# Number as quantity, order, and identity



## millionrainbows (Jun 23, 2012)

We all know what quantity is. Let's say I have 53 sheep. If I sell you one sheep, I have 52 sheep left. Easy enough.

However, if I number each sheep 1-53 in order to keep track of them, then they have been given a number identity. For example, if I sell you sheep number 52, then I will have 52 sheep left, although sheep #52 is gone. So now we can see the difference between number as quantity and as identity.

These two ideas used to get mixed up in the olden days; we had no "zero" because people tended to see numbers as representing actual objects, and when they counted their sheep, for instance, each number corresponded to an actual sheep. There was no "zero sheep;" the concept was useless to these lamb-eaters and traders, who dealt in concrete terms.

This is part of the reason time is usually measured in numbers without zero; there are 7 days in a week, but there is no "zero" day of the week; they are 1 thru 7, as identities. There is no "zero year;" Christ was crucified in the year 1 A.D., and the year before that was I B.C., not "zero."

From WIK: Astronomical year numbering, used by astronomers, includes a year zero (0). Consequently, the first century in these calendars may designate the years 0 to 99 as the first century, years 100 to 199 as the second etc. However, in order to regard 2000 as the first year of the twenty-first century according to the astronomical year numbering, the astronomical year 0 has to correspond to the Gregorian year 1 BC.

According to WIK:
Start and end in the Gregorian Calendar

According to the Gregorian calendar, the 1st century A.D./C.E. started on January 1, 1 and ended on December 31, 100. The 2nd century started at year 101, the third at 201, etc. The n-th century started/will start on the year 100×n-99 and ends in 100×n . A century will only include one year, the centennial year, that starts with the century's number (e.g. 1900 is the final year in the 19th century).

1st century CE and BCE

There is no "zeroth century" in between the first century BCE and the first century AD. Also, there is no 0 AD[1]. The Julian calendar "jumps" from 1 BC to 1 AD. The first century BC includes the years 100 BCE to 1 BCE. Other centuries BC follow the same pattern.

Arthur C. Clarke gave this analogy (from a statement received by Reuters): "If the scale on your grocer's weighing machine began at 1 instead of 0, would you be happy when he claimed he'd sold you 10 kg of tea?" This statement illustrates the common confusion about the calendar. If one counts from the beginning of A.D. 1 to the ending of A.D. 1000, one would have counted 1000 years. The next 1000 years (millennium) would begin on the first day of 1001. So the calendar has not 'cheated' anyone out of a year. In other words, the argument is based on the fact that the last year of the first two thousand years in the Gregorian Calendar was 2000, not 1999.

So, in our non-zero system, the first century consisted of the years 1 B.C. thru 100; the second century was 101-200; and so on, until we get to the eighteenth century, 1701 to 1800, and the twentieth century, 1901 to 2000. That's why many experts were telling us that the "millenium" was not actually the year 2000, but January 1, 2001.

Part of the reason for avoidance of zero was religious, and goes back to the Church doctrine of "privatio boni"... Look it up if you're interested.

So, now that we understand the difference between number as quantity and identity, let's look at the idea of number as order, and see how this can apply to music.

Arnold Schoenberg came up with his "12-tone" system, in which "rows" or series of notes were the basis of the entire composition. For example, a tone-row could be C#-C-Eb-G-Bb-D-F-F#-G#-B-A-E, with each note occupying a place, numbered 1 thru 12. The notes had to be used in that order.

The row could be transposed up or down, and start on D instead of C#, which would give the row D-C#-E-G#-B-Eb-F#-G-C-Bb-F. The relations or intervals between notes remain constant; this is like a "template" which can be moved up or down in pitch. What Schoenberg was interested in preserving was the interval relations between the notes, not the pitches themselves.

The same thing happens when we transpose a major scale; C major becomes G major, essentially the same scale, only on a different pitch. The "major-ness" is preserved.


----------

