Thursday, July 18, 2013

Equal and Just Temperament -- They Have Nothing to Do with How Well You Behave

Written by Matt Lammers of Fein Violins

We've all been there: you've practiced your quartet part well. You know the notes and can play them. You head into rehearsal confident that the opening chords will slot beautifully into place, but are greeted instead by beat frequencies, colleagues frowning at each other, and self-depricating, frustrated remarks like, "well, I guess this'll never be in tune". So you go home and sit in front of a tuner...and it gives you a nice little beep and green light, as if to say, "congrats! You're perfect!" Alas, you know you aren't. Rather than putting your temper to the test, let's put temperament to the test.

Simply put, temperament is the spacing between the pitches in a scale; it defines the sizes of intervals. Take the most notorious temperament issue for example: the third in a major or minor chord. Let's say the third of either chord is A (this means our major and minor keys are F-Major/FM and f#-minor/f#m). If you were to play an A in an FM chord at the pitch your tuner says is correct, you would find that your A sounds sharp (see video). If you were to play that same exact tuner-tuned A in am f#m chord, it would sound flat. In either case your A, which is supposedly in tune, sounds out of tune in reality. Don't blame your colleagues, blame physics. If you haven't encountered this issue before, watch our video that demonstrates the problem:

Before we get into the nitty-gritty good stuff let's define some units of measurement. The two relevant ones for the purposes of this discussion are often thrown around as synonyms, which is far from the case. Hertz (Hz) measures the frequency of a pitch, or how many times an air particle through which the sound (energy) travels vibrates every second. High pitches have a higher frequency than lower ones, and there are a certain number of Hz between two pitches. It is an absolute measurement of the physicality of sound at any given pitch. Frequency, however, has an odd characteristic. Take, for example, the Hz between C and C# on the C-string of a cello and compare it to the Hz between C and C# on the E-string of a violin. The Hz between low pitches on the cello is much less than the Hz between higher octave pitches on the violin. This is because Hz become "denser" the higher a pitch's register rises. This is problematic because the unit of measurement across and within octaves is not constant and would require octave specifications when talking about pitch distances. To overcome this issue, the unit "cent" is also used. The cent is defined on a logarithmic scale, which compensates for the variability of Hz. One octave is defined as 1,200 cents, regardless of the starting/ending pitches or their registers. As a result, cents are a means of uniformly measuring pitch distance, as opposed to Hz, which are useful when measuring the absolute identity of one pitch.

Now that terminology is straightened out we can clarify something. The terms A-440, A-442, etc. that you see on a tuner are not random designations. They are the exact pitch of the reference note (usually A) in Hz, where A-442 is a slightly higher pitch than A-440. This system is great for defining a reference pitch, because Hz are an absolute quantity, but when it comes to determining how far away from that reference pitch you are, Hz are not practical (due to the variable "size" of a Hz). It is for this reason that the distance the tuner needle moves is measured in cents; each tick mark to the right or left of your reference pitch (0 cents, the start/end point of each 1200-cent cycle) is equal. In short, a tuner illustrates your distance sharp or flat relative to a reference pitch, which is most commonly A-440 Hz.

Our shop manager, Amy Tobin, demonstrates the use of a chromatic tuner

Our shop manager, Amy Tobin, demonstrates the use of a pitch pipe. The calibration of a pitch pipe is not variable, usually A-440.

Our shop manager, Amy Tobin, demonstrates the use of a tuning fork. Like a pitch pipe, the calibration of a tuning fork cannot be adjusted.  

Alright, let's dig in to temperament itself, starting with equal temperament. Equal temperament is what it sounds like, and most people, musicians and otherwise, assume it is synonymous with "in tune". The octave is split into twelve equal partitions of 100 cents each. These twelve increments define where each half step lies in the chromatic scale of an octave. We'll return to this definition and the ups and downs of equal temperament after we define just temperament.

First, however, something essential to remember: Electric chromatic tuners decide what is "in-tune" using equal temperament.

Before attempting to understand just temperament, it's important to understand the harmonic series. A vast majority of instruments are designed to achieve maximum resonance. Resonance, as used in regular conversation, is a somewhat ambiguous term. A significant component of resonance is the overtone series. Overtones are pitches generated by the interacting vibrations of the fundamental pitch (the one you intentionally play) and supplement the tone color and quality of the fundamental pitch tremendously. Every pitch class has a distinct overtone series built into it, and a resonant instrument is considered one that projects overtone series' effectively. The overtone series produced by any pitch contains the following succession of overtones:

This notates the intervals above a fundamental pitch from which an overtone series is produced. The fundamental illustrated here is C, but the intervals above it can be transposed above any other pitch. Note that the space between successive overtones shrinks (from octave, to fifth, to fourth, to third, etc.)

Keep in mind as you continue reading that the overtones produced are naturally "tuned" in the way our ears perceive to be in tune relative to the fundamental; they define our sense of intonation. This is the basis of just temperament and is contrary to equal temperament (since the overtone series is not produced in 100-cent increments).

This video demonstrates the audial differences between just temperament (referred to as "Justonic" or just intonation) and equal temperament (referred to as simply "tempered")

Alright, so how is just temperament structured? It is a tuning system that takes into account the overtone series' of multiple pitches when they interact. Let's take our earlier instance, A played with F versus F#, for example. The first eight overtones (those with appreciable presence in sound) of the three pitches are as follows:

F: F, F, C, F, A, C, Eb, F
A: A, A, E, A, C#, E, G, A
F#: F#, F#, C#, F#, A#, C#, D#, E# 

First off, how is A tempered to sound in tune with an F? Notice that A appears in the overtone series of F. In this case, the A that is played (fundamental A) must be made to match the overtone A. If these do not match, the A and F will sound out of tune played with each other due to overtone interference. If the A is altered to match the overtone of an F that is tuned with a tuner, you would find that the A is lower than if it were tuned with a tuner (equal temperament). 

Second off, how is A tempered to sound in tune with F#? Now these two overtone series' must be compared. Notice that C# appears in both series'. If A and F# are to sound in tune these C#'s must match. Altering the A alters its overtones too, so tuning two overtones is as easy as tuning a fundamental and an overtone. If the A (and its overtone) is altered to match the overtone of an F# that is tuned with a tuner, you would find that the A is higher than if it were tuned with a tuner. 

Based on these findings it's easy to understand why tuners can't use just temperament; the definition of A, as with all other notes, varies dramatically depending on the pitch(es) it's played with! This is fine and dandy for string and wind players, since we can adjust our tuning with the movement of a finger or by lipping something up or down, but this presents a problem for pianists; while there are some pretty unusual, modern extended techniques for piano that involve plucking around in the piano, I've never seen someone reach in there and adjust a pitch on-the-go during a performance. In other words, it is completely infeasible to tune a piano using just temperament, so they're forced into equal temperament. Since equal temperament is used, pianos don't have to be retuned between pieces in different keys, which presents a simplified alternative definition of equal temperament: an average of the just temperaments of all keys.

And now the time you've spent reading all of this pays off with some practical advice!

Dr. Beat, the king of drones. While you likely don't need this many toys or this much volume, a tuner that produces a loud, audible drone is a good investment for intonation work.

When doing intonation practice, be sure to use a reference drone. The alternative, matching each pitch individually to a tuner, trains you to play in equal temperament (our nightmare scenario at the beginning of the blog). If you're not used to practicing with a drone, begin first by acquainting yourself with a scale. Play slowly up and down the scale with the tonic drone, and scrutinize each pitch. Eliminate beat frequencies by adjusting slightly, taking note of how much higher or lower the in-tune pitch is than you were expecting. Train yourself to automatically adjust these tendency tones in the major and minor modes of all keys. By doing so you can consciously adjust your intonation in the context of any piece, and it will be incorporated into muscle memory with time and repetition. Aside from the practical value of playing in tune, this process hones active listening skills and improves other aspects of playing, practice, and ensemble work. However,

CAUTION: It's essential to keep in the back of your mind that pianos are equal tempered! For passages in sonatas, for example, where the piano part is in unison or similar to yours, the tendencies of tones should be ignored. In larger ensembles with piano the issue becomes more complex. In a piano quintet, for example, the strings should employ just temperament when blending together. They should, however, be continuously aware of passages where just temperament clashes with the equally tempered piano part. This careful balancing act takes practice playing with piano to perfect and vigilance to maintain.

There is one more tendency to consider: the leading tone. It is fairly common knowledge that the leading tone (the major seventh scale degree of the major, harmonic minor, and melodic minor modes) should be played sharp, as if leading to the tonic. While this is similar to the adjustments made playing with just temperament, the raised leading tone is not a function of the overtone series and therefore unrelated to temperament. Let's take, for example C and B (leading tone in the key of C). Their overtone series' are as follows:

B: B, B, F#, B, D#, F#, A, B
C: C, C, G, C, E, G, Bb, C

There are no overlapping overtones, so just temperament is ruled out as a rationale for raising the seventh. Reasoning can, however, be found in music theory instead. See the basic V-I (GM-CM) cadence below. The GM chord consists of G, B, D, and the CM consists of C, E, G. As you can see, the B (degree three of chord V) ascends to the C (tonic of chord I) by a half step, the closest relationship between V and I (excluding the common tone G). As such the B "longs" to become the C over a very small interval, which generates tremendous harmonic energy and drive. By sharpening the leading tone and making the gap even smaller, the drive towards chord I is emphasized, and cadential anticipation is exaggerated. 

After a somewhat hefty exploration of just and equal temperaments, I'm afraid to say there are many more out there. Very few of them enjoy widespread use on the scale of equal and just, so a summary of these two is the most relevant discussion for practical purposes. This topic seems nitpicky and unworthy of the time needed to understand it, but it pays MASSIVE dividends on stage and saves you from shot-in-the dark tuning procedures in rehearsal. Good luck and happy tuning from Fein Violins!

While not exactly relevant to the temperament issue, a fascinating example of the tuner's sister at work

Are you a String musician or interested in becoming one? Take a look at our Fine Violins, Violas, and Cellos!

One of our staff members highly recommends the following publication:

A Violinist's Guide for Exquisite Intonation by Barry Ross

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