Why intervals are not named after distance [duplicate]

I was wondering about the foundations of the way we name intervals.

For example, the interval between C and G is a fifth because there are five notes from C to G. But it's a common mistake of the person learning intervals for the first time to count four: one step from C to D, one from D to E, one from E to F and one from F to G. This is what I call "distance", because one intuitively counts the steps.

It seems to me that both approaches are valid if one was to rebuild the principles of music theory. So the question is, why is it the way it is? I think that advantages and disadvantages are worth mentioning but I also think that the truth behind this must be on historical reasons.

Edit: to sum up, I see there are (were) two options for naming intervals. One was counting note by note, and the other, counting the steps. My question is how and when the first one became the standard.

Please take into account that why this option was “chosen“ is relevant but not the main interest, so in my opinion this shouldn't be considered a duplicate of Why aren't intervals zero indexed. It's rather a variant, with a different point of view.

Regarding motivation for the question: I saw little children getting confused because they counted four steps from C to G, and I realized that it was perfectly natural to make that interpretation. As children, they don't have their minds pre-programmed as grown ups do. We consider that calling that interval a fifth is natural exactly because of that: we got used to it.

• Comments are not for extended discussion; this conversation has been moved to chat. – Doktor Mayhem Dec 25 '16 at 21:18
• @DrMayhem I think this comments shouldn't have been moved to chat. They contained a useful link to a related post, and they didn't contain one extended discussion, but several very different comments with my replies. And they weren't a lot of comments. This motivated meta.stackexchange.com/questions/289025/… – Emilio Dec 29 '16 at 21:19
• Unfortunately the end result is the same: difficult to get to the answers. If there are bits in those discussions that you think should be in answers or added to the question, please do that. Remember comments are not meant to be permanent. – Doktor Mayhem Dec 29 '16 at 21:21
• @Dr Mayhem I also think this shouldn't have been closed as duplicate. I edited the post adding clarifications. I don't consider this question is the same. It's a variant, and as long as I read this can be useful. Moreover, it had a lot of answers, upvotes, and even an accepted answer – Emilio Dec 29 '16 at 21:22
• None of those would stop something being a duplicate, Emilio. And as you can see, 5 people did think it is a duplicate. – Doktor Mayhem Dec 29 '16 at 21:26

This question is probably more about linguistics than music. This "illogical" counting system goes back at least as far as the ancient Romans, who used it for dates. In Latin, "the day before X" and "the second day before X" both refer to the same day (i.e. "day X-1") and "the third day before X" means "day X-2". This counting convention also appears the Bible, where "the third day" after Friday is Sunday, not Monday.

Also, the notion of "zero" as a number did not reach Europe until about 1200AD, and the origins of western music theory predate that mathematical innovation.

• I like the linguistical point of view and the example about the Romans. But I think that musicians didn't start talking about intervals until a few hundred years ago – Emilio Dec 19 '16 at 15:27
• @Emilio: The seven-tone Pythagorean scale dates back a couple thousand years. The Greeks called the perfect fifth the "diapente", which literally means "across five" [notes]. Similarly, the perfect fourth was the "diatesseron" ("across four"). So the counting system for intervals was established well before the European Renaissance. – Michael Seifert Dec 19 '16 at 15:45
• Yep- you all have it: it's a linguistic (or counting) convention, pretty obviously. Equally obviously, we should rather call a fifth a 3/2, but as long as everyone knows what's meant, what's the big deal? – Scott Wallace Dec 19 '16 at 16:26
• @Emilio the earliest Western writing about musical temperament and intervals (dating back to the 13th century or earlier) was linked to the Catholic church, and in Latin. The church still uses the term "octave" to mean the time interval between a two days with the same name, e.g. Wednesday to the following Wednesday, which we would call seven days not eight. – user19146 Dec 19 '16 at 19:29
• Yet in music 'an octave' works in the same sort of way; eighth note away, (hence octo) but seven spaces. – Tim Dec 20 '16 at 17:03

It's the same reason why scale degrees start on one as the root rather then start at zero which is due to the basic ideas of counting rather than distance. The counting always starts on the first note in the interval which is considered "1". In most other fields and with a more modern approach to it, it would be called 0. So for your example for C-D the C is 1 then we go up to 2 on the D. While counting from 0 may make more sense, it wouldn't make sense to change the terminology as every single musical text would have to change or else you would make them useless.

There are more modern approaches to intervals such as the one in set theory where the distances between notes are just enumerated and based on semitones. If you are doing any computations with distances between notes, they make things much simpler.

• Distance is not a modern concept. Although you provide an excellent explanation and start numbering in 1 may be natural, the distance to the starting point could never be defined to be 1. It's again distance vs. counting. Of course once a terminology is settled changing it is not worth the effort – Emilio Dec 19 '16 at 15:18
• @Emilio I'm wondering if thinking of an increase in pitch in the same way as an increase in distance makes sense before the idea of frequency is fully understood - which inspired this question : music.stackexchange.com/questions/51265/… – topo Reinstate Monica Dec 19 '16 at 15:58
• @topo morto In this context, distance is a word for addressing more precisely the action of counting steps instead of counting notes. So it makes perfect sense to me without the concept of frequency – Emilio Dec 19 '16 at 16:16
• Despite what I said about distance, after thinking about it I like more the idea of counting steps, which seems more natural. Whether zero is or not a number for start counting notes, is not key here. – Emilio Dec 21 '16 at 5:40

I wonder even whether 'interval' is the apposite word. An interval is the space in between, I think, so maybe it is a misnomer. However, the first note is always called 'one', etc., and it's probably too late in the day to change things. I remember reading somewhere that scientists actually proved that the positive terminal on a DC battery is negative, but sometimes we have to let sleeping dogs lie.

C>G is 1st to 5th, so I guess it was easier to say C>G IS a fifth.

Because one is not interested in the distance itself, but in the relationship between the two notes.

Seven semitones distance tells me nothing, but an interval of fifth rings me a bell, because it relates the root note of the interval with the fifth grade of its scale:

`C - D - E - F - G - A - B - C`
`I ------------- V`

Also worth noting, you can alter the name of the interval by appending or prepending attributes (minor, major, augmented, diminished...): minor third conveys more information than just three semitones, even if the two definitions are equivalent.

Ok, by experience I immediately know that "3 tones and a half" are a (just) fifth, but the current nomenclature helps having a better visual clue of what's going on, especially if mentally displayed on a piano keyboard.

It's like saying that the town library is 10m walking from where I live, rather than 997m far ^____^

• 'Seven semitones' means a lot to me as a guitarist, 'a fifth' less so. As a muso generally, both are significant. – Tim Dec 20 '16 at 16:58
• @Tim good point, the fretboard is much more linear than the keyboard. That's why I am learning to play guitar at the moment : )) – moonwave99 Dec 20 '16 at 18:21

The name of the interval can be thought of as derived from the second note's relationship to the first note in terms of scale degrees, and where the first note is taken as "1". So for a fifth we can think of the root of a temporary, imaginary, scale which begins on the first note of the interval. The name of the interval then tells us scale degree of the second note (as well as the quality of the interval).

This is really just another way of saying what another answer says, which is that we start our counting by naming the first note of the interval as "1". But we extend that concept slightly here to say that intervals names relate to the scale degrees of a scale that would begin on the first note of the interval.