Musical Temperament Visualizer

How do you tune your viol? How did musicians of centuries past tune their viols (or other instruments)?

I’d like to offer one (new) contribution to this complex and ongoing conversation: a way of visually comparing temperaments for use on viols or other instruments (but especially viols, as I’m a viol player and the principle historical source I’ll be discussing is a treatise on tuning and temperament by a viol player: Thomas Salmon’s 1688 A Proposal to Perform Music in Perfect and Mathematical Proportions).

To make this post as accessible as possible, I’ve placed references and links to some of the rich, ongoing conversation on tuning and temperament for viol players at the bottom. Go there if you want to get in the weeds. Feel free to scroll down to the visuals and video and then tack back up to my explanations if you’re feeling impatient!

Who was Thomas Salmon and how did he think we should tune our viols?

Thomas Salmon (1647-1706) was an English clergyman, music theorist, and amateur musician and viol player who published several treatises on music theory and got involved in a music-theory pamphlet war with composer Matthew Locke during the final decades of the 17th century. In his A Proposal to Perform Music in Perfect and Mathematical Proportions (1688), Salmon lays out a fascinating tuning system for the viol da gamba that he would later, famously, demonstrate before the Royal Society in 1705 with the help of several notable musicians. In his short article in Oxford Music Online, Michael Tilmouth writes:

“Salmon’s later work on temperament, published in his Proposal (1688) and in the Philosophical Transactions for 1705, stemmed from his interest in mathematical acoustics and involved him in contact with Wallis, Newton, and other members of the Royal Society. It was by no means purely theoretical, however; Salmon built relationships with performers and instrument makers, and persuaded the viol players Frederick and Christian Steffkin, and the violinist Gasparo Visconti, to participate in a performance before the Royal Society in 1705 demonstrating a special form of just intonation on customized instruments.”

So, what is this “special form of just intonation” that Salmon developed for “customized” viols?

Without getting too technical or history-encumbered (again, head to bottom of this post for references, etc.), Thomas Salmon used three different ratios to calculate where on the neck of a viol one should place frets (and, consequently, how one should tune open strings). Drawing on music theory from Marin Mersenne and others, Salmon decided that the best way to get as many intervals as close to perfectly in tune as possible on the viol was to establish two sizes of whole step (1/9 and 1/10) and one size of half step (1/16).

The process Salmon describes for determining the pitches of a natural minor scale (A-B-C-D-E-F-G-A) goes like this: 1) measure the string length of your viol (Salmon settles on a string length of 28 inches from nut to bridge, but explains that his system will work equally well on any given string length); 2) then, place your second fret (not the first!–that’ll come later) 1/9th of the total length of the string from the nut to the bridge; 3) Then, place your third fret 1/16th of the distance from your second fret to the bridge; 4) then, place your 5th fret 1/10th of the distance from your 3rd fret to the bridge; etc.

Here’re some graphics from Salmon’s treatise to help visualize this process [p.9]

The numbers between each note of the A (natural minor) scale at the top of this diagram are the denominators of the fractions of the string length as measured from each subsequent fret, while the numbers below are the denominators of the fractions of the total string length that yield each corresponding interval. (Proposal p.9)
Salmon proposed a set of interchangeable fingerboards, each with frets positioned to favor a particular key. Above is his diagram for the fingerboard that would produce “mathematical” intonation for the keys of A minor and C major (Proposal p29)

[Brief digression into Just Intonation and the mathematics of Salmon’s system: Salmon drew on earlier discoveries by Mersenne and others to show that “stacking” ratios of 1/9 or 1/10 (whole steps) with half steps of 1/16 of some given string length [“X”] actually yields “pure” (i.e. “Just”) versions of larger intervals such as the minor (“lesser”) third (1/6, it turns out, is equal to (1/9X + 1/16(8/9X))/X, where X is the total string length from nut to bridge). The same is true for nearly all the other “diatonic” intervals between the seven pitches of the major or minor the diatonic scales Salmon provides. Using only stacks of ratios 1/9, 1/10, and 1/16, you can generate “pure” (i.e. “just”) whole steps, major and minor thirds, perfect fourths and fifths, and major and minor sixths. For a thorough and readable deep dive (with proofs!) of how simple ratios can be used to generate a usable tuning system, see Propositions IV-X in The First Book of Stringed Instruments of Mersenne’s Harmonie Universelle (1636) [published in English translation by Roger Chapman in 1957].]

There is much more to Salmon’s treatise that is worth considering (such as his proposals for interchangeable fingerboards for different keys and the use of split frets or tastinos (“fretlets”) to solve some of the issues that inevitably arise when you place horizontal frets across multiple strings separated by BOTH imperfect (3rds) and perfect (4ths) consonances, etc.etc.). But, the very brief discussion above provides enough of a basis to allow a VISUAL COMPARISON of Salmon’ temperament for the viol with several temperaments in common use today, EQUAL TEMPERAMENT, QUARTER COMMA MEANTONE, and THIRD COMMA MEANTONE (ok, this one isn’t that common but I like it for its pure minor thirds!).

The following video captures a VISUAL RENDERING of the differences between Equal Temperament, Quarter-Comma Meantone, Third-Comma Meantone, and Salmon, in which the ‘beating’ of all the diatonic intervals can be viewed simultaneously. The idea is that by seeing all the ‘beating’ (remember, the faster the beating, the less “pure” the interval!) simultaneously, you can get a unique sense of the “gestalt” of the given temperament in a way that facilitates comparison between temperaments. What I’m after here is a way of experiencing the UNIQUE QUALITIES of SALMON’S TEMPERAMENT!

In the renderings below, made using max/msp, the different regions of color pulse at the speed that each particular interval beats as we compare temperaments. The blue horizontal rectangles represent 5ths, the squares represent minor and major 3rds (the red squares are major, the blue/pink ones are minor), AND the vertical bars to the right of the screen represent the 5 intervals (4ths with one major 3rd) that separate the six open strings of a viola da gamba. For someone curious about what it’s like to tune the open strings of the viol in each of these temperaments, this “open string” visualization is particular interesting (spoiler alert! Salmon’s temperament allows all the open strings to be tuned pure to each other except 1!)

For best results, select quality = “1080p60HD” from the settings menu at the lower right.

For those curious to compare pitches (in Hz) from temperament to temperament, below is a screenshot of the max/msp patch I built to do the calculations (this patch feeds pitch information to the visualizer). The subpatchers (e.g. “p 1/3 C -4th”) contain the actual math [in this particular case, to calculate a 1/3 comma meantone descending 4th, you multiply your starting frequency by ((3/2) * pow((80/81),(1/3)))) / 2)]…

Below is a video that demonstrates the acoustic “beating” of 3 different versions of a C-E major third made using the visualizer, above. If you have trouble hearing beats, START HERE! The video “isolates” the beating from the rest of the sound (by fading out [filtering] everything except the beating and then fading everything back in).

The following video demonstrates how CHORDS (major triad, minor triad, major seventh chord) sound (and look!) in our four chosen temperaments.

So, to return to the question at the top of this page, what do we learn about tuning the viol by comparing these animated “gestalt” visualizations of these four temperaments?

If you focus on the pulsing red, vertical lines on the right side of the video, you can see precisely how rapidly each pair of adjacent strings beat in each of the given temperaments. Considering how difficult it is both to hear beats (for many people), as well as how hard it is to tune consonances that are _almost_ in tune but not quite (by some very precise measure), PURE INTERVALS ARE MUCH EASIER TO TUNE for most players.

That’s the magic of Salmon’s temperament! For each key for which Salmon provides fret placements, the 4 of the 5 pairs of strings on the viol can be tuned pure! Those who have experimented with different temperaments on the viol will immediately see a host of complicated implications, here, which I invite readers to explore on their own time!


What about accidentals? Why don’t your visualizations show all 12 pitches, as opposed to only 7?

In brief, playing the viol in tune entails using various techniques to “micro-tune” accidentals based on the specific musical context. Unlike keyboard players, who get fixated on how a given temperament propagates itself way out in the remote reaches of the circle of fifths, viol players can focus on the “scaffolding” of the diatonic intervals and then place interstitial notes “in tune” according to criteria of their choice. The fact that most writing on temperament has been done by keyboard players has DISTORTED our understanding of what’s actually important, as string players, when we consider temperament.

Why don’t you show 1/6 comma meantone? Viol players love 1/6 comma meantone!

It’s true, we do. In my consort LeStrange Viols, we routinely tune our frets and open strings to 1/6 comma (though we don’t actually _play_ in 1/6 comma, of course, but rather use the tuning as a “foundation” to play as “just” as possible). But, 1/6 comma has no acoustically pure (just) intervals! For that reason, it doesn’t look that different from equal temperament in the visualizer and is less interesting to look at than the other temperaments I have included.

What do these temperaments actually sound like?

They sound quite different from each other, once you become accustomed to listening for temperament! At some point, I’ll post some comparative audio, but there are already many such resources available on youtube and elsewhere if you want to hear how polyphony sounds in different temperaments. [UPDATE:]

What’s happening with that pink tritone? How do you calculate it and why does it beat so fast in all four temperaments?

Tritones are complicated, and there are numerous competing systems for defining them and for determining what “pure” means for a tritone. I define a pure (or “just”) tritone as the interval you get when you multiply a starting frequency by 11/8, so the beating you see in my video reveals the difference between the F-B interval as calculated in those four temperaments and the interval you’d get by multiplying the frequency of F by 11/8. I’d love to know how you would have to split up a comma to generate a temperament comprised of acoustically pure tritones…anyone?

Resources and links

Thomas Salmon’s A Proposal to Perform Music in Perfect and Mathematical Proportions (1688)

Thomas Salmon’s article “The theory of musick reduced to arithmetical and geometrical proportions” (1705) in The Royal Society’s publication Philosophical Transactions.

Pollens, Stewart. “A Viola da Gamba Temperament Preserved by Antonio Stradivari” Eighteenth-Century Music 3/1, 125–132 © 2006

Rene Descartes’ lute fret placement chart from the Compendium (1653)

Ross Duffin’s very useful article (with musical examples) “Just Intonation in Renaissance Theory and Practice” in MTO (a journal of the Society for Music Theory) © 2006