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I have a question about theoretical music theory...

So say whoever first invented/discovered the musical alphabet decided to make it an 8-letter alphabet, consisting of A, B, C, D, E, F, G, and H. The major scale pattern is now changed to "WWHWWWWH" so the C major scale starts and ends on C.

So if this was true and all we ever heard was music composed with the 8-letter music alphabet, would C6 and C7 (or any other notes with the same name on different octaves) sound like the same note in the same way they do with current music theory?

This seems to boil down to: do we hear notes of different octaves as being the same note simply because we're used to music being composed that way or because it's the fundamental nature of music and would stay that way even if we had a different music alphabet?

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    Octave equivalence is based on the frequency being doubled, so it has a basis in physics. However, you can choose to ignore this; e.g. the Bohlen-Pierce scale uses tritaves instead of octaves: en.wikipedia.org/wiki/Bohlen%E2%80%93Pierce_scale The fact that you've probably never heard of it gives an indication of how successful tritave equivalence is compared to octave equivalence. – Your Uncle Bob Oct 4 at 2:04
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    @YourUncleBob Another interpretation of the Bohlen-Pierce scale's obscurity is that octaves are culturally ingrained in our minds so deeply as to appear universal. Octave equivalence is a principle that for one reason or another seems to pervade human civilizations, but it's entirely possible that this is simply a human construct. If the past had been different, perhaps we'd recognise an interval ratio of 7:11 as our basic building block of harmony. – user45266 Oct 4 at 3:41
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    @Your Uncle Bob: I would vote your comment up if it were an answer. I think it's the answer tot the OP's question: octave equivalence and the obscurity of actual music that uses scales without octaves. – Tim H Oct 4 at 8:59
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    @user45266 It's not a human construct: octave-separated pitches resonate perfectly, and this is physics (mathematics, if you wish): it has nothing to do with cultural context. A specific cultural context (e.g., Bohlen-Pierce's tritaves) may ignore this frequency doubling in favour of another convention, but the pitches would still resonate. – Simone Oct 4 at 10:30
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    @CarlWitthoft The question is not about the division of the octave, but about octave equivalence. I agree that it is confusingly worded. (The upvotes are probably the result of the question being featured in the HNQ.) – Your Uncle Bob Oct 4 at 17:01

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Yes absolutely. An "octave" is all about a doubling of the frequency of the note, not the letters commonly used to refer to them. The octave can be split into any number of tones, which may or may not be equally (in the logarithmic sense) spaced.

We use a system of 12 equally spaced "semitones" (as we call them) in most western music, called "equal temperament tuning", but more or less sub-divisions are possible.

It's also worth noting that the use of letters is also not universal. Some places use "doh re me". Computers use integer numbers across multiple octaves. It's just a labeling system and there is nothing sacred about it.

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    It would be interesting to see if octaves are used by other kinds of music than the western twelve-tone one. – marcellothearcane Oct 5 at 20:16
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    @marcellothearcane Yes sure, you could say it is the single most intuitive interval. Indian classical music uses octaves with 22 intervals of varying size. – smcs Oct 7 at 8:32
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There is one misconception here: Music practice does not follow theory. Theory follows practice, and practice follows physics. You cannot just invent a note-naming system and expect to write pleasing music in it, the note-naming system must follow what is actually pleasing to hear, and what is pleasing to hear is dictated by the physics of sound.

You see, the point about the octave is not that it is some kind of convention. The point is, that one octave equals a factor 2 in frequency. This factor 2 is the reason why all the overtones in the spectrum of the higher note fall squat on the overtones of the lower note, allowing the two notes to sound together as one.

The fourth and fifth intervals are likewise fixed by very simple fractions of frequency (the fifth is almost 3/2, the fourth 4/3). The fact that we have 12 semitones is due to the fact that 2^19 = 524288 is roughly the same as 3^12 = 531441. This allows us to close the circle of fifths. This is the first point where a power of 3 comes close enough to a power of 2 to not sound way off. You cannot just add a 13th/14th semitone to the octave without destroying this relationship.

Furthermore, the circle of fifth is the basis for choosing which of the semitones to use within a scale. I won't go into details here, as they are a bit too involved for this answer.


Bottom line is, you cannot ignore physics. Our 12 semitones to an octave "convention" follows directly from physics. So...

If you add your H by relabeling one of the sharp/flat notes, the octave would remain unchanged, and thus sound the same.

If you instead add your H by putting more semitones within an octave, the "octave" will sound really off.

  • Comments are not for extended discussion; this conversation has been moved to chat. – Dom Oct 7 at 16:30
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Do we hear notes of different octaves as being the same note ... even if we had a different music alphabet?

Yes. Dodecaphony has twelve "letters." When we ask humans, or computers, to listen to music and write it down, both suffer from octave transposition errors, and it doesn't matter if what they're transcribing is 7-letter Beatles or 12-letter Schoenberg.

By the way,German notation used your eighth letter H.

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    By "Bach's culture" do you mean "German-speaking countries" (and sometimes elsewhere), or something else? – Richard Oct 4 at 10:05
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    I feel your remark about the h is a bit misleading; there is no mysterious new note in use, just the b is called h and the b flat is called b in German (it’s still this way). This stems from the unreliable printing process in the early days of music printing. B natural would be an angular b, of which the lowest line would sometimes not show up in print which made it look like an h. The round b for b flat didn’t have this problem obviously. – 11684 Oct 4 at 11:15
  • I spotted this for myself when I was young and I heard blink-182's I Miss You - listen to the scale during the chorus (around 0:57 in that video) and count. Seven notes, but it definitely ends on the same note it started on. Being new to music theory, I had to ask my teacher how this could possibly work when an octave should have eight notes. – anaximander Oct 4 at 12:57
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    'H' is still very much alive and kicking in several countries even now, centuries after Bach left us. Germany being one, where the use of H is substituted for our B, and their B is the only black key single letter name on a piano, referring to Bb. – Tim Oct 4 at 14:15
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    @Tim Actually, the H is used in all of Eastern and Northern Europe, not only in Germany (e.g. Poland, Czech Republic, Slovakia, ex-USSR and ex-Yugoslavia countries, Scandinavia). The only countries in Europe which have predominated letter designation but using B instead of H are the UK, Ireland and the Netherlands. – trolley813 Oct 7 at 6:29
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Yes, just as Esteban Gutiérrez said.

But a hypothetical musical alphabet containing not 7 letters but some other number is not so far-fetched. Guido of Arezzo invented a system of naming pitches. This system was made up of hexachords. As the name suggests, a hexachord has 6 pitches. If Guido had designed his system differently, we might have had to make do with only 6 letters for pitch names today.

Now the names for a hexachord's 6 pitches, ut, re, mi, fa, so, la, denote only the 6 pitches relative to that hexachord's bottom note ut. And Guido's system had hexachords based on G, C and F. A G hexachord's mi is B; an F hexachord's fa is B♭. To distinguish them, Guido called B "B durum" and B♭ "B molle". He might instead have chosen two different letters, which indeed the German system actually does, as Camille Goudeseune pointed out. In that case, there would have been 8 letters for pitch names.

But, no matter what our notation is, octaves would still sound the same.

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It actually is the fundamental nature of music. More precisely, what we perceive as a note is a sound whose frequencies form some particular pattern. This pattern has the peculiarity that there is one of those frequencies that is in some way the fundamental one (any other frequency is actually the fundamental frequency times an integer number), and what we perceive as octaves are sounds whose fundamental frequencies are related by a multiplication by a power of 2. For example, it is well known that in most popular music the fundamental frequency of A is 440hz, so 2*440hz=880hz and 2*2*2*440hz=3520hz correspond to the fundamental frequencies of an A as well.

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(I provide an answer of my own, since what I consider crucial is mentioned mostly in comments or indirectly hinted to.)

As indicated by user45266, the octave is the basis, since most cultures agree on the fact, that the same note in different octaves shares something important.

The partitioning into substeps is far more arbitrary and different cultures/times made other choices. The resulting scales sound different but the octave as interval is unaffected.

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A recent study of a tribe that has not been exposed to other cultures' music shows that they do not have a natural preference for any sort of musical interval. This is significant, as it tells us that even strong harmonics like octaves are a cultural reference, not an innate preference.

An article about this work with the Tismane' tribe, isolated from other cultures in rural Bolivia, can be found here: https://www.earth.com/news/interpretation-musical-pitch-culture/

That said, you can break down an octave any way you choose. Many cultures do. Ever notice the irregularly spaced frets on a sitar? They correspond to patterns of quarter- and half-steps in Indian music, just to cite one example.

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A musical octave is a precise physical relationthip between two notes where the second note is double the frequency or half the frequency of the first. For example Middle A (A4) is 440Hz (vibrations per second). Low A (A3) is 220Hz (half). High A (A5) is 880Hz (double). This is also the case for all other notes, High/Middle C for example being 261.63Hz/523.25Hz.

This is manifested in physical instuments as twice or half the length of a plucked string, twice or half the length of the tube of a blown horn.

In the (equal temperament) diatonic scale, the relationship between individual notes (half-steps in the western usage) is 2^(1/12) ~= 1.05946.

Some fun math:

Middle A (A4) is 220Hz.
Middle Bb (Bb4) is Middle A * 1.05946 = 233.08Hz. Middle B (B4) is Middle Bb * 1.05946 = 246.94Hz. Middle C (C4) is Middle B * 1.05946 = 261.63Hz. Bah-bah!

12 half steps is 1.05946^12 = 2. Double!

Major and minor scales skip some of the half steps as the original question notes but the WWHWWWH scales are all made from the 12 half steps between octaves.

If the scale is separated into more or fewer divisions which is the case in many cultures across the world and history, the octave is still the same because it is the physical halving or doubling of the frequency. To your ear, octave notes have the same sound because they are perfect harmonics of each other, ie the vibrations line up if the two notes are played together.

As an interesting side note, if you were to play A3 on a piano at 220Hz and A4 on a tunable instrument that was tuned (sharp) to 441Hz, you would hear a "wobbling" called a "beat" that wobbled once per second, corresponding to the mismatch in harmonics between what would otherwise be a perfect octave.

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We detect sound through the movement of stereocilia, which are tiny hairs that can be set to vibrate by sound. Each one has a different fundamental frequency, and detects sound most strongly when it is at that frequency, but also detects sound that is an integer multiple of its fundamental frequency. Thus, there is a biological sense in which a sound that is twice the frequency is perceived as being similar. In signal processing terms, there's aliasing in the Fourier transform of signals whose frequency differs by an integer multiple of the sampling frequency. Another place where this concept comes up is wagon wheels in film: if each spoke in a frame is in exactly the same place as another spoke was in the previous frame, it will look like the wagon wheel is standing still. If we call the frequency required to get a spoke in one frame to the same place as the adjacent spoke in the previous frame the "fundamental frequency", then all integer multiples of the fundamental frequency will look the same. The fact that we perceive a wagon wheel going twice the fundamental frequency the same as one going the fundamental frequency is physics, not cultural conditioning.

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The audio spectrum is divided into octaves, in which the individual notes within each octave precisely double the frequency of their counterparts in the octave immediately below. Within each octave, the spectrum is divided into notes; what we perceive as a uniform progression of semitones (chromatic scale) is in fact a logarithmic progression. the result of this is a seemingly continuous uniform progression of semitones from the lowest possible note to the highest.

Musical notes are rarely pure tones. All instruments produce a fundamental with some harmonic structure - a superposition of frequencies occurring at integer multiples of the fundamental frequency, thus selected pairs of these harmonic frequencies will be related as simple fractions/ratios (2:1, 3:1, 3:2, 3:4), etc.

As it happens, some pairs of notes in the familiar 7-note scale (12-note chromatic) have very nearly the same simple ratios in frequency we see in the harmonic structures of notes produced by physical instruments. Consider that any two pure frequencies, when superimposed, will create the effect of a third frequency - the difference between them. If this difference is small, we hear a "beating" effect. Depending on what that difference frequency is, the effect may be pleasing or it may be jarring. We use this effect in selecting intervals and building chords. The system we know provides a framework for this and so we have major and minor chords, in which the underlying note frequency value have certain arithmetic relationships to each other.

If you divide the spectrum differently - either changing the frequency interval between "cycles" (we can no longer call them octaves), or change the spacing between notes to increase or decrease their number, you will create different relationships between pairs of notes and thus things will certainly sound different.

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