Barbershop Arranging — Part 1: The Science of Sound
How the physics of the human voice affects ensemble unity
This is Part 1 of a 10-part series on barbershop arranging. The full guide is here.
The first step to becoming a comfortable barbershop arranger is to understand how singing and sound actually work. The introduction to this series described barbershop harmony as a synergistic experience. This article will describe how that synergy comes to be, starting from first principles. In particular, we’ll look at how barbershop’s “expanded” sound slowly formulates as we add each successive singer. We’ll start with a lone voice.
The Science of One 🙋♂
One singer may be able to serve expression and artistry by the truckload, but for our purposes, the first singer simply provides a pitch.
A pitch is a sound wave measured in wavelengths per second (Hz). Higher pitches have higher frequencies; lower pitches have lower frequencies. The human ear can hear pitches roughly between 20 Hz and 20,000 Hz. (Dog whistles sound anywhere from 23,000 Hz to 54,000 Hz, far beyond our capacity to hear). Collectively, the human vocal range stretches from around 40 Hz for the lowest basso profundo to around 1300 Hz for the highest sopranos.
Listeners typically perceive a single pitch at a time when listening to the human voice, but the voice actually produces many frequencies at once. We can demonstrate this with a spectrogram, a 2D visualization of the frequencies in a recording over time. In a spectrogram, time is shown on the x-axis, frequencies are shown on the y-axis, and frequency “loudness” is shown as color, on a spectrum from muted blues to bellowing yellows.
Here are the spectrograms of a pure, computer-generated tone at 440 Hz, compared to a human voice singing the same pitch on an Ah vowel:
Notice that the computer-generated sine wave produces one frequency at 440 Hz, while the human voice produces a whole stack of frequencies!
What are these extra frequencies in the human voice? The fundamental is the pitch the singer is actually singing. It is the lowest frequency in the spectrogram and the pitch we hear most audibly: if a singer is singing a middle C, the fundamental will be that middle C. But due to the passage of sound waves through our oral cavity, there are higher, weaker frequencies latent in the sound as well, called harmonics.
Harmonic frequencies are an integer multiple of the fundamental frequency (the fundamental is thus the first harmonic, since its multiple is 1). The more general term partial refers to all frequencies above the fundamental, including non-integer multiples. Overtone is a synonym for partial: all harmonics are overtones, but not all overtones are harmonics.
Together, the integer-multiple partials form a harmonic series, demonstrated in the video below. Note that the distance between harmonics decreases the farther up you go in the series. This is due to the logarithmic nature of frequencies: each octave in the series has twice the frequency of the previous octave, thus frequencies are growing at an increasing rate and it takes more of a “jump” to ascend to each successive pitch the higher you go.
Concretely, this means that when a singer sings the lowest note shown above — in this case, C2, the C below the bass clef — the singer’s vocal tract will also amplify the later pitches in this series to varying degrees, to the point that listeners can hear them.
The exact amplitude of each harmonic will depend on the vowel the singer is singing. Indeed, the human voice is perhaps the only instrument that changes shape while the musician is “playing” it! Higher harmonics will also become less prominent if a singer sings more breathily. Singers can be trained to maximize the audible harmonics in their voice, an objective known as singing with resonance. Below is a comparison; note how much less amplified the harmonics are in the breathy singer’s sound.
There is a boundless number of different pitches we might sing, each with a distinct series of harmonic frequencies above it. The “E-natural” in the harmonic series built on a C will be a little different from the “E-natural” in the harmonic series built on A. Rather than deal with all of these small variations between near-identical pitches, Western music typically quantizes frequencies into the twelve different pitch classes or keys that we find on the equally tempered piano:
Quantizing the pitch space is just a compromise that we live with to limit the number of keys we need on a piano. However, the human voice is technically capable of singing much more in tune than the piano, since the pitch space of the human voice is not discrete, but continuous!
The Science of Two 👨👧
When we add a second singer to the mix, the two voices can sing pitches that form harmonic intervals with one other. Here are the intervals our two singers can sing within one octave, starting on C:
When two voices sing two pitches at the same, the harmonics of each voice start to interact. The interactions of their harmonics affect our perception of the interval being sung, such that the interval may seem consonant or dissonant.
Consonant intervals
Consonant intervals are those that share many harmonic frequencies, amplifying their common partials via constructive interference. A perfect unison is maximally consonant by this definition, since the harmonic frequencies of each fundamental will be identical. Perfect 5ths and 4ths also overlap significantly, and Major 3rds and 6ths overlap significantly enough to sound pleasing as well.
Consonant intervals also tend to have neat, simple ratios between their two fundamental pitch frequencies. For example, an E sung at 660 Hz and an A sung at 440 Hz — together forming a Perfect 5th in just intonation (i.e., non-quantized, non-equally tempered intonation) — have a frequency ratio of 3:2. A just Perfect 4th has a ratio of 4:3, a major third 5:4, and a minor third 6:5. The ratio can be read as: “every xth wave of the first pitch’s fundamental, will align with every yth wave of the second’s.”
The sum of the A440 and E660 waves shown above is plotted in the chart below. When waves meet crest to crest, the sum’s crest will be extra high, and when waves meet crest to trough, they will cancel each other out. Above all, note that the wave pattern produced is regular and consistent, indicative of a stable, consonant relationship.
Here’s a sample of a Perfect 5th sung by two voices. Note how strong and pleasing the relationship is between the two pitches, all due to the aforementioned traits of a consonant relationship:
Dissonant intervals
In contrast, dissonant intervals are those with much more complex pitch ratios, and whose component fundamentals’ harmonic frequencies overlap much less. Dissonant intervals tend to sound grating, as if they’re in a competitive fight to the death.
The most notorious dissonant interval is the Tritone, the equal-tempered compromise between the Augmented 4th and the Diminished 5th (which are distinct pitches in just intonation). The Tritone’s frequency ratio is 45:32 or 64:45, depending on your tuning preference. Other dissonant intervals include the Major 7th and the Minor 2nd.
Here are sung samples of the tritone and the minor second. Note that both are harsher to the hear than the Perfect 5th, as if the two pitches are rubbing against each other in perfect disharmony:
If we plot the sum of two frequencies a Tritone apart, we see a much more irregular wave that reeks of instability. Here, we show a wider domain on the x-axis so you can see just how stubbornly unique each wave form is:
Note: Consonance and dissonance are actually a fair bit more nuanced than we can capture with these quick definitions. See this video from YouTube channel 3Blue1Brown for a longer exploration.
Tuning, vowel matching, and vibrato
Tuning can of course affect the interval ratio between two singers’ fundamentals. An interval that is slightly out of tune will have minimal harmonic overlap between the two pitch fundamentals, which can sometimes yield a sound that is more unpleasant than even a perfectly tuned dissonant interval.
Case in point: here is the sum of A440Hz sung against A436Hz, a pitch that is just 4 cents (4% of a half step) flat:
The result is an extremely slow, unnerving wobble—like a minor-second rub slowed to a snail’s pace. Nevertheless, tuning issues like this can be difficult to diagnose until you hear the interval tuned properly.
Mismatched vowels have a similar but more subtle effect. Different vowels have different harmonic patterns. Thus, two singers singing perfectly tuned pitches on different vowels will actually tend to sound out of tune, or at least out of sorts — because although their fundamentals are consonant, their harmonics are not!
Vibrato can affect perceived tuning as well. To be sure, two operatic singers with rich vibrato can produce incomparably beautiful harmonies (the “Flower Duet” from Léo Delibes’ opera Lakmé is a wonderful example), but at the limit, the small fluctuations in pitch that produce vibrato can cause partials not to align consistently. This is why barbershop errs toward a narrower vibrato band, if not full-blown straight tone.
Addendum: Scales
While we’re talking about intervals, let’s touch on one topic that is sure to come up as we start arranging: scales. A scale is an ascending sequence of specific intervals arranged in a unique and well-defined order. One of the more common scales is the Major Scale — defined as the sequence Whole (Step), Whole, Half, Whole, Whole, Whole, Half — but there are others, too. This sequence can start on any key, meaning there are twelve unique major scales available (C Major, C♯ Major, D Major, etc.). Here is a visualization of the intervals comprising C Major:
Pitches in a scale may be referred to by absolute name or by scale degree, their ordinal index in the scale. In the example above, C is scale degree 1, D is 2, E is 3, F is 4, G is 5, A is 6, B is 7, and the high C is 1 again. Scale degrees also help us refer to notes that are not included in the scale. For example, since C♯ is between scale degrees 1 and 2 in C Major, we can refer to it as “1-sharp” — or equivalently, “2-flat”—as needed. Scale degrees will be pivotal to our analysis of the ensemble sound when we add the third and fourth voices.
The Science of Three 👨👩👧
When we add a third singer, the focus moves from intervals to chords, namely three-part chords called triads.
The Major Triad
The Major Triad is built from the unique pitches we encounter in the first six frequencies of the harmonic series (in which the fundamental is the first harmonic).
One theory of music history posits that European polyphony first developed with singers of Gregorian Chant. Per this theory, monks began to mimic the harmonics they heard reverberating around the church while they sang — with the fifth and third being the most audible given those intervals’ lower positions in the harmonic stack. See if you can pick out the subtle Fifths and Thirds floating above the sung pitches in the following recording, particularly at the ends of phrases:
In western music, we think of the Major Triad as the most stable chord, because it is constructed from consonant intervals whose partials overlap significantly.
Inversions
If we hold the pitches in the chord constant, the stability of a chord can still be affected dramatically by the order in which the pitches are arranged. These different pitch orderings are called inversions.
In music theory, an inversion is uniquely identified by its lowest note. For a chord consisting of N pitches, there will be N possible inversions, since any of the pitches may be placed at the bottom. We can measure the stability of each inversion based on (a) the distance from its lowest note to (b) the lowest harmonic that that note shares with another note in the chord.
The default inversion is called root position, which places the root at the bottom. The root is the pitch after which the chord is named, and is thus considered the “home base” of chord inversions (though technically, root position is the default, rather than an inversion.)
Here’s what a root-position Major Triad sounds like when sung:
For a root-position C Major chord spelled as C, E, G, the first shared harmonic will be 13 scale degrees above the lowest note in the chord (the C):
If we move the C an octave up and leave the E as the bottom note, we get first inversion, the inversion stacked on scale degree 3.
Here’s what it sounds like:
For a first-inversion chord spelled as E, G, C, the first shared harmonic is a whopping 20 scale degrees above the lowest note in the chord (the E). Harmonics that high are fainter, so singers experience much less reinforcement from the shared harmonics:
Finally, if we move the E an octave up as well, we’re left with a C Major chord in second inversion, the inversion stacked on scale degree 5.
Here’s what it sounds like:
For a second-inversion chord spelled as G, C, E, the first shared harmonic is 16 scale degrees above the lowest note in the chord (the G)—more robust than first inversion, but not as strong as root position.
These results match the conventional wisdom often espoused in rehearsals: a triad in root position is the most stable, followed by a triad in second inversion, followed by a triad in first inversion, the least stable. The perceived stability of the root inversion is why songs typically end on root-inversion chords (unless the composer is abandoning them to make an artistic choice). And the perceived instability of first-inversion chords is why classic barbershop rarely uses them—and why barbershop basses run from them in fright.
Spelling
Even though a root-position chord spelled as C, E, G is the strongest of the three inversions above, it isn’t necessarily optimal for voices, particularly in low registers.
Harmonics higher in the harmonic series are generally less perceptible than those lower in the series — especially if a singer is singing without much resonance. Amplifying higher harmonics gives singers a greater sense of stability via “reinforcement from the rafters.” To achieve this, an arranger can adjust the chord spelling: which singers sing which pitches and in which octaves.
In the “collapsed” root-position chord spelled as C, E, G, the low E doesn’t actually reinforce any of the C or G’s harmonics until 2 octaves up, so it isn’t contributing much to the stability of the chord. However, if we move the E an octave up to “expand” the chord into a C, G, E spelling, exactly the same harmonic frequency will be amplified, but it will be much louder. This is because that frequency will now be lower in the harmonic series of the new, octave-higher E fundamental.
All told, this increase in harmonic amplification is why barbershop tends to prefer “expanded” spellings with the third at the top of the chord, rather than collapsed spellings with the third buried low in the stack. Here are both chord spellings for comparison (the effect may be more apparent with headphones):
Other triads
Technically, any three pitches on the piano can be sung at the same time and formed into an arbitrary chord, but the most common chords historically are Minor Triad, Diminished Triad, and Augmented Triad. These other triads are less stable, but they are no less important to the chord vocabulary.
The Science of Four 👨👩👧👦
Enter the fourth singer. What does this final voice enable for a quartet?
Octavization in triads
First, the fourth singer can empower an existing triad through octavization—that is, singing an octave above or below another singer’s note in an otherwise 3-part chord. Octavization reinforces the harmonics of the lower pitch in the octavized pair, because pitches an octave apart belong to the same pitch class and thus have the same harmonic series as each other (albeit offset by an octave).
Consider the shared harmonics between two octavized Cs. The harmonic series of the higher C starts an octave higher than the harmonic series of the lower C, but four of the lower C’s first seven harmonics are still reinforced. Further, the specific frequencies that are not necessarily reinforced by both fundamentals are nevertheless consonant with both, since most of them form Perfect 5ths and Major 3rds with both fundamentals.
Sung well, two octavized pitches will be consonant enough that they “lock in” to one another—precisely because their harmonic series will align so perfectly. As it was for medieval (apocryphal?) chanters, this effect will be most pronounced in reverberant acoustic settings. Here’s an example with the reverb and ambience dialed to 11. See if you can hear the hint of a faint third or fifth above the sung pitches:
Ultimately, octavizing pitches allows a four-voice quartet to sing much more “expansive” major chords in which shared harmonics are much more audible. Building off the 1–5–3 spelling from before, here’s an example of a quartet singing a major chord as 1–5–1–3, with roots octavized:
Other effective spellings of major chords for four voices are 1–1–3–5 and 1–5–3–1 because they spread the pitches nicely while remaining in root position. These and the 1–5–1–3 spelling are all used throughout the barbershop style, particularly as the last chord of a song, when stability is key.
Seventh chords
The second and primary enablement of the fourth voice in a quartet is to sing seventh chords. A seventh chord, generically, is a four-part chord—meaning it consists of four distinct, non-octavized pitches—constructed from scale degrees 1, 3, 5, and 7. There are several different types of seventh chords that we can make by applying accidentals to various combinations of scale degrees 3, 5, and 7.
Let’s return to the harmonic series for a moment. Recall that the first six harmonics (starting with the fundamental) will spell out the Major Triad. If we move our focus to the seventh harmonic—pardoning the ordinal coincidence — we discover the minor seventh of the scale:
The chord built from these first seven harmonics, scale degrees 1, 3, 5, and 7b (“flat seven”), goes by several names. Barbershoppers eponymously dub it the Barbershop chord. Classical musicians call it the Dominant 7 chord, since its root is often the fifth scale degree, or dominant, of the scale; or they may call it the Major-Minor Seventh chord, owing to its construction as a minor seventh tagged onto a Major Triad. Jazz musicians may simply call it the seventh chord, tagging a number “7” onto the root on which the chord is built (e.g., “C7”), since the chord is so fundamental to jazz-chord progressions. We’ll call it the Barbershop chord for now.
The Barbershop chord: A harmonic amplifier
The Barbershop chord is special for two reasons.
First, the Barbershop chord is virtually engineered to amplify harmonics. The chord is hiding in plain sight within the first seven frequencies of the harmonic series, and when sounded all at once, its component pitches thus enjoy extensive sharing and amplification of their respective harmonic series.
Here is a sample of a quartet singing a barbershop chord in the order that the pitches appear in the harmonic series (root, then 5th above that, then 3rd above that, then flat seventh at the top):
And here’s another, perhaps even more exciting flavor of the Barbershop chord, spelled in second inversion with the root and seventh co-located a Major Second apart at the top:
Each of these chords achieves that hallowed end of lock-and-ring. Deftly tuned and rightly spelled, they will cause overtones to scream from the rafters with sublime efficiency, enriching the harmonic complement to the four voices even more so than an octavized major chord can.
The Barbershop chord: A harmonic mover
The second reason why the Barbershop chord is important has to do with harmonization, the process of adding accompanying chords to a given melody.
The Barbershop chord has in fact served as the foundation for chord progressions in tonal music since the 1600s. This is a big statement that we must unpack in a separate article altogether, by looking at the chord not as four stacked pitches, but as two overlapping intervals.
Reflection
You’ve just read a several-thousand-word treatise that explains how ensemble singing comes to pass from a barbershop perspective. Congratulations!
Before we go on, let’s synthesize our learnings thus far:
- One singer. Each singer’s voice produces a fundamental pitch that we hear most audibly, as well as a predictable series of harmonics above that fundamental—all at the same time. The single singer’s objective is foremost to maximize resonance, such that higher harmonics are amplified as much as possible to create a fuller sound.
- Two singers. When we add the second singer, we must start thinking about the alignment of their two contributions. Two pitches sung at once will be consonant if their frequencies have a simple wavelength ratio and more shared harmonics, or dissonant otherwise. Tuning, vowel matching, and the minimization of vibrato all help the singers lock in to each other.
- Three singers. A third voice enables the singing of three-part chords called triads, consisting of a root, a third, and a fifth. Triads—and indeed all chords—can be sung in different inversions that seem more or less stable depending on how far removed the lowest shared harmonic is from the lowest note in the chord. Holding the bottom note constant, the remaining pitches of a triad can also be sung with different spellings that reinforce shared harmonics with varying degrees of efficiency.
- Four singers. Finally, the fourth voice enables octavization in three-part chords, which reinforces shared harmonics even more dramatically, as well as the singing of four-part seventh chords. Notably, the fourth voice allows the ensemble to sing Barbershop seventh chords constructed from the first four unique pitches of the harmonic series, which enables a method of harmonic progression that has been foundational to Western tonal music since the likes of Bach and Handel in the Baroque era.
Now, we ourselves will progress onward to the most important topic in barbershop arranging…the Circle of Fifths!
Next: Part 2: The Circle of Fifths
Full guide: Barbershop Arranging: A Modern Guide
