How do we produce more tones without going beyond the 3-limit? The simplest solution is to use each of the new tones to produce another new tone. Recalling that the first new tone is related by 3, we use the relation to produce two new tones symmetrically around the source tone. This means creating relations of 3 times 3, which is 9. On the one hand we have 9/1 above the source tone and on the other we have 1/9 below the source tone. To justify these tones, we find duplicates by successively multiplying and dividing by 2. This gives 9/8 above, and 16/9 below the source tone.

These two new tones are *inversions* of each other. The figure above shows how these new
tones are produced symmetrically, and the number line below reflects how we hear these new tones
along with the others within the boundary tones.

9/8 is equal to 1 and 1/8, or 1.125 as a decimal fraction, and 16/9 is equal to 1 and 7/8, or 1.77 as a decimal fraction, as shown below on a typical number line.

The names of these two new tones are derived from their proximity to the tonic. One is directly
*above* the tonic and the other is directly *below* it. The prefix *super* means
*above* and, as stated earlier, the prefix *sub* means *below*; hence, these two
new tones are called the *supertonic* and *subtonic* respectively. Just like the dominant
and subdominant, the names of this pair of tones reminds us that they are *inversions* of
each other.

The figure below summarizes the names of the tones derived so far.

At this point, we have derived a collection of tones which is in fact found in a wide range
of folk musics from around the world. These five tones are known as *pentatonic*; *penta*
means *five* in Greek. Pentatonic music has a recognizable sound, resulting directly from the
*intervals*, the distances between the tones, in the system. The smallest interval between any
two tones in a pentatonic system is 8:9, which appears in three places.

Notice that an *interval*, the distance between two tones, is found by *dividing* the
higher tone by the lower tone, and the result is expressed with a colon in the form *lower : higher*.
This form is used to distinguish *intervals* from *tones*, which are expressed with a slash
in the form *higher / lower*. The interval 8:9 is significant, and a bracket-like figure is often
used to represent this interval.

Why stop at five tones? We can still produce more tones. Again, the simplest method is to use each
of the new tones to produce another new tone. We use the relation 3 to produce two new tones
symmetrically around the source tone, which means creating relations of 3 times 9, which is 27.
On the one hand we have 27/1 above the source tone and on the other hand we have 1/27 below the
source tone. To justify these tones, we successively multiply and divide by 2. This gives 27/16 above,
and 32/27 below the source tone. These two tones are *inversions* of each other.

27/16 is equal to 1 and 11/16, or 1.6875 as a decimal fraction, and 32/27 is equal to 1 and 5/27, or 1.185 as a decimal fraction, as shown below on a standard number line.

The names of these two new tones correspond to thier proximities to the tonic tones and the dominant
tones. The tone 32/27 is roughly in the *middle* between the tonic and the dominant, while the
tone 27/16 is roughly in the *middle* between the upper tonic and the subdominant. The word
*mediant* means middle. 32/27 is the called the *mediant* while 27/16, being between then
tonic and *subdominant*, is called the *submediant*. Again the names of this pair of tones
reminds us that they are *inversions* of each other.

The figure below summarizes the names of all of the tones we have derived.

At this point, we may notice that the number line is getting fairly crowded. A small interval appears between 9/8 and 32/27, and also between 27/16 and 16/9.

The appearance of this smaller interval, 243:256, marks an arrival point, and a temporary stopping place for the derivation of our system. A wedge-like figure is used to represent this interval.

These seven tones are considered a complete group. The two symbols of the bracket and the wedge can be used to show two different intervals from one tone to the next.

These seven tones are called *natural* tones, or simply *naturals*. The brackets and
wedges designate intervals which at this point we can call *steps*. The two sizes of steps
are given the following names which roughly describe their *approximate* relationships to
each other.

Sounding all seven of the natural tones in sequence is considered to have stepped
*through the tones*. The term *diatonic* literally means *through tones* in Greek;
hence, this is known as a *diatonic* arrangement of tones. The modern ordering of tones from
lowest to highest is known as a *scale*; hence, this arrangement is also known as a
*diatonic scale*.

Beginning with the 3 Tones, we have
derived four new tones in a symmetrical arrangement, creating a *diatonic scale* of seven
tones.

Our next task is to come up with simple ways to represent the tones using letters and symbolic notation.

NEXT: Letters & Notation