(at)SNAFU.DE
Cc: auditory(at)lists.mcgill.ca
Subject: Re: Pitch evoked by spectral regularity (inharmonic components)

>My question always is "what is the pitch of a square wave with a missing
>fundamental?".

My study really started with some complex tones like “a square wave with a missing fundamental”.
Unlike a square wave, the stimuli have five equal-amplitude harmonics.

{1400, 1800, 2200, 2600, 3000}, f = 350 Hz
{1800, 2200, 2600, 3000, 3400}, f = 435 Hz
{2200, 2600, 3000, 3400, 3800}, f = 525 Hz

The empirical equation f = e/4 gives f = 350, 450, 550 Hz.

>three frequencies, 800+d, 1000+d, 1200+d.
>The virtual pitch frequency heard is approximately (1/3)*((800+d)/n + (1000+d)/(n+1) + (1200+d)/(n+2)), where n = 4 or 5.

My results are in line with this formulas. Thank you very much for indicating the effect of “Shift of Virtual Pitch”.
I have not read [Schouten et al 1962], but this formulas and the empirical equation f = e/4 appear to support the predominance of the 4th (perhaps also 5th and 6th) harmonics in pitch extraction, but not support that the virtual pitch is simply extracted from the envelop of the beating resulting from adjacent components (sometimes they are unresolved harmonics in the auditory channel).

Chen-Gia Tsai
gia(at)snafu.de

```

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