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Basics of Fundamental Frequencies, Formants, and the Singer's Formant

[Editor's Note: Originally, this post was a reply to a ChoralNet Forum request from Marisa Makowski, an undergrad student from an unnamed college. Here's the key part of her post: 
“Rereading some of my vocal pedagogy books and I'm having a really hard time understanding the Singer's Formant…. From what I am reading it's a specific ring that occurs at a certain frequency. But isn't the higher the frequency the higher the pitch? So you can't have the 'ring' unless you are able so sing the pitches that lie within those frequencies? And how does frequency change with vowels?...I feel dumb for not understanding, but every source I look at says the same thing.”]
Dumb does not exist in any human being.  Period! 
After reading your post, dear Marisa Makowski, I’m prepared to declare to the world the following ABSOLUTE TRUTH
You are, indeed, a human being (cue the ‘duhhh, no kiddin’ track). 
So, therefore, ‘dumb’ does not exist in you and never has, and…there never have been any ‘dumb questions.’  Evidence?  You have publicly declared that you do not know everything, AND a leading Wisdom-Freak of the Ages has declared: 
The admission of ignorance is the beginning of knowledge
Congratulations, Marisa!  You are hereby declared to be an official Knowledge Freak!  This is a BIIIIG honor!  Welcome and enjoy.  Now go to the website and get your T-shirt.
[All seriousness aside, now, Marisa]
Your confusion about frequencies, formants, and ‘ring’ leads me to believe you may need some information about those three concepts.  [And believe me, at least 50%-plus-one of the choral conductors in the world are confused or incompletely ‘knowledged’ about the interrelationship of those three concepts (my estimate), as was I into my late 40s (as are maaaany singing teachers, speech pathologists, and ear-nose-throat physicians).] 
As Charles Livesay, Brian Holmes, Ernie Scarbrough, and Kentaro Sato have written or implied, I’m pert dang sure you know that with every single pitch that human vocal folds produce, they create a fundamental frequency (abbreviated as f0) and a series of what are often referred to as overtone frequencies (frequencies that are produced simultaneously with, but higher than, a fundamental frequency). 
To get clarified about the interrelationships between frequencies, formants, and ‘ring,’ we gotta put into memory some basic knowledge about how voices are made and how they are ‘played’ in skilled, expressive singing.  That ride starts now, and will come in this Part I.  The next installment will be posted soonly…it’s more than halfway done.  I’ll also post these Parts for members of the ChoralNet Community, Voice Education for All Types of Choral Singing.
In order for perceivable sound to be created, a physical object must be set into some kind of ‘back-and-forth’ vibratory motion.  In the knowledge domain of acoustic physics, there are three basic categories of vibratory motion: chaotic motion (perceived as noise), simple harmonic motion, and complex harmonic motion (here, harmonic refers to vibratory motions that, in essence, are patterned and regularly timed).
In complex harmonic vibratory motion, a physical object (surrounded by air) vibrates, and when it does so, it creates chain-reactions of higher-pressure to lower-pressure wave-like motions within the air molecules that surround it.  These air-embedded chain reaction waves radiate away from the vibrating object in all directions and they have been labeled as sound pressure waves.  The shorter expression issound waves
If you could see air-embedded sound waves in an open space, they would resemble the waves created in a still pool of water when a pebble is dropped into the middle of it.  Sound waves radiate away from an original vibrating source, within its ‘pool’ of surrounding air molecules.  [Just So You Know (JSYK):  In sea level units with dry air, and an air temperature of 68ºF (20ºC), the speed of sound (waves) is 1,125 feet per second (767-mph, 1,235 km/h).]
[Also JSYK:  When complex sound pressure waves impact on the two eardrums of a human being, a series of energy transductions begins.  First, all the sound wave patterns are transduced into corresponding vibratory motions in the eardrums, from which they then travel through the middle ear bones and then the inner ears’ structures (the cochleae in both ears)through, in the hair cells of the cochleae, the vibratory motions are transduced into corresponding nerve impulses that are then conducted through neurons of the vestibulocochlear nerve (8th cranial nerve) into the central nervous system’s brainstem.  Several ‘relay nuclei’ later, the ‘edited’ nerve impulses arrive in the primary and secondary auditory cortices and then on to other cortical areas for ‘interpretation’ and possible action decisions.  All of this neural activity happens in hundreds of milliseconds of time.  Interpretations and possible actions are, of course, influenced by stored past experiences.  We human beings are awesome creatures, aren’t we?]
A musical example of simple harmonic motion would be a tuning fork; it only vibrates at one single frequency (no overtones).  An example of complex harmonic motion would be the vibration of piano strings.  Between their attachment points, each single piano string vibrates as a whole string (fundamental frequency, abbreviated as f0) and in halves of itself (f1), thirds of itself (f2), fourths of itself (f3), and on and on.  All of those vibratory frequencies are produced at the same time, with each section of the string producing patterns of higher and higher frequencies of vibration (the overtones), and all of them together are referred to as the harmonic series.  Does all that make sense?
Intensityrefers to the amount of acoustic pressure power that a vibrating object generates into all of the component frequencies of the sound waves that it makes.  Intensity variation in thefundamental frequencies (f0)of a sung melody, for example, is related to perceived changes of vocal volume in that melody, be they subtle or obvious (i.e., musical dynamics).  Intensity variation in the overtone frequencies that are produced along with the fundamental frequencies of a whole sung melody contribute significantly to the perceived voice quality or timbre of the tones produced in that melody.
When our human auditory systems listen to vocal music, we almost always hear—in our conscious awareness—the fundamental frequencies that voices produce.  But overtones?  We rarely hear overtones in our conscious awareness.  That’s because the intensityof fundamental frequencies is great enough to pass the threshold of our brain’s conscious auditory awareness.  Our brains, however, do take in and process all the overtones, but they do so outside our conscious awareness.  In a manner of speaking, brains ‘average’ all of the incoming frequencies to produce a perceptionof voice quality or vocal timbre—what a voice sounds like to us. 
[Just So You Know (JSYK):  You also may have read or heard about two other nearly synonymous terms:  harmonics and partials.  While overtones refer to the frequencies that are produced above a fundamental frequency, harmonics refers to all the vibration frequencies that happen simultaneously, including the fundamental frequency.  Partials also refer to allparts or frequency components of a tone that is produced by complex harmonic motion.  So, the fundamental frequency (f0) is the first harmonic or partial.  The first overtone is the second harmonic or partial (f1), and so on.  Voice scientists hardly ever use the term overtones.]
All partials/harmonics of a vocal tone (fundamental frequency and all the overtone frequencies)—along with the different intensities of each one—form what is called a sound spectrum.  Perceived voice quality or vocal timbre in vocal music is derived from the varying sound-wave spectra that happen in the course of a song.  When voices produce increases or decreases of intensity within any of the partials/harmonics of a sound spectrum, the perceived voice quality (vocal timbre) is changed
HERE’S WHERE IT GETS INTERESTING, Marisa!!  [Part II is the next installment.]
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