The musical interval, the perfect fifth (7 semitones), is called perfect because Pythagoras held it to be a manifestation of Divine perfection. The same is true for the perfect fourth and the octave. Musically, they fit together perfectly, the perfect fifth and the perfect fourth together, or side by side on the keyboard, forming an octave. They also fit together perfectly in terms of the arithmetical ratios associated them.
3/2 (perfect fifth) X 4/3 (perfect fourth) = 2/1 (octave)
Musically, an inversion is made by taking any chord, and then taking the lowest note and raising it in octaves. When the root or key note of the harmony is the lowest note, the chord is not inverted, but is in its root position.
A perfect fourth, or a perfect fifth, on its own, lacks a third note, and so doesn't have a harmonic root. But it can still be inverted in the same way.
So the perfect fourth is just an inversion of the perfect fifth, and vice versa. Any musical interval has a complementary one which is its inversion. For example, a minor third is the inversion of a major sixth. And always, any interval less than an octave, and its inversion, one on top of the other, or side by side on a keyboard, make an octave between their lowest and highest notes.
So musical intervals and their inversions are related to each other through the octave (or multiples thereof). All musical intervals are. That relationship, as an entire network, is represented by the Great Circle of Fifths:
This particular picture of the Great Circle of Fifths (above) shows enharmonics (alternative note names) not including double sharps or double flats. Really, there should be enharmonics shown all round the circle.
Around the circumference in the clockwise direction are notes separated by rising perfect fifths, or their falling inversions, perfect fourths. In the anticlockwise direction they are falling perfect fifths, or their rising inversions, perfect fourths.
The straight lines across the circle connect notes separated by the other musical intervals. And there, the same principles apply. In one direction is one interval rising, or another falling, and in the other direction along the line are the inversions, the other way around.
Complicated? Maybe, but it's an elegant representation of how musical intervals are related to each other. You can see that clearly, what you are looking at here, is a network.
You don't usually see the great circle of fifths represented in this way, in music theory books. And that's because music theory doesn't ordinarily go deep enough into the nature of the great circle of fifths.
The reason all those enharmonic note names appear is that going clockwise round the circle actually gives different notes to going anticlockwise round the circle. Going anticlockwise round the circle the notes shown only go as far as F flat. Beyond this we will start to run into double accidentals. Even going clockwise around the circle, by the time we reach the 12 o'clock position where we started on C, we have come not back to C but to B sharp.
This isn't just a question of musical grammar. Musical grammar - using the correct note name basis, or five steps in the alphabet A to G for a perfect fifth - actually reflects something deeper, that standard music theory doesn't deal with. It reflects an actuality about the relationship between arithmetical ratios and musical intervals.
Pythagoras is said to have known that the perfect fifth has associated with it the ratio 3:2, and the octave, the ratio 2:1. We see these ratios (or their inverses, 2:3, and 1:2) in pipe and string lengths in musical instruments, when they are producing the corresponding intervals. It also occurs in other ways, such as in frequency ratios and wavelengths. Other ratios are associated with the other musical intervals.
As a mathematical object, the great circle is like a multidimensional equation between many arithmetical ratios, but in order for the circle to arithmetically work out, the ratios have to be compromised. For example, those perfect fifths around the circumference cannot all have the ratio 3:2, whilst the octave has the ratio 2:1. Going clockwise round from the C, there is a difference between the size of the interval made by those 12 perfect fifths with a ratio 3:2, round the circumference of the circle, and the interval of 7 octaves with a ratio 2:1, that would otherwise end on the same note.
Basically, if you want to express that mathematically:
[(3/2)^12] > (2/1)^7
That's why going round in perfect fifths we end up back at the 12 o'clock position on a B-sharp, and not on a C. If we are sticking to our ratio 3:2 for every perfect fifth, and 2:1 for an octave, then going up in 12 perfect fifths doesn't bring us to the same note as going up in 7 octaves. Going up in perfect fifths, we arrive at a B-sharp, whilst going up in octaves, it is of course still a C that we arrive at.
In short, B-sharp is not the same note as C. The B-sharp is, musically speaking, a little sharp of the C. The same thing is true of an E-sharp and an F. The E-sharp is a little sharp of the F.
You might think that cannot be, because on the keyboard there's only one white key for both notes, but that's because in the tuning of the keyboard, those ratios we were talking about, are compromised, in order to allow this to happen.
The difference between a B-sharp and a C, or between an E-sharp and an F, as a musical interval, is called the Pythagorean comma. It's a micro-interval, but that doesn't mean it's insignificant. To dispell any doubt about its significance, here's a "perfect" fifth tuned on a piano, not between an F and a C, but between an E-sharp and a C, without any compromises. Of course, in terms of musical grammar, that's not a perfect fifth, but a diminished minor sixth:
There is a highly technical term for this musical interval :). It's usually called a wolf interval. No one really knows how the term originated, but we can guess easily enough. You could tune any perfect fifth on the piano so that it sounded not like this, but optimally beautiful and harmonious. But you can't tune all the perfect fifths on the keyboard in this way, unless some of the octaves sound equally as bad as the wolf interval. (So what happens on modern instruments is that all the perfect fifths are compromised a little, by a twelfth of the Pythagorean comma, or thereabouts).
There was once a whole science concerning this incompatibility between musical interval ratios, called the science of harmonics. In the Middle Ages it constituted the academic subject of music which was studied together with arithmetic, astronomy, and geometry, at Oxford, as the quadrivium.
The reason it was so important as to be part of the quadrivium, was that the universe itself was thought to be structured on musical principles and the arithmetical ratios associated with them. Astronomy itself was based on this idea.
Today, in our modern scientific world, things have moved on. Firstly, we now know that the motions of the planets and stars are not based on the arithmetical ratios associated with musical intervals. Secondly, we also know from modern acoustics that defining a musical interval simply by a single arithmetical ratio, is an idealisation and an oversimplification, that only really works in practice as an approximation, and doesn't reflect the true complexity of sound or the psychoacoustics of how we hear things.
Nevertheless, even as an approximation, defining musical intervals by arithmetical ratios does remain meaningful and useful in musical acoustics. It still represents the basic structure on which all musical intervals are built.
You will often see these ratios in relation to pipe lengths and string lengths in musical instruments, but really, the basic mechanism in nature that drives all this, is a structure called the harmonic series.
The harmonic series is like a set of ingredients on which many musical sound recipes are based. Most sounds we hear are like complex recipes made from a mix of sound ingredients. If the sound seems to us to have a definite musical pitch, then the chances are its ingredients fall into a harmonic series, or something close to it, or to part of it. It is the harmonic series, as a set of sound ingredients, that our ear and brain unconsciously recognises as creating a definite musical pitch.
It's not that musical pitch itself is a fixed, objective thing, like the properties of a sound ingredient, such as frequency and amplitude. Rather, musical pitch is a perception, and is psychoacoustic. It is subject to how the brain and ear is performing. So it can't be scientifically measured, except as an inference.
People talk about pitch intervals as being measured in cents, where a cent is a hundredth of a semitone, expressed as a ratio (actually, as it happens, it's an irrational number), as if that's a measurement of pitch. It's often referred to as a pitch measurement. But in fact, it's no such thing.
What we perceive as a musical pitch remains a subjective perception, which is not something that a simple arithmetical ratio or number can define. None of our perceptions can be scientifically defined in such a simplistic way, because they arise through the complexity of the brain.
The cent, as a scientific unit, is actually just a particular single frequency ratio. That's how it's defined. It's a single frequency ratio, and most musical sounds do not consist of a single frequency. Nor do they consist of a mathematically perfect harmonic series.
Human beings are creatures who commonly imagine that their perceptions are always of things outside their own mind, separate from their self. But the truth is that nature doesn't work like this. Musical pitches don't exist as things separate from our mind.
The sound ingredients in a musical sound we perceive as having a musical pitch, are usually called harmonics or partials, and each one is effectively a musical note in its own right, "hidden" inside the overall sound of the musical tone. The harmonic series of ingredients is theoretically infinite, but in practice is limited in a sound recipe. A given sound recipe is determined by the amount of each ingredient present.
So, our brain responds to the harmonic series, as a sound recipe, usually unconsciously, and from it, recognises a musical tone with a musical pitch. We can still recognise pitch in some simple sounds that aren't made from the harmonic series, but if a sound is rich and complex in its timbre, then it will be the harmonic series in its ingredients, or something close to it, or to part of it, that enables our perception of a definite pitch.
In practice, most musical tones have ingredients that vary from a true harmonic series, from just a little, to being completely unrelated. Nevertheless the harmonic series remains as a mathematical "ideal" for tones with a clear musical pitch. There is actually a mathematical relationship between any complex waveform or vibration that repeats itself regularly, or is periodic, as it is called, and the harmonic series. One mathematically equals the other.
The further away a waveform or vibration is, from being periodic, the more its ingredients deviate from being in a distinct harmonic series. There's a mathematical relationhip there, too.
As ingredients, harmonics are numbered 1, 2, 3... and so on. Each one has its own musical pitch, depending on the musical note in question, for which this particular set is the ingredients. But the relationships between them, as musical intervals, is the same for all notes.
Between harmonics numbers 1 and 2 is an octave. Between 2 and 3 is a perfect fifth. Between 3 and 4 is a perfect fourth. And so on. The musical intervals between them go on getting smaller and smaller, the higher the numbers. By the time you get above number 10, you start to run into intervals unfamiliar in most Western music.
Listen to that "perfect fifth" - or wolf interval - again:
What's happening here, is that the two individual notes, the E-sharp and the C, are individually still notes with distinct pitches, and are made from recipes of partials that are pretty much in the harmonic series. When we mix them together, by playing both notes at once, as we are here, there are some partials or ingredients of one note that are almost the same as some in the other note. But because they are close, and yet not identical, when they mix, they sound like this:
This kind of sound is called beating. It happens when ingredients that are similar, but mismatched, are mixed together.
Together, these ingredients sound like this:
And if you put them together with these ingredients:
You can see how they contribute to the unpleasantness of the interval itself:
It's because the interval itself is not tuned to the ratio 3:2, a ratio that occurs in the harmonic series, that mixing the two notes, each itself made from the harmonic series, creates the mismatch.
You'll see from this that the Great Circle of Fifths is very much related to the harmonic series. In fact, without the harmonic series as a facet of nature in the first instance, the Great Circle of Fifths would be pretty much meaningless, although you wouldn't know it, just from most music theory books.