Bloody Hell!
The post above containing the wrongful & spurious claim

"A Helmholtz resonator has no associated overtone series."

which as an apparent indisputable 'statement of fact' now has been referenced by Google and the misleading crap will now continue to spread, thus misleading others!!!

Just to clear things up, if anybody is actually following this thread!, here are some dispellings of the oversimplification 'lies to children' approach...

"Overtones is mental garbage"

From Wikipedia

QUOTE
In Hermann von Helmholtz's classic "On The Sensations Of Tone" he used the German "Obertöne" which was actually a contraction of "Oberpartialtöne", or in English: "upper partial tones". However, due to the similarity of German "ober" to English "over", a Prof. Tyndall mistranslated Helmholtz' term, thus creating "overtone." This created unfortunate confusion, adding an additional term that is somewhat unclear and has unfortunate mystical connotations. This has also led to the idea that if there are overtones, perhaps there are "undertones" - which is a term sometimes confused with "difference tones". In contrast, the correct translation of "upper partial tones" does not have any problematic implications. Alexander Ellis, on pages 24–25 of his definitive English translation of Helmholtz, makes clear all the unfortunate confusion of this mistranslation which entered common usage. Ellis strongly suggests the avoidance of the term "overtone". [1]
UNQUOTE


Some more useful and correct quotes follow:

An overtone is any frequency higher than the fundamental frequency of a sound. The fundamental and the overtones together are called partials. Harmonics are partials whose frequencies are whole number multiples of the fundamental (including the fundamental which is 1 times itself.) These overlapping terms are variously used when discussing the acoustic behavior of musical instruments. Due to a translation error in its coining, Alexander J. Ellis strongly suggested avoiding the term overtone in deference to upper partial (simple) tones.[1] (See etymology below.)
Vibrational modes of an ideal string, dividing the string length into integer divisions, producing harmonic partials f, 2f, 3f, 4f, etc. (where f means fundamental frequency).

When a resonant system such as a blown pipe or plucked string is excited, a number of overtones may be produced along with the fundamental tone. In simple cases, such as for most musical instruments, the frequencies of these tones are the same as (or close to) the harmonics. An example of an exception is a circular drum, whose first overtone is about 1.6 times its fundamental resonance frequency.[2] The human vocal tract is able to produce a highly variable structure of overtones, called formants, which define different vowels.

An overtone is a partial (a "partial wave" or "constituent frequency") that can be either a harmonic partial (a harmonic) other than the fundamental, or an inharmonic partial. A harmonic frequency is an integer multiple of the fundamental frequency. An inharmonic frequency is a non-integer multiple of a fundamental frequency.

Some musical instruments produce overtones that are slightly sharper or flatter than true harmonics. The sharpness or flatness of their overtones is one of the elements that contributes to their unique sound. This also has the effect of making their waveforms not perfectly periodic.

Musical instruments that can create notes of any desired duration and definite pitch have harmonic partials. A tuning fork, provided it is sounded with a mallet (or equivalent) that is reasonably soft, has a tone that consists very nearly of the fundamental, alone; it has a sinusoidal waveform. Nevertheless, music consisting of pure sinusoids was found to be unsatisfactory in the early 20th century.

Most oscillators, from a guitar string to a bell (or even the hydrogen atom or a periodic variable star) will naturally vibrate at a series of distinct frequencies known as normal modes. The lowest normal mode frequency is known as the fundamental frequency, while the higher frequencies are called overtones. Often, when an oscillator is excited by, for example, plucking a guitar string, it will oscillate at several of its modal frequencies at the same time. So when a note is played, this gives the sensation of hearing other frequencies (overtones) above the lowest frequency (the fundamental).

Timbre is the quality that gives the listener the ability to distinguish between the sound of different instruments. The timbre of an instrument is determined by which overtones it emphasizes. That is to say, the relative volumes of these overtones to each other determines the specific "flavor" or "color" of sound of that family of instruments. The intensity of each of these overtones is rarely constant for the duration of a note. Over time, different overtones may decay at different rates, causing the relative intensity of each overtone to rise or fall independent of the overall volume of the sound. A carefully trained ear can hear these changes even in a single note. This is why the timbre of a note may be perceived differently when played staccato or legato.

A driven non-linear oscillator, such as the human voice, a blown wind instrument, or a bowed violin string (but not a struck guitar string or bell) will oscillate in a periodic, non-sinusoidal manner. This generates the impression of sound at integer multiple frequencies of the fundamental known as harmonics. For most string instruments and other long and thin instruments such as a trombone or bassoon, the first few overtones are quite close to integer multiples of the fundamental frequency, producing an approximation to a harmonic series. Thus, in music, overtones are often called harmonics. Depending upon how the string is plucked or bowed, different overtones can be emphasized.

However, some overtones in some instruments may not be of a close integer multiplication of the fundamental frequency, thus causing a small dissonance. "High quality" instruments are usually built in such a manner that their individual notes do not create disharmonious overtones. In fact, the flared end of a brass instrument is not to make the instrument sound louder, but to correct for tube length "end effects" that would otherwise make the overtones significantly different from integer harmonics. This is illustrated by the following:

Consider a guitar string. Its idealized 1st overtone would be exactly twice its fundamental if its length was shortened by ½, say by lightly pressing a guitar string at the 12th fret. However, if a vibrating string is examined, it will be seen that the string does not vibrate flush to the bridge and nut, but has a small "dead length" of string at each end. This dead length actually varies from string to string, being more pronounced with thicker and/or stiffer strings. This means that halving the physical string length does not halve the actual string vibration length, and hence, the overtones will not be exact multiples of a fundamental frequency. The effect is so pronounced that properly set up guitars will angle the bridge such that the thinner strings will progressively have a length up to few millimeters shorter than the thicker strings. Not doing so would result in inharmonious chords made up of two or more strings. Similar considerations apply to tube instruments.

In barbershop music, the word overtone is often used in a different (though related) way. It refers to a psychoacoustic effect in which a listener hears an audible pitch that is higher than, and different from, the four pitches being sung by the quartet. This is not a standard dictionary usage of the word "overtone." The barbershop singer's "overtone" is created by the interactions of the upper partial tones in each singer's note (and by sum and difference frequencies created by nonlinear interactions within the ear). Similar effects can be found in other a cappella polyphonic music such as the music of the Republic of Georgia.

String instruments can also produce multiphonic tones when strings are divided in two pieces. The most developed instrument for playing multiphonic tones is the Sitar in which there are sympathetic strings which help to bring out the overtones while one is playing. The most well-known technique on a guitar is playing flageolet tones. The Ancient Chinese instrument the Guqin contains a scale based on the knotted positions of overtones. Also the Vietnamese Đàn bầu functions on flageolet tones. Other multiphonic extended techniques used are prepared piano, prepared guitar and 3rd bridge.

Overtone singing, also called harmonic singing, occurs when the singer amplifies voluntarily two overtones in the sequence available given the fundamental tone he/she is singing. Overtone singing is a traditional form of singing in many parts of the Himalayas and Altay; Tibetans, Mongols and Tuvans are known for their overtone singing. In these contexts it is often referred to as throat singing, though it should not be confused with Inuit throat singing, which is produced by different means.

A similar technique is used for playing the Jew's harp: the performer amplifies the instrument's overtones by changing the shape, and therefore the resonance, of their mouth.

~~~~~~~~~~~~~

Most of the musical references you will see relating to 'harmonics' and 'overtones' refer only to the oversimplified 'lies to children' approach involving vibrating lengths of air or solids, in which the 2x 3x etc stuff applies. None of this applies to 'Helholtz resonators' (HRs) which involve ONLY the property of a resonating fluid VOLUME.

Consider the 'embouchure flute', you not only have the HR at the cork end, you also have one at the other end, which is also subject to 'fluid flow' modifications. Both of these HRs are also acting simultaneously with the 'vibrating length of a fluid' model.

If the volume of the cork end HR is X, then the volume of the other HR end is ~ nX. Since the fundamental Frequency of the cork end HR is F, the Fundamental Frequency of the other HR end approaches ~ nF.

Since I don't have one in front of me to measure, lets just say x is (for simplicity, and for those who like to think with real numbers instead of algebra variables!) 10.... so you see that the longer HR starts to impose significant effects on the modifications of the higher produced tones and partials!


"I can't imagine why you would want to invoke Helmholtz resonators as components of a flute. These days anybody who wants to get detailed answers can just simulate the flute directly, doing numerical solution of the relevant partial differential equations"

... and those simulations make assumptions that compensate for the HR effects by making guesses and including constants... ignorance is not bliss!

I suppose you as an expert are to trying tell me that Ocarinas produce NO PARTIALS, ie produce ONLY SINUSOIDAL WAVEFORMS?