[This post is the third in a series on Introductory Music Theory. The series begins with A Musical Theory. The previous post in this series is Stepping Stones. The next post in this series is Intervals.]
Music is itself a form of communication, but communicating about the music that we play and create is a vital part of being a musician. I am referring not only to communication with other musicians, but also to the internal dialogs we have with ourselves when thinking about the music that we play. A necessary part of being able to communicate effectively is to have fluency with an appropriate vocabulary. The musical vocabulary is comprised not only of words but also of numbers. Numbers are used in various ways in different musical contexts, and this at first can seem quite confusing. In this post I will introduce some of the ways we use numbers to communicate about music.
In the last post we described the interval of two half-steps as a whole-step. Now since ½ + ½ = 1, we will also refer to a whole-step, simply, as 1 step. We can now describe any musical distance numerically as some number of steps. For example, when we looked at the Do-Re-Mi scale we saw that to get from Do to Fa we moved two whole-steps and a half-step. This is more succinctly described as 2½ steps. Now it is very important to bear in mind that the basic unit of measurement (the musical inch) is the half-step. In other words, notes that are right next to each other – as from one fret on a guitar to the next higher or lower fret (on the same string) , or from one key on the piano to the closest neighboring key on either side of it – are ½ step apart. So when we describe the interval from Do to Fa as 2½ steps, we can see that this is the same as five frets on the guitar, or five keys on the keyboard!
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Numbers are used for more than just describing intervals. We will now look at using numbers to describe the position of notes on a scale. Specifically the position of the notes on the Major Scale, which, as we know, is another name for our old friend, Do-Re-Mi. (We refer to the Do-Re-Mi scale so much in music theory that one could reasonably think it is called “Major” because of its great importance in the way we describe music!) Using numbers for scale position looks like this:
Do Re Mi Fa So La Ti Do
=
1 2 3 4 5 6 7 8
In other words we can refer to Do as 1, Re as 2, etc. When using numbers to describe scale position, it is crucial that one not get confused with using numbers for distance (intervals). The distance from Re to Mi is 1 step, and the distance from Mi to Fa is ½ step (to review this, see the post Stepping Stones). Using numbers for these scale positions, we can say the same thing the following way: The distance from 2 to 3 is 1 step and the distance from 3 to 4 is ½ step. It bears repeating that the size of the interval from 2 to 3 is different from the size of the interval from 3 to 4. So you can’t subtract position numbers to get the distance between notes.
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There are another set of numeric terms used in music theory that combine the concept of position numbers with the concept of distance values by describing the interval from the first note on the Major Scale (Do or 1) to each other note on that scale. Ordinal numbers are used for this purpose (i.e. second, third, etc., instead of two, three …). Below is a table describing this usage:
Do to Re (1 to 2) = a 2nd (Whole step) Do to Mi (1 to 3) = a 3rd (Whole + Whole, or 2 steps) Do to Fa (1 to 4) = a 4th (Whole-Whole-Half, or 2½ steps) Do to So (1 to 5) = a 5th (W-W-H-W, or 3½ steps) Do to La (1 to 6) = a 6th (W-W-H-W-W, or 4½ steps) Do to Ti (1 to 7) = a 7th (W-W-H-W-W-W, or 5½ steps) Do to Do (1 to 8) = an Octave (W-W-H-W-W-W-H, or 6 steps)
Looking at this sequence, it is easy to see where the term “octave” comes from. The oct (as in octopus) means eight, so an octave is literally an eighth (although it is rarely, if ever, called that). It is important to note that these ordinal numbers are used to describe intervals no matter where those intervals occur. In other words, the interval of 2 steps can always be called a third no matter what note you start on. So, since the distance from 4 to 6 (Fa to La) is also exactly two steps, it can also be called a third. However, by starting with 1 (Do) it is easy to see where the terms come from. For example, the interval of 4½ steps is also called a sixth because that is the distance from 1 to 6 on the Major Scale. There is another special term for the distance from Do to itself, or 1 to 1 (no distance at all!). Instead of being called a 1st, this interval is called “unison” (“uni” meaning one, “son” meaning sound; or “one sound”). When two instruments or voices are playing the exact same pitches, we say they are in unison. There are also additional terms if we were to extend the above table beyond an octave.
Do to second Re (1 to 9) = a 9th (7 steps) Do to second Mi (1 to 10) = a 10th (8 steps) Do to second Fa (1 to 11) = a 11th (8½ steps) Do to second So (1 to 12) = a 12th (9½ steps) Do to second La (1 to 13) = a 13th (10½ steps)
The intervals greater than an octave are sometimes referred to as compound intervals because they can be seen as an octave plus one of the simple intervals (less than octave) that we described in the first table. Thus a 9th is also seen as an octave plus a 2nd, and a 13th is seen as an octave plus a 6th. Using ordinal numbers to describe intervals is not used beyond the 13th. So although one could conceptually refer to the (large!) interval of 14-steps as a 17th, this terminology is never heard in practice.
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At this point we have been using numbers in multiple ways, so let’s review by looking at the example of singing or playing Do followed by singing or playing So. We know from the definition of the Do-Re-Mi scale that these notes are 3½ steps apart. We can also refer to this scale as the Major Scale and then refer to these same notes as 1 and 5. Another name for the interval between 1 and 5 on the Major Scale is a 5th. If we play these notes on one string of the guitar we can see that 1 and 5 are 7 frets apart. And if we count how many piano keys it takes to get from Do to So we find that So is 7 keys above Do. Therefore:
The interval called a 5th = 3½ steps = 7 frets or 7 piano keys
Here we see three different numeric values used to describe the exact same thing. It may be confusing at first, but internalizing this information will provide a vital foundation for understanding the upcoming topics as we elaborate upon this terminology and learn to put it to use in various musical contexts. As we progress we will see how the use of these numbers gives us a powerful tool to aid our understanding of music.
