This article was originally published on March 14, 2014 on my now defunct blog.

Some Starting Definitions

Pythagoras, the Ancient Greek polymath, devised a standard unit for measuring the size of perceived intervals resulting from two frequencies vibrating at a given ratio. Today, this unit is called a cent because, in a way that’s similar to American money, it equals 1/100th of a half-step. A half-step is the smallest interval in the western musical system. It is the distance between any two adjacent keys on the piano (regardless of their color). There are twelve half-steps in an octave; thus, one octave equals 1200 cents.

Equal Temperament and Just Intonation

All of the normal intervals on the modern piano are divisible by 100 cents. Thus:

• minor second (C up to Db) = 100 cents
• major second (C to D) = 200 cents
• minor third (C to Eb) = 300 cents
• major third (C to E) = 400 cents
• perfect fourth (C to F) = 500 cents
• augmented fourth / diminished fifth (C to F#) = 600 cents
• perfect fifth (C to G) = 700 cents
• minor sixth (C to Ab) = 800 cents
• major sixth (C to A) = 900 cents
• minor seventh (C to Bb) = 1000 cents
• major seventh (C to B) = 1100 cents
• perfect octave (C to C) = 1200 cents

The above list represents Equal Temperament. It has this name because the distance between each interval is all based until the common interval of 100 cents in a minor second.

Meanwhile, back in reality, physics has some issues with the scientist’s rounding formulas. Unfortunately for equal temperament, the physics of sound are measured in hertz, not cents. These two units of measurement are not equal. Hertz measures the actual number of vibrations a wave makes in space. Some complicated math is involved for the reasons why a minor second is not 100 cents, which aren’t important; however, interesting. It is important to know that a minor second is close to 100 cents, but is actually only 88 cents (plus change). So the minor second you hear on the piano is actually 12 cents too wide!

But how can this be?!? We listen to pianos all of the time and they sound great!

The human brain is OK with the tuning being a bit strange as long as it is consistent throughout the octaves. The brain wants, nay, requires that octaves be perfectly (within a cent or two) in tune or it starts to think that something is wrong. The other interval that we really like to hear in tune is the perfect fifth. In Equal Temperament the perfect fifth is only two cents low, which is difficult for even the most golden-eared, hot-house flowers to notice. So, the most important intervals—the octave and the perfect fifth—are in tune on the piano. Combine that with the piano’s relatively quick decay, and the fact that we’ve been listening to it like this since we were a fetus and we don’t mind a bit. Crazy organ, that brain.

Here is the natural scale, as it occurs in just intonation:

• minor second (C up to Db) = 88 cents
• major second (C to D) = 204 cents
• minor third (C to Eb) = 316 cents
• major third (C to E) = 386 cents
• perfect fourth (C to F) = 498 cents
• augmented fourth / diminished fifth (C to F#) = 600 cents
• perfect fifth (C to G) = 702 cents
• minor sixth (C to Ab) = 814 cents
• major sixth (C to A) = 884 cents
• minor seventh (C to Bb) = 971 cents
• major seventh (C to B) = 1088 cents
• perfect octave (C to C) = 1200 cents

How far off is it?

• minor second (C up to Db) = -12 cents
• major second (C to D) = +4 cents
• minor third (C to Eb) = +16 cents
• major third (C to E) = -14 cents
• perfect fourth (C to F) = -2 cents
• augmented fourth / diminished fifth (C to F#) = 0 cents
• perfect fifth (C to G) = +2 cents
• minor sixth (C to Ab) = +14 cents
• major sixth (C to A) = -16 cents
• minor seventh (C to Bb) = -29 cents
• major seventh (C to B) = -12 cents
• perfect octave (C to C) = 0 cents

That means that a major third must be lowered 14 cents from where we hear it on the piano to get it into tune. This system is called Just Intonation.

Why do we care?

Wind bands and orchestras are comprised of flexible tuning instruments. They use Just Intonation on a daily basis. Anyone who plays a non-fixed pitch instrument must know the tendencies of each interval and adjust accordingly to put the notes properly into tune.

How to tell when you are in tune

When two or more pitches are played at once, a crazy phenomenon happens inside the human brain. Your brain measures the amplitude of each pitch (in hertz) and subtracts those numbers from each other. The resulting number will be another pitch that your brain will trick your ears into hearing, even though it is not occurring in the physical space. This phenomenon is called the difference tone. It is a valuable tool for achieving perfect tuning.

When we subtract an A♭3, which cycles 207.652 Hz, from a middle c, which cycles at 261.626 Hz we get 53.974 Hz. 53.974 Hz = A♭ two octaves below the sounding A♭!

To get the interval perfectly in tune, all we need to do is adjust the pitches so that the two A♭s are perfectly in tune. Doing that not only makes the interval sound great to our ears, but adds depth to the ensemble sound. Imagine a quartet that is perfectly in tune while playing only major thirds. It would sound like there are two more players in the group!

Practicing Tuning

The ability to play in tune comes from practicing with both your instrument and your ears. As saxophonists we must know the tendencies of every pitch on the instrument and which adjustments are available for any pitch. We must also know the tendencies of the interval we are attempting to tune and where our pitch fits into the harmony. This is why ear- training courses are extremely important.

The best method to practice tuning alone is to use a standardized drone. Many tuners can drone pitches for tuning practice. These are fine, but I found them too quiet and with a very annoying tone. I created a set of my own tones that you can download for free. My tones are simply sine waves generated by a computer. They are in mp3 format and will show up perfectly tagged in iTunes. I use them all the time for my tuning/ear-training practice. The method book that I recommend for tuning practice is: Exercices d’intonation (Pour Tous Saxophones et Tous Niveaux) by Jean- Marie Londeix. It works chromatically through the instrument and progressively increases the difficulty and the speed at which the saxophonist must tune. However, owning the book isn’t completely necessary. You can always use drones to practice tuning any interval. I usually warm-up (scales, intervals, melodies, etc.), then do long-tones with drones. This way, I’m working on my tone, embouchure, and air support, along with training my ears and muscles to play in tune.

Resources

Computer Generated Sine Waves (A2 to A6).zip