Architectural and Musical Scales in Sound Room Design

by Wes Lachot

This paper originally appeared as a handout for a lecture of the same name at Taliesin in Spring Green, Wisconsin, on July 19, 2007.

The purpose of this presentation is to describe the close relationship between architectural scales and musical scales, and to show the usefulness of understanding and exploiting this relationship in order to create musically pleasing and accurate sound rooms. We’ll be describing room proportions in terms of musical intervals, and learning how to use the piano keyboard (see Fig. 3, last page) as an exponential “tape measure”.

For simplicity’s sake, we’ll look first at three-dimensional rectangular room shapes, and we’ll define musical tones in terms of the standard western twelve-tone scale, although these principles can be extended to other architectural shapes and musical systems.

One reason for exploring the mathematical and geometric similarities between architectural and musical scales is that there are many opportunities to take the formal insight gained from one of these disciplines and find uses for these forms and geometries in the other, by way of analogy. But the more immediate reason for doing so is that there is one area where architectural and musical scales actually meet face to face, and that is the realm of modal resonance, in which the room designer looks at the room as a musical instrument, and attempts to create a musically balanced room by tweaking its size and proportions.

Interference and Modal Resonance

Interference occurs any time a sound is produced inside an enclosed environment; actually, two opposing walls are all that’s required to produce this phenomenon. As sound waves travel back and forth between two opposing surfaces, a portion of the sound is reflected back and forth, recombining with itself, causing what is known as constructive interference (louder tones) and destructive interference (softer tones). This interference is the cause of most acoustic problems, creating unevenness of volume up and down the musical scale. Constructive and destructive interference occurs with any sound frequency played between opposing surfaces of any length.

The worst case of interference is when the room dimension is a multiple of the sound’s half wavelength, creating standing waves, which is to say that the room itself acts as a resonant instrument and reinforces these specific notes at the expense of others. This is what is referred to as modal resonance, and it can cause extreme unevenness in the loudness of different notes in musical scales. If you want to hear the modal resonances created by just one set of opposing surfaces, find a building with a large overhanging roof and no opposing walls, and clap your hands. With a roof overhang at 6’-9”, you’ll hear a fundamental tone approximately E₂, an octave and a 6^th below middle C. This is because the speed of sound—1130 ft. per sec.—divided by 83 Hz (E₂) equals a wavelength of 13’-6”, and half of this wavelength is 6’-9”. (The reason it’s a half and not a full wavelength is that modal resonance is created by round trips between the surfaces.)

In the example above, in addition to hearing the fundamental frequency (which we’ll label f ) you’ll also hear the overtone series, or harmonic series -- higher tones that are multiples of the fundamental, namely f x 2, f x 3, f x 4, f x 5, and so on. These frequencies are inversely proportional to their corresponding wavelengths, so the wavelength of these harmonics would be 1/2, 1/3, 1/4, and 1/5 the length of the round trip between the opposing surfaces. An easy way to hear the harmonic series is to pluck an open guitar string (fundamental), then lightly place your finger at the positions 1/2, 1/3, 1/4 and 1/5 length of the string, and so on. What we hear are the pure or just intonation harmonics, which, translated to the equal tempered piano keyboard would be E an octave above E₂, B a perfect fifth above that, E a perfect fourth above that, and D# a major 3^rd above that. (For brevity’s sake we’ll ignore the difference between just and equal temperament for now.) You can see that the harmonic series forms a major chord, which in fact becomes a dominant 11^th chord when extended up to the 11^th harmonic. So the point is: Any pair of parallel surfaces will create a resonance at frequencies whose half-wavelengths divide evenly into the distance between the two surfaces.

The Harmonic Series and Musical Intervals as Ratios

Following is a chart of the Musical Harmonic Series (Fig. 1), from the 1^st harmonic (fundamental) through the 16^th harmonic. We can use any frequency as a starting point, but to simplify things mathematically, we’ll make the fundamental frequency 100 Hz, with harmonics at 200, 300, 400, etc. Let’s call 100 Hz G₂ (G₂ is actually 98 Hz), so that the harmonic series would look as shown, with lower tones toward the bottom of the chart.

Below that is the Musical Harmonic Series chart (Fig. 2), from which we can see that just-intonation intervals have the very precise, simple proportional relationships as shown in the chart.

It’s interesting to note that these ratios can be flipped upside-down while maintaining the same musical interval—a 3/2 ratio or a 2/3 ratio is still a perfect fifth, for instance. This comes in handy when we begin looking at room dimensions and their corresponding frequencies; for example a 19 ft. x 38 ft. room would have fundamental frequencies of 60 Hz and 30 Hz, as frequency is inversely proportional to length, but in it’s simplest terms, it’s a 1/2 or 2/1 relationship—an octave.

The most important thing to note from the Musical Interval Ratios chart is that all of these musical intervals involve simple, low order, whole number relationships. It is apparent that the human brain finds these simple ratios to be pleasing and musical—harmonious, as it were. Note that the intervals involving the more complex, higher order ratios are the ones considered my most to be dissonant, like the minor second (16/15) and the major seventh (15/8). On the other hand, the simplest ratio of all, the octave (2/1) sounds to the ear like the same note, in higher and lower form. You can’t get any more harmonious that that (except for a unison, of course).

Balancing Three Dimensions for Evenness of Musical Scale

Modal resonances can be tamed somewhat with the use of acoustical traps, but when designing a sound room, the first order of business is to balance the resonant chords created by the three dimensions, so that you have as even a distribution of resonances as possible as a starting point. For example, if the room were a perfect cube, say 19 ft. x 19 ft. by 19 ft., you’d have all three dimensions supporting a Bb chord (approximately), with lots of Bb and F tones, and a few others thrown in, but essentially the room would be strongly biased toward the key of Bb. This perfect unison between the three dimensions would be an acoustic nightmare. So, if we were to leave one dimension at 19 ft. (key of Bb), what musical keys would be desirable for the other two dimensions, in order to achieve a smooth musical response in the room? Modal resonances below 200 Hz (G below middle C) are the most worrisome, because higher frequency resonances occur closer together, and are more easily treated with acoustical materials and room furnishings. Accordingly, we want to proportion the room so that bass resonances occur evenly as we move up the scale, as opposed to having them bunched together with wide intervals between them. So again, what two other dimensions (musical keys) would work best with our 19 ft. (Bb) dimension, to give us this even modal spacing below 200 Hz?

This question of modal proportions is a realm that has regularly been left to mathematicians, who’ve used complex algorithms and computer programs to come up with room ratios, typically displaying the modal distributions in modal graphs, or in long lists of numbers. While this can be a useful approach for the mathematicians themselves, designers are left feeling it is better to choose room dimensions from a list of “preferred” ratios rather than to strike out on their own and try to understand the math. But the truth is, it’s possible to bypass all this and go straight to the piano keyboard in order to look at the problem. The piano keys are laid out left to right just as a modal graph is, but has a built-in logarithmic scale, which many modal graphs lack (meaning that if the first octave represents 100 Hz – 200 Hz, the second octave represents 200 – 400, the third 400 – 800, and so on). Also, the piano octave is divided into 12 musically relevant 1/2 steps, which, while being a limitation of sorts, is ultimately a powerful aid to understanding modal spacing. Modal graphs and lists of numbers, although they support all possible ratios and not just musical ones, are difficult to digest, as they take room ratios out of the musical context where they properly belong. And as we will see, the limitation of twelve equal divisions of the piano octave -- a musical grid, if you will -- may not turn out to be such a limitation after all.

Many of the “good modal ratios” developed by mathematicians over the years are indeed low number ratios, or musical intervals, that can be easily seen on the piano keyboard. For instance, F. Alton Everest has compiled several sets of well accepted room ratios developed by various mathematicians and acousticians.¹ Of the 24 ratios given, 18 of them (75%) are either right on or very near musical intervals, and most are right on the money. We don’t know if these mathematicians developed their ratios at the piano, but it’s quite possible that after a lot of computing and number crunching, what they came up with could have been worked out just as easily at the piano.

Musical Intervals and Room Proportions

Let’s look at some of the musical ratios that would not create even modal distribution.

The 1/1 ratio is one such ratio, as in the case of the cubic room.
The next simplest ratio, 2/1, is also undesirable, for similar reasons. Since the harmonic series repeats itself in every octave, having two dimensions an octave apart will create many coincidences, in which the same tone is being reinforced by two dimensions.
A 3/2 ratio has the same problem, to a lesser extent. With a 3/2 ratio, the two dimensions are a perfect fifth apart and contain many coincidences, though not as many as in the case of the octave. There will be a coincidence between the 3^rd harmonic of the longer dimension and the 2^nd harmonic of the shorter dimension, and this is too low in the bass range to have such coincidences occur.

Now let’s look at some good ratios.

One commonly used ratio, 7/5, is well known as a very desirable ratio. It’s no wonder, as 7/5 represents a diminished fifth, meaning the two dimensions are six keys apart from each other (or half way around the circle of fifths). These keys are far, far distant cousins, and there will be no coincidences until the 7^th harmonic of the longer dimension meets the 5^th harmonic of the shorter dimension -- well above the critical low frequency range.
Another well known good ratio is 8/5, a minor sixth, which is four keys, or one third of the way around the circle of fifths. One recommended set of room ratios is 1 : 5/4 : 8/5, or a series of major thirds making an augmented chord. If you look at this chord and its harmonics on the piano, you’ll see that these ratios offer the smoothest possible arrangement of bass resonances in the lower octaves, although there are some coincidences in the upper octaves. Major sixths (5/3) are also good, as we’ve seen.
Major sevenths (15/8) are very good, as are minor sevenths (7/4).
Extending these principles to the other musical intervals, we see that a minor third (6/5) is a good ratio, since the first coincidence would be the 6^th harmonic and 5^th harmonic of the two dimensions, well above the low bass range. Major thirds (5/4) are also good, with reasonably high order coincidences.
Major and minor seconds (9/8 and 16/15) are good intervals, with coincidences coming very high in the harmonic series.

The bottom line is that keys immediately adjacent on the circle of fifths, that is perfect fourths and perfect fifths, are undesirable, as are unisons and octaves. But any other interval will work, as long as we are careful not to stack them so that they create perfect intervals, like stacking a minor third on top of a major third, creating a perfect fifth between the outer dimensions.

The Architectural Harmonic Series

The nice thing about using the piano to look at room proportions as musical intervals is the fact that these musical intervals can always be expressed as low whole number ratios. And, these low whole number ratios can translate directly into room dimensions, if you are using a modular planning system.

For instance, if we use a 4 ft. x 4 ft. grid to lay out a room, we will automatically wind up with low order ratios between the dimensions. For instance, the room might be 4 grids by 5 grids (16 ft. x 20 ft.), which is a good musical interval (major third), whereas 4 grids by 6 grids would give us on a 3/2 ratio -- not so good. But the point is, as long as the module is large enough relative to the space to yield small whole number multiples, the dimensions are by definition going to be musically related. If you design with a grid, you create dimensions that are musically related.

Notice that when we are dealing in architectural lengths like 20 ft./16 ft., the larger number in the ratio refers to the large dimension, whereas when we are dealing in frequencies like 200 Hz/160 Hz, the larger number refers to the shorter dimension, since frequency and length are inversely proportional. But it makes no real difference whether we are looking at lengths or frequencies, since we are dealing with ratios that can be flipped upside-down without consequence, and this leads us to what I call the Architectural Harmonic Series. This upside down harmonic series uses the module as the “fundamental”, as it were, and multiples of this length as the harmonic series. For instance, if we choose 2 ft. as the module, the series would be 2, 4, 6, 8, 10, 12, 14, etc. Choosing dimensions from this series, while staying away from perfect intervals, will yield good room ratios. If we take 8 ft. as the ceiling height and 10 ft. x 14 ft. as the dimensions, we get a good sounding room for its size, for example. In this musical way of thinking about room dimensions, the module is the “key”, or fundamental. As a visual aid to thinking about room proportions in this way, it’s helpful to learn the harmonic series upside-down (from right to left) on the piano keyboard.

Sometimes there are good reasons for using intervals (proportions) that fall outside the musical and architectural grid system. In some cases, the vertical grid system may use a different module than the system used in the plan, as is the case with many Usonian houses that use a 13 inch vertical system along with a 2 ft. or 4 ft. plan module. Still, in many such instances a harmonic convergence can be arranged. One example is the Rosenbaum house, whose living room proportions are approximately 10 ft. x 16 ft. x 26 ft., or 5, 8, 13 on the harmonic series. (It’s also worth noting that these “sixth” intervals also happen to fall within the Fibonacci series.) But even when dimensioning outside the natural musical intervals, it’s always good to keep in mind the principle of avoiding very low order harmonic ratios.

Modal Density

For small and medium size rooms, like those commonly found in residences and recording studios, good modal ratios are very important if you want to avoid bass boominess and modal ringing. This is because the smaller the room, the fewer resonances there are below 200 Hz, so their spacing becomes critical. Many small rooms have only a half dozen or so resonances in this range, and if they are all bunched up together the sound can become quite obnoxious. In order to achieve good modal density, which means enough modes (even spaced) below 200 Hz, experience has shown that it’s best to stay within a 3/1 ratio between the largest and smallest dimensions. So let’s say for a small room with an 8 ft. ceiling, you’d want the largest dimension to be no more than 24 ft. Otherwise, the spacing between modes just becomes too large, and there aren’t enough of them to evenly fill the range between 30 Hz and 200 Hz.

Larger rooms, on the other hand (like worship spaces, for example) may have 40 or more resonances below 200 Hz, so that there is much less chance of wide spaces between modes, no matter what the dimensions are. Still, all else being equal, even modal spacing is a good idea when smooth bass response is desired.

Conclusion

If one desires to create rooms that are in tune with the nature of sound, one need only look into the nature of music itself, as the inevitable architecture of music’s integral geometry will provide us with the key to proportioning our musical spaces in a way that is in harmony both musically and architecturally.

Fig. 3 ² Piano Keyboard – Lower Octaves

¹F. Alton Everest “The Master Handbook of Acoustics" McGraw-Hill (1981)

² Ibid.