Harmonics

(Fig 1.)

Note	Frequency	Harmonic - Interval
F#4	370 Hz	Fundamental
F#5	740 Hz	First Harmonic - Octave
C#6	1110 Hz	Second Harmonic - Fifth
F#6	1480 Hz	Third Harmonic - Octave
Ab6	1850 Hz	Fourth Harmonic - Third
C#7	2220 Hz	Fifth Harmonic -  Fifth
Eb7	2590 Hz	Sixth Harmonic - Dominate Seventh
F#7	2960 Hz	Seventh Harmonic - Octave

As you can see that harmonics are a specific “multiple” of the fundamental note. While you can hear harmonics (partials) they are most likely to be heavily incorporated into the primary tone. We hear this combined tonal pattern as the timbre of the instrument.

To get a better idea of what we are discussing I will use an mp3 file so you can hear it as well. One of the purest tone generators is a single oscillator synthesizer that uses a simple sine wave. We hear these in the form of electronic devices like the warning bell in an auto when you put your keys into the ignition with the door open. It has low amplitude or no partials. Buzzers are like Saw waves, these waves are made up of  the fundamental plus many higher amplitude partials. In both cases below only one note was played and was the same pitch as well.
 
 *** Click Here To Play Sound Sample ***
 
To visualize the sound graphically I ran the mp3 above through a spectral analyzer and took screen snaps of the resulting graphics.
(Fig 2.)
The first is the sine wave where you see a single peak wave. This is a simple sound note there is no harmonic content.

(Fig 3.)
The second is the saw wave where you can see the fundamental starting at the same point but has multiple waves following it at regular intervals (keep in mind the graphic scale is log not linear). Although you may not care for this tone is it however very rich and complex in tone.

Partials whose frequencies ARE integer multiples of the fundamental are called Harmonic and more pleasant and those that ARE NOT integer multiples are referred to as enharmonic and generally perceived as unpleasant. A single tone may be have harmonic partials and still have enharmonic content as well.

During the Flute Spring gathering in 2007 I gave a little talk about harmonics and used that opportunity to do some of many different flutes by different maker and recording the fundamental pitch and capture the harmonic waveforms for comparison of tonal and visual content. I tried to get a representative sample of flutes that had similar fundamentals. Although we tested many flute of varying pitches for this primer I will display only those of which were F#4.  To prevent any potential negative conclusions by this study alone the particular makers graphic will not be revealed, except my own. The makers were those who’s flutes were available from the interested players at the gathering these included in random order here and below; Light, Gomez, Hebrec, Young, Miller, and myself. If the other makers happen to read this and would like to know which wave form was theirs I would be happy to share that information via private email. I will not however send a response to a general email address such as a gmail, yahoo etc. Readers please keep in mind that the analysis was based on one flute one note and may vary flute to flute by the same maker, however if the maker has a consistent design, the elements effecting harmonics will be present but to varying degree. The samples were normalized to RMS (short definition mean of the wave forms) to allow an apples to apples comparison and minimize the effect of variable player breath pressures. The players did practice blowing with an even breath pressure and coached to achieve an audible comparable volume flute to flute across the sampling. Flutes that are blown too hard can cause any irregularity in harmonics to become even more pronounced. It is like increasing the noise floor of a recording where the hiss becomes more evident. The simple recordings were made using a Mac iBook using Bias Peak editing and mastering software. The Spectrographic images are recorded and freeze framed with Elemental Audio’s Inspector XL spectral analysis module. The Graphics are showing peak values only with the same zoom ratio for all samples. Greater detail is available but for our basic purposes peak values will suffice.
 
To read the chart the X axis is the frequency scale (Hertz) which is logarithmic and the Y axis is the amplitude (Decibels) of the wave form. The white tracings are the actual peaks and valleys of the fundamental and each harmonic interval (left to right). We will concentrate on the fundamental through the 7th Harmonic (eighth peak) but comment on those above in general. While glancing at the graphic one might think they all look alike but the audible tonal differences were profound and as such the detail must be considered. Because the sound is made up of ALL the partials, to gauge the volume of the flute by these graphics one must sum the amplitude of each wave. Without getting into a lot of math you can quickly generalize which might be loud or soft. The other volume consideration is perceived loudness. The overall tone will sound louder with strong well formed harmonics and few distorted or enharmonic waves. These out of place, out of form waves will mask the purer tones like muddy water. The masking might be heard as breathiness or dullness to the ear. Harmonic content is always present in any flute but the amplitude and uniformity of them is dependent on ability of the flute to support and or enhance them. Bottom line the more the wood and design supports resonance the more refined the tone, richer and louder. Keep in mind this may not be the sound you are looking for, you may like a dryer sound, soft breathiness, or a quiet flute. Science aside you must like it. To support resonance the wood must be able to move and should not impede or dampen it.

Here are some considerations:
Wood Density – The softer the wood the more likely it will support resonance the higher the density the less support it will give.
Bore Thickness – The thinner the walls the greater the ability to move hence support, thicker walls are rigid move less so do not support.
Grain Patterns – Straight grain moves in a more uniform way and supports, Figuring, Knots, Inlays, Laminations all impede the resonance and dampen the tone.
Coatings – A totally unfinished flute, not even oiled, with give the greatest support to it resonance while this may be unreasonable it is important to note. The greater the density of any coating(s) inside and out the less support to resonance it will have.
Sound Mechanism – This grouped consideration is actually made up of many contributing sub components but in summary the move artifact free the mechanism the greater the support to harmonic production. More artifacts will mask or distort the harmonics, or introduce enharmonics.
Bore Shape/Diameter/Length - In order that the intervals of the harmonics be correct the air column must also support the formation of the series of waves to produc them. While in the making of the Shukuhaci master flute makers will make subtle changes to the interior of the flute's bore to change the location of where nodes and antinodes will exist. This alows them to tune the harmonics to greater presence, making an otherwise dull flute rich in tone. The opposite can be true as well adjustments to bores outside the optimal size or shape can cause a weakening or mispositioning of the harmonic structure. By example an oversized bore generally decreases harmonic structure while undersized bore will increase it. 

As one plays the scale other factors such as hole placement, hole size, and hole shape come into play but will not be discussed here to keep the primer shorter.
 

This first sample (Fig 4.) is my own White Crow Flute, it is an F# flute made of recovered Chestnut with Rainbow Poplar end caps. It tends to be my go to F# flute when I perform. It has a rich tone and quite even timbre across its range. It is not an optimized tuned flute which makes this more likely with range optimized flutes you sacrifice some of the timbre for the increased range. Chestnut is generally a medium density wood but being recovered from a build with outside exposure it had about a 100 years of drying out so in this case is a little less dense. The wall thickness is about 1/8” uniform over the length of the bore. The grain is straight and absent of any figuring, knots, inlays, or laminations with the exception of the end caps. The flute has a thin poly finish with one coat inside and one oil and three fine coats (sanded and steel wooled) of poly outside. The intervals are labeled and you can reference the first table (Fig 1.) as to what each represents. Typically the fundamental is the widest and strongest hence the one we identify with hearing that pitch. Those that follow add to the color or character. You will note that the 1st, 2nd, and even 3rd harmonics are very high in amplitude and well defined these are add much to the volume. The definition of the waves does not start to breakdown until the 7th and even then it is a minor widening with low pronounced enharmonics. The conculsions from the graphic alone would be that the flute is loud (summed amplitudes), very resonant (amplitude of harmonics), has clarity (definition of impulses), this note was well tuned for this temperature, and quite free of artifact or inharmonic content (some at the 10th). This is exactly the way this flute performs. About the only thing that could make this flute have even stronger harmonic content would have been to not use end caps (reversed grain and additional glue joints).  As a reminder I analyzed just the fundamental note in this study, each note would have its own set of harmonic frequency patterns.

(Fig 5.)

This next flute (Fig 6.) shows that the fundamental is quite well defined by its narrower width and good strength as well. Although the peak of the fundamental is well tuned the harmonics are all slightly flat which would add to a more perceptible flatness. The harmonics also very quickly deteriorate quickly becoming flattened, widened and irregular with the introduction of a full enharmonic at the 4th. Above the 7th the amplitude of the impulses is only fair. This flute might be perceived as very slightly flat, average to slightly quiet volume, fair to average clarity, with a complex sound but may include artifact content.
(Fig 6.)


This sample (Fig 7.)  is nearly identical to the graphic above only having a slightly more definition to the impulses but slightly flattened in frequency like fig 6. The amplitude of the harmonics severely falls away after the 11th harmonic. Perception of this flute would be as indicated in fig 6 as well.
(Fig 7.)


This flute (Fig 8.) has a very wide fundamental wave form. All the harmonic intervals are at the precise frequency expected. There are two enharmonics introduced at the 4th and 5th. There is average definition and amplitude of the lower harmonics while the uppers are very good all the way to the end of the graphic scale. This flute would be of average volume with average to good clarity especially in the higher harmonics which would make this flute sound “sweet” as apposed to complex because the lowers are reduced. My impression with the reduced amplitude in the lower and more in the higher that this flute may have been made of an exotic wood or had thick bore walls. It would have been a drier sounding flute, less complex with only minor artifact presence. 
(Fig 8.)


This last flute (Fig 9.) has a very well defined fundamental wave form. All the harmonic intervals are at the precise frequency expected. Enharmonic content begins at the third harmonic level and persist throughout the remaining harmonics. This flute would be of average to quiet volume, cool in tonal color, but would have complex elements of artifact. I know this flute well and indeed it has a pleasing quiet tone and fewer strong harmonics. It is a thick walled flute. It is made of ash which by the nature of its grain structure can introduce some interesting but peculiar harmonic content. The flue of this flute is also very shallow adding a breathy nature to the tone, quieter voice, and less resonance.
(Fig 9.)

Well that was fun!

So what can YOU do with this information. You can use this information to reasonably predict what a flute might sound like, know what is required to give you the sound you want, and use this information help you get the sound you want in a given key. There is no correct way and what might be perceived as bad for one key by one person may be an attribute to another. Given what we have already observed if a flute maker wanted to have all his flute have a similarity in tone and timbre across the range of the key he/she makes, wood and technique are important factors. By example my own preferences in sound are that I like my low flutes to have a rich complex sound, a sense of mystery to them. My mid range flute I like a mix of clarity and resonance giving them a lot of interest. Lastly on my high flute I like them very clear with fewer harmonics to keep them from sounding squeaky. The choices for the mid range are easier, keep the wood a mid density, straighter and moderate grain, little or no figure and walls thinner. On the bottom end I prefer to use softer low density woods, very straight grain with no figure, walls proportional thinner. However on the top end to reduce harmonic content I like to use dense woods like exotics or very hard woods, smooth or figured, and proportional thicker walls.  Now some situations you might require the opposite as the most desirable choice. If you wanted to play say a low flute with a number of other instruments if the flute were made of a soft wood, un-mic’ed it would get lost in the mix of sound. In these conditions you would be willing to sacrifice some of the character for greater clarity and opt for a harder wood that would be drier but broadcast the sound much better. This could be the case as well with mid range flutes to a lesser degree. For high flutes by their nature are already loud little buggers but if you wanted to get a little more character or color to the tone you might want to move to a mid density wood or thin the walls but chances are not a low real low density wood. So the next time you go to a flute gathering and see a mid range flute made of curly paduck, with ebony laminations, stone inlay, with a thick bore, full body wraps, and ten coats of finish you can pretty well figure that flute is going to be quite dry sounding, have a broadcast presence, and much less character.