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I found this list of octave clefs on Wikipedia. Here, there's a series of octave clefs listed as using 8 and 15 to alter the clef by one or two octaves respectively.

My understanding of music theory is that there are 12 notes in an octave, and only 7 whole notes in a typical scale. Why was 8 chosen?

Similarly, given that the gap of one octave is 8, shouldn't a double octave change be 16, not 15? Why was 15 chosen?

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Let's try some (algebraic?) maths. C-C. CDEFGABC 8 note names for an octave. CDEFGABCDEFGABC 15 note names for two octaves. Count to check!

Just like other intervals, the first note is note number one, not zero. It's not like measuring with a rule, or timing with a stopwatch. The first note is number 1.

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  • then why 8 and not 12/13? are the black keys just not counted?
    – Corsaka
    Commented Sep 12, 2023 at 10:22
  • @Corsaka - of course the black keys are counted - just not in key C! 8 notes are counted because they constitute the scale - the diatonic notes commonly belonging to a particular key. In key C, they happen all to be white, but in, say, key F#, there are 6 black keys (on piano), that get counted, leaving only a couple of those white ones to be counted/played, as far as this is concerned.
    – Tim
    Commented Sep 12, 2023 at 11:00
  • so octaves are entirely built around scale? should i open another question to dive into how/why?
    – Corsaka
    Commented Sep 12, 2023 at 11:01
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    @ojs covered in that answer 1+7 = 8, 1+14 = 15
    – haxor789
    Commented Sep 12, 2023 at 12:02
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    @Aaron "At the time the octave was invented, there were no black keys, as such": when hard and soft B were invented, western European music theorists were still using diapason to mean "octave," so that is unlikely to be true.
    – phoog
    Commented Sep 12, 2023 at 17:26
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Interval naming is a bit non intuitive.

It starts with the interval between two identical notes being a perfect unison, or perfect prime. The word prime comes from the number one. In maths, we would normally describe the difference between two identical things using the number 0. But in music we use 1, which may lead to off-by-one mistakes.

It continues. The interval between adjacent scale steps is a second, coming from number 2, rather than 1. And so on. Octave, the word originating from 8, describes an interval of 7 steps apart.

If you add another 7 steps to an octave, you get two octaves, but the resulting name will be quint-decima, originating from the number of 15 = 1+7+7, rather than 8+8.

The interval names in western theory are based on steps of the western 7-note scales, rather than the pool 12 chromatic notes used to build these scales.

So to answer the question "why?" – because certain names were invented long before more modern theory and approach to conceptualize music, and now nobody dares to change it. Note that early music often didn't even span beyond a single octave. Moreover, when analyzing western music, it is more natural to think in terms of scale steps, rather than semitones.

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Play a C major scale, two octaves. Count as you play, C, D, E will be 1,2,3 etc.

What number is reached when you get to C one octave up? Keep going, up to the next C.

That's why.

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They're the interval of an eighth (in Latin, octava - this is where the term "octave" even comes from) and a fifteenth (quintdecima) respectively.

Intervals are named by ordinal numbers. There's a first (better known as unison in English; in Latin the word is "prima" and literally means first), second, third... seventh, eighth (aka octave), ninth... and so on. There's no musical interval called a zeroth.

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