Philip Tagg (2014-08-10)

The interval between **c** and **f** is a *fourth* (**4**) because, starting on **c** as **1**, **d** counts as **2** (a second), **e** as **3** (a third) and **f** as **4**.

There are, however, *only ***3*** steps* —three intervals— between **1** and **4**: [1] from 1 to 2; [2] from 2 to 3; and [3] from 3 to 4. The interval between **c** and **f** should therefore more logically be a *third* (**3**) because ‘interval’, when applied to music, means *difference in pitch* and because the difference between 4 and 1 is **4-1 = 3**.

Counting confusion continues with the *octave* (*octava* = ‘eighth’). That **8th** pitch is boundary for the *heptatonic* or **7-note** scale, e.g. c=1, d=2, e=3, f=4, g=5, a=6, b=7. The heptatonic scale then starts again an octave higher.

Take the note **c** in the second octave as an example: it’s **c²** (65.4 Hz., 2 octaves below middle c or **c ^{4}**). Counting

**Tip**. If you’re confused by 11ths, 13ths and 15ths, just subtract 7 (the *hepta*tonic octave). An **eleventh** chord contains the **fourth** (11-7=4) and a **thirteenth** chord the **sixth** (13-7=6). Th **twelfth** is an interval of one octave plus a **fifth** (12-7=5) and the **fifteenth** an interval of two **octaves** (15-7=8).

As if all that weren’t enough, the **octave (8)** *can also be equal to* **nine (9)**. That's because the octave’s three internal *complementary interval pairs* add up to nine —the *second* and the *seventh* (**2+7=9**), the *third* and the *sixth* (**3+6=9**), and the *fourth* and *fifth* (**4+5=9**). In other words watch out: **the octave (8) can ‘equal’ both 7 and 9***.*

All these inconsistencies are inevitable because *we use no zero* as starting point for counting intervals and because we mix cardinals with ordinals. We are dealing with the same anomaly as when we count years and centuries. Just as we had no year zero, starting instead in the year 1 —only 99 years in the first century but 100 in all the others— we call an interval of *zero* (**0**) steps (no difference of pitch, total absence of interval) a *prime* (**1**), as if it were an interval of one step. We then call a **1**-step interval a second (**2**), a **2**-step interval a third (**3**) and so on. There’s no point in trying to bring order into this ingrained confusion but it’s certainly worth bearing in mind.