# Why aren't intervals zero-indexed?

Note: italics such as "2nd" refer to my proposed notation; existing notation is written without italics: "2nd".

Why is the interval between C and D called a 2nd, as they are 1 note apart? Surely this should be called a 1st? Similarly, the interval between C and E should be a 2nd, from C to F should be a 3rd, and so on; rather than an "octave" we have rather a 7th. ("heptave"?) Naturally, the interval from middle C to middle C is a 0th, as the same note is at a distance of 0 from itself.

This would greatly simplify matters. For instance, if one makes a triad by stacking two 2nd's, then the resulting interval between the top and bottom notes is a 4th, as one would expect from 2 + 2 = 4. The same would be true for any addition of intervals. In contrast, in the currently-used notation if one stacks two 3rds, the resultant interval is a 5th. This is clearly the sort of off-by-one error that results from one-indexing. Are there any reasons besides existing tradition (which is of course quite a compelling argument) why this notation persists?

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It definitely would be better, but what can we do? Heck, even mathematicians still use 1-based indexing (albeit not for something that's nothing but a difference), we're not going to convince a majority of the far more traditionalist musician occupation that zero-based is sufficiently advantageous to switch... – leftaroundabout Feb 10 '14 at 1:03
Because only programmers count starting at zero. I like your system, especially since I made an off-by-one error today asking (erroneously) about 16va instead of 15. – Mike Feb 10 '14 at 1:13
@Mike Actually, that question was the motivation for this question. – Istvan Chung Feb 10 '14 at 1:28
One is the note where you start – Neil Meyer Feb 10 '14 at 9:41
I don't see how it makes any difference as long as the convention is well documented and well understood. VB programmers start arrays at index 1, most others at zero. Americans call the ground floor the first, Brits don't. It is not a problem. – dumbledad Feb 10 '14 at 10:59

I really think the answer to this question has most to do with how music is composed. Tonal composers are not really thinking at all about the math behind the intervals; they're thinking about the sounds.

Another way of looking at this is that all tonal music is scale-based, and when playing a scale from bottom to top you number the notes starting from 1. The names for intervals come right out of these scale formations. i.e. The "first note of the scale" and the "fifth note of the scale" create an interval of a "fifth". (And in order to figure out the quality of the interval--major or minor--you always need to construct a new scale starting on the bottom note, so the off-by-one issue doesn't really arise.) No one before the advent of computer technology would ever think of calling the first element of any ordered set of things the "zeroth" item.

That's as far as I'm going to try to go on the "why 1-indexing makes sense" route, but I want to give you this additional nugget to chew on:

We DO have zero-indexed intervals in modern music called 12-tone technique. Not only is this zero-indexed, but it's in base 12! (When you think about it, using zero-indexing in tonal music doesn't completely solve the problem of mathematical consistency, because you're still in base 10 and not base 7...)

12-tone technique is a MUCH more mathematical approach to composition, so it makes sense to use a number system that is more mathematically consistent (or in this case, completely consistent with 12-tone-equal-temperment).

It's interesting to watch musicians without a mathematical background try to get their heads around 12-tone analysis -- thinking with zero-index is hard enough for a non-programmer, but then switch out of base 10 and give equal importance to all of the chromatic notes and all of a sudden nothing makes sense anymore. ;-)

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Just a quick addendum I've thought of since writing this: Since nearly all western music theory is learned with a keyboard (or at least with one close by), there is a strong STRONG link between counting on one's fingers and playing the first five notes of a scale. I'd bet that's where a good portion of this really comes from. – NReilingh Feb 11 '14 at 1:18

Let's think of a the C major scale. What note do you start on? You would start on a C. Would it make sense to call it the first note of the scale or the zeroth note of a scale? Most people would call that the first note of a scale as do musicians. That is why a D would be a second away because it is the second scale degree and then E is a 3rd away because it is the 3rd scale degree. This is why intervals are the way they are because the naming is more based on the position of the note on a scale (scale degrees) then the actual distance between the notes.

Let's just do a comparison with zero based note indexing and one based indexing:

```    Notes:  C D E F G A B C D E  F  G  A  B  C
Zero:   0 1 2 3 4 5 6 7 8 9  10 11 12 13 14
One:    1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
```

While it is easier to figure out compound intervals (intervals above an octave), I think it would be harder to think about the scale degrees that the intervals represent. To think that the second note of a scale is a 1st seems just if not more confusing to me than the distance of one note to the next note being a 2nd.

Since music itself revolves around the use of scales it makes more sense to look at it from a scale degree point of view than an actual distance point of view. Sure counting intervals would be easier, but at least I feel it would be more difficult to look at from the perspective of a scale.

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Yes ... but no. An interval is inherently about the distance between two pitches. In a logical sense a distance of zero would unambiguously mean the same pitch. However, the tradition is that an interval of a 1st means "the same pitch" and an interval of a second means the next consecutive pitch in a scale. In practice we don't often talk about an interval of a "first" so the confusion only arises when we talk about intervals larger than an octave, e.g. what is an interval of two octaves called? – Richard Padley Feb 10 '14 at 2:02
@RichardPadley My point was the names of intervals relate to the scale degree. An F is the 4th scale degree of a C scale and is a 4th in distance. In music those two ideas are intertwine. – Dom Feb 10 '14 at 2:07
Right, but if we had just always called the initial note the 0th scale degree, that wouldn't seems weird at all. – Pat Muchmore Feb 10 '14 at 2:12
It's probably appearing confusing because we're so used to the original numbering system. Dom, from C to F, distance-wise, is actually 3 in distance. I don't mean a third, but look - C-D-E-F ;there are only 3 spaces.Space = distance. Was there ever a year 0 A.D. Doubt it. The first note in a key (root note) needs to be 1, just like the winner of a race (the first) is 1. However, when we talk about which octave a note is in, don't we use , for example, Co ? – Tim Feb 10 '14 at 9:42

A slightly different answer from the others is this: the issue comes up when you think of intervals in cardinal terms rather than ordinal terms. While it isn't "wrong" to do so, it isn't "wrong" to think of them ordinally either.

Historically, intervals have been perceived as a musical entity in their own right, rather than a quantification of a distance between two notes. Therefore, the classification of intervals involves their being arranged in order, from smallest to largest.

This point is important to the reason for ordinality. Two people singing the same note are singing an interval called a "unison", or "singing in unison". This interval is also called the Prime interval, or interval of a first. Therefore, the interval that is one more than the first is called the interval of a second, and so on.

I won't get into why there are only 8 intervals in a 12-note scale...

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Yes!! This is why I prefer set theory. In the definition of interval and sets used in that system, 0 is always the starting point (usually, nowadays, C is arbitrarily defined as 0. So what used to be called a M3, is just pi4 (pitch interval four), and a m7th is pi10. The old system however, is just too well known to eradicate, and that's probably just as well...

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These off-by-one errors don't stem from 1-indexing of the notes but from the fact that instead of defining the (numerical) interval from X to Y as the number of increases from X to Y (which is Y-X), we define it as the number of notes from X to Y inclusive (which is Y-X+1). This is perfectly natural from a musical point of view: play three successive notes of the scale and you have a third.

Using the Y-X -definition for an interval does not greatly simplify the mathematics either. For example even with the traditional definition you can add any successive intervals A1, ..., An together by (A1-1)+(A2-1)+...+(An-1)+1 (proof: exercise).

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The intervals come from note positions in the scale. The primero is the tonic, so the C in C major. The D would be the secondo, and the interval is a second. etc...

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