As a teenager I was intrigued how my history teacher could refer to the Victorian period as being both in the 1800s and in the 19th century! It wasn’t till more recently that I fully understood the two methods of counting numbers humans have mysteriously been using over the years and the interesting impact that has on the world of music. Curious? Let me explain.
What Are Ordinal Numbers?
The ancient Romans and Greeks employed a method of counting we now call “Ordinal.” This means that they didn’t have a concept of zero in their maths and therefore they would start their counting from one. Think about Roman numerals around a clock face; where’s the numeral for zero? Do you know of the Roman numeral for zero? No? That’s because there is none! An ordinal number expresses it’s position in an order, for example first, second and third prize. That is why we can say that the Victorians lived in the nineteenth century, using the old ordinal counting method.
What Is Cardinal Counting?
If ordinal numerals is the “old” method of counting then cardinal is the “new” way. Following through our example of the 12 hour clock, the noticeable difference of the 24 hour clock is that the hours run from 0:00 to 23:59. A cardinal numeral expresses amount as supposed to ordinal which expresses order. Employing this modern cardinal method, zero is presumed and full numbers are counted when reached. For instance, when measuring a piece of wood our rule will show zero at the end and we count full inches or millimetres to determine the length. That is why we can say that we can say that the Victorians lived in the 1800s, using the new ordinal counting method.
We take for granted the use of both methods of counting in music but new students can find the differences a little bewildering at first and careful thought to our explanations can go a long way to avoid ambiguity.
We use ordinal (order) counting in music to describe intervals, degrees of the scale, bar (measure) numbers, finger numbers, frets etc. and yet use cardinal (amount) to count semitones (half-tones) and tones (whole-tones). For example, when a student is counting an interval they need to realise that they count the starting note (ordinal) but when they count three semitones (half-tones) lower than the keynote of a major key to find the keynote of the relative minor, that they should not count from the starting note but start counting from the next note on (cardinal).
Can you think of any other examples where understanding and explaining the difference between ordinal and cardinal counting can help music students?