Where Are There 12 Notes in an Octave?

One thing many musicians take for granted is the choice we make to divide the octave into 12 equal pieces.

In fact, many musicians don’t realize that this is a choice. That’s how much this idea is taken for granted.

But there are many more than 12 potential pieces in the octave. And many tuning systems have been used or proposed beginning with the 12-tone system we use, spanning to systems which divide the octave into more than 50 notes.

There is a small community of composers, performers and researchers involved in this area. However, the development of microtonal music, whose tones are smaller than the 12 pieces we normally use, has been slow. Perhaps the reason is that the majority of instruments are designed for the 12-tone octave. But audiences are now trained to hear and appreciate 12-tone music. There are also relatively few musicians creating microtonal music. And while the 12-tone system is standard, systems using more than 12 notes are experimental and not as well understood. Using any other system makes you a pioneer. Finally, the people most interested in creating microtonal music are generally not interested in making music more consonant. They tend to see a potential for creating unusual and cutting-edge sounds. Thus, microtonal music has a PR problem because its most accomplished practitioners are an advertisement for novelty, not for mainstream music.

A Little History

Hundreds of years ago, before we had tools that could measure musical pitch, there was not much of a basis for understanding how musical sounds work. For this reason, even though human beings could potentially sing rich harmonies without special equipment, the concept was slow to develop, and in early music harmony was relatively simple.

Many early instruments, especially the human voice and unfretted string instruments such as violin, could produce any pitch within the octave. But with the development of instruments that have rigidly “fixed” pitches, especially the piano (and the guitar), it became necessary to develop a system of determining what those pitches would be, exactly.

In order to craft instruments that can be played together, it became necessary to have a universal system of tuning.

However, in order to develop a universally-accepted tuning system, many compromises had to be made. Some people even believe these compromises have done serious damage to music. In our modern world, all musical instruments and equipment are based around the 12-tone system. But what if the most pure-sounding music has 19 notes, or 22? Or 51?

A (Practically Infinite) Scale?

In order to understand the different possibilities for dividing the octave, we need to use some unit of measurement to talk about the differences in frequencies between different tones within an octave.

In music, the term cents is used to describe a difference in pitch. An octave has 1200 cents, and therefore in our 12-tone scale consisting of equally-spaced notes, each note is 100 cents apart. This unit of measurement is relative and useful for comparison only. Frequency can be used to measure the absolute values of pitches, when needed.

But what if our notes were 50 cents apart? Then we would have a scale with 24 notes, and it would sound quite different.

For this discussion, we propose a new unit of measurement called Pitch Audible Difference, or PADs. One PAD is equal to 5 cents. That’s because 5 cents is the approximate amount of pitch difference needed for the average human to detect a difference between two tones. Since any other pitches may fall beneath the average person’s perceptive ability, we’ll limit our discussion only to pitch increments that can be detected. Now, an octave is made of 240 PADs, and a single note in our 12-tone system is made of 20 PADs.

The Basis of Harmony

The most important concept in understanding harmony is knowing how two notes fit together. Consider this: take any note, randomly, and use it as a root pitch. Now, add a second, higher note to the first in order to harmonize it. The resulting harmony is best represented as a division of wavelength; how many times the vibration of that second wave will vibrate within the span of the first. This is what determines consonance and dissonance, and this is how harmony is created. Human beings have probably already discovered the most important harmonies. Although there are musicians working with more than 12 notes, this lesson is concerned with those primary 12, why they exist and how they are tuned.

The lowest “C” will be the reference point. The other notes are in quotation marks because, as you will see, their definition is not fixed.

Take the tone called Middle C and produce a harmonically perfect major chord

As you recall, one note in our 12-tone system is equal to 20 PADs. The fact that the “true” major third above C is 76 PADs and not 80 means that our 12-tone system does not accurately represent that note. It is a little bit off. That’s because our system is tempered, or adjusted so that all 12 notes are exactly equal to each other. Although the major third is not perfectly correct in our system, the perfect fifth basically is. (Every perfect fifth in 12-TET is almost exactly tuned to a ratio of 3:2. The result is off by less than one half of one PAD, a difference not noticeable to most human ears.) If we decide to tune every E on the piano up by 4 PADs, now the major third is correct but the perfect fifth from A to E would be incorrect. Temperament is a compromise, and nobody’s perfectly happy.

Why 12 Notes?

All human music is based upon harmony which stems from apparently natural laws.

For instance, sing a “C” note. Ask your friend to sing a perfect fifth above, or “G”. Now, play the same two notes on any modern instrument. This interval, the perfect fifth, is the strongest and most consonant interval. Its frequency ratio is 3:2. That means the G will vibrate at a frequency of three times per every two times that the C vibrates. And actually, in our 12-tone tuning system, fifths are represented very well. The difference between a true perfect fifth (3:2) and a fifth in our system is so slight that it falls beneath one PAD. Therefore, the 12-tone system represents this interval quite accurately. However, every interval is different. And some intervals are so poorly represented in the 12-tone system that they are actually 10 PADs away from the note they are intended to be. The distance between two notes is 20 PADs. So a discrepancy of 10 PADs is exactly halfway between two notes, and is very serious. It means that the interval is grossly inaccurate.

So why 12 notes? During the course of history, humans established harmonies for any given note based on a second note corresponding to a certain ratio of the first. Opening the debate of how many notes should be in an octave does not necessarily suggest there are more than 11 pleasing ways to harmonize a single note.

How Many 12-note Systems Are There?

Throughout history, many 12-note systems have been proposed. Unless you really want to learn a lot about this topic, don’t worry too much about the differences between these systems. Here’s what you need to know:

Cumulative Damage

The differences between the notes of a key played in 12-tone equal temperament may not be big enough to bother most people. But they begin to widen when one considers the possibility of repeating the calculation for another note.

Consider that we are in the key of C major and we play an E note, which as we know is already 4 PADs off of its true center. This difference may be slight. But now we want to use that E as a V/VI secondary dominant chord. So we raise its third to G♯. But the source of the G♯ is not that of a note 5:4 above the root, E. It is also equally tempered. The E was already four PADs flat, and we now know every major third in our system is four PADs inaccurate. So now the new G♯ is actually 8 PADs different than the correct note that would correspond to the leading tone that resolves to A. When we consider this type of accumulation, we can see the big problem with equal temperament tuning.

In summary, the difference between equal temperament and the “true” notes it attempts to approximate may not be easy to hear in just one or two examples. But over the course of a longer piece of music, it will become clearer and more noticeable.

Listening Examples

You may find it difficult to hear the difference, especially in such a short sample. If you think the files are exactly the same, try playing them both at the same time. Then you will be able to hear the difference.

More Examples

• Comparison of Three Tuning Systems
• Acapella demonstration: Just Intonation vs. Equal Tuning
• Canon in D in 3 Different Tuning Systems

As the creator of Hub Guitar, Grey has compiled hundreds of guitar lessons, written several books, and filmed hundreds of video lessons. He teaches private lessons in his Boston studio, as well as via video chat through TakeLessons.