Basics: Derivation and Discussion of the Natural Pentatonic and Heptatonic - tschw/perfect-harmony GitHub Wiki
As previously discussed, stepping in fifths lets us derive the musical notes. Five notes derived from four of such steps resemble a natural pentatonic tonality, while the first note is the root of its major mode. So starting at C we get C, G, D, A, E, depicted in green in the figures. Ordering by pitch we can derive C-Major-Pentatonic as: C, D, E, G, A. Because of equal temperament, we can derive a similar tonality starting at any note and it sounds the same just higher or lower, in other words; only its absolute pitch differs while relative pitch in invariant under transposition - in contrast to historic musical instruments before equal temperament was commonly being used which had to be tuned to a specific key.
More interestingly, the symmetry we get from equal temperament lets us change keys rapidly within a piece of music. It has been used extensively throughout many different styles of music, and even defines some of those, such as for example traditionally Blues and Jazz. This so called modal mixture is loads of fun in improvisational music and a celebrated key ingredient to freer forms of music, such as modern Jazz and Jazz-inspired fusion, which in turn nowadays influence artists serving the entire range of musical genres from contemporary to popular.
Further, the natural tonalities (as e.g. reflected by the piano keyboard) have beautiful, combinatorically fractal traits. Throughout the history of music, musicians have exploited these, some more and others less consciously, but as for an exhaustive, scientific discussion, clearly deserve close examination:
Extending above derivation of the natural pentatonic on both ends - with one fifth below and another above yields the natural heptatonic. In above example the respective notes are F and B, our tonality spans all the white keys, and (keeping the C we started at as the tonal center) is commonly called C-Major.
At the opposite end of the circle of fifths are all the remaining notes, the ones not in part of the natural heptatonic we derived, and following from the symmetry, resemble another natural pentatonic, which in case of our example are all the black keys on the piano keyboard, depicted in blue in the figures. The major mode of their pentatonic tonality is rooted at G♭/F♯.
Since the natural heptatonic consists of seven consecutive fifths, this means three subsets of five consecutive notes each, in other words; overlapping natural pentatonic structures! In case of the example, the respective major modes are at F, depicted in yellow, C, depicted in green, and G, depicted in red in the figures. As we extended C-Major-Pentatonic with two notes on both ends, F-Major-Pentatonic added the F- and G-Major-Pentatonic added the B.
Below figure shows the natural pentatonic structures of the piano keyboard on the circle of fifths. It also shows the musical intervals relative to C, but because of aforementioned symmetry, this depiction could be based starting with any other note - the structure is invariant under transposition:
Another observation is that C-Major(-Heptatonic, and also C-Major-Pentatonic as a subset thereof) is completely defined by F- and G-Major-Pentatonic, the predecessor and successor of C-Major-Pentatonic. Following from the symmetry of the circle of fifths, any natural heptatonic can be seen as the overlap of the predecessor and successor of its corresponding natural pentatonic.
In practice, because musical notes repeat in octaves, two consecutive fifths sum up to a major second, or, by another name; a whole tone step. So the two overlapping natural pentatonic tonalities that define a natural heptatonic are just two semitones apart! Since the natural pentatonic does not contain any semitone steps, all notes not in the natural heptatonic - the pentatonic inverse represented by the black keys in case of C-Major - must lie in their middle! These three natural pentatonic structures have a respective distance of 5-7 semitones from the one that corresponds to the heptatonic; a fifth down, 5 semitones, same as a fourth up, a tritone, 6 semitones, a fancy name for a flat or diminished fifth or sharp or augmented fourth (both having the same distance of six semitones in both directions), and a fifth up, that is 7 semitones ahead (see below for a complete discussion os musical intervals). These distances are depicted by the yellow, blue and red arrows in above figure.
The following figure shows how the piano keyboard reflects these natural pentatonic structures and how the note of C-Major, reflected by the white keys, exemplary for the natural heptatonic, can be either be found in F-Major-Pentatonic, the previous, or in G-Major-Pentatonic, the next pentatonic structure, respectively shown in yellow and red, or in both of them. C-Major-Pentatonic is shown in green all notes not in C-Major(-Heptatonic), that is F♯ or G♭-Major-Pentatonic, reflected by the black keys, in blue:
As discussed so far, the natural pentatonic fits with all its notes into the natural heptatonic at three different positions. Again, because musical notes repeat in octaves, additional notes end up in the gaps resulting from the overlap. These superpentatonic structures are self-similar, as they are all supersets of the natural pentatonic, but are not identical, for the additional notes turn them into degrees of the heptatonic depending on their positions. Taking a closer look at these differences brings us to the combinatorically unique property of the natural tonalities: they can be tranposed in their entirety by moving certain individual notes just by a semitone; the whole tonality moves by one step along the circle of fifths, this way jumping by four scale degrees - a fifth, that is 7 semitones, or by three degrees, a so called fourth, that is 5 semitones counting in opposite direction. In case of the natural heptatonic, its corresponding natural pentatonic, the green one in the figures, changes places with its predecessor or successor, respectively depicted in yellow or red, including the additional notes found in the gaps.
That in itself is no secret, as it is fundamental to score notation, however, the fact that the natural heptatonic and the natural pentatonic as its logical inverse are indeed the only out of all 4096 possible tonality subsets drawn from the set of twelve notes with this trait (ignoring the trivial singular edge cases of one single note and, inverse to that, eleven note structures) may come as more of a surprise! It can be proven with a computer program implementing an exhaustive search, or constructively, reasoning that by pidgeonhole principle, the structure of the black keys could not look any different within a system of twelve notes.
The following figures illustrate an alternative view on stepping around the circle of fifths that presents this fact more intuitively: understanding the black keys as a visualization of the inverse set of C-Major, that is all the notes not present, let's play an F♯ instead of F, and as a thought experiment, build a new keyboad where all the keys we play are white. Now it becomes obvious from the structure of the black keys, that the root of the Major mode, previously C has shifted and become G. We can play that scheme on repeat, rise C to C♯ and end up with D-Major, and so on - this way walking clockwise around the circle of fifths:
Similarly we can walk counter-clockwise, lowering the B to B♭ going to F-Major, lowering E to E♭ to B♭-Major, and so on:
Naming and Counting Conventions in Music
Scale, Tonality and Key
Music jargon can be very ambiguous. Scale might be the worst word, as it is commonly being used to mean several related but not quite identical things, depending on the context it is being used in. It could mean a) tonality - the subset of the twelve notes which is actually being used throughout a piece of music or part thereof, also b) tonality type, that is the structure of a tonality, without pinning it to an absolute starting point, c) key - that is tonality plus mode (tonal center), e.g. talking of C-Major could mean that the harmony of the piece is structured around C; it's the same tonality as A-Minor, as it consists of the same notes, but the harmony is centered around a different note, so it's a different key. Ultimately it could mean d) a sequence of somehow adjacent notes ordered by pitch you play on a musical instrument.
As discussed above, we can reach any natural heptatonic tonality by walking the circle of fifths, rising or lowering one note at each step, moving the tonality up or down by a fifth, depicted in clockwise or counter-clockwise directions respectively. This process is used to notate tonalities in score notation, by setting up to seven sharps or flats after the clef.
| Major Key | Minor Key | Alterations |
|---|---|---|
| C♭ | A♭ | B♭ E♭ A♭ D♭ G♭ C♭ F♭ |
| G♭ | E♭ | B♭ E♭ A♭ D♭ G♭ C♭ |
| D♭ | B♭ | B♭ E♭ A♭ D♭ G♭ |
| A♭ | F | B♭ E♭ A♭ D♭ |
| E♭ | C | B♭ E♭ A♭ |
| B♭ | G | B♭ E♭ |
| F | D | B♭ |
| C | A | |
| G | E | F♯ |
| D | B | F♯ C♯ |
| A | F♯ | F♯ C♯ G♯ |
| E | C♯ | F♯ C♯ G♯ D♯ |
| B | G♯ | F♯ C♯ G♯ D♯ A♯ |
| F♯ | D♯ | F♯ C♯ G♯ D♯ A♯ E♯ |
| C♯ | A♯ | F♯ C♯ G♯ D♯ A♯ E♯ B♯ |
Intervals and Scale Degrees
Intervals are distances between notes, but they are not counted as differences but traditionally on a specific musical scale, depending on a certain starting point. The starting point already counts as one, so the smallest possible step is that of a single semitone said to be a minor second, denoted as ♭2. As you may be guessing now, there is also a major second, denoted as ♯2, in other words; a whole tone step, and whether the (natural) second (note in the scale, that is - the first step) is a major or minor one depends on where and in which scale we are.
In C-Major, for example, a second starting at C - its first scale degree, brings us to D, two semitones ahead - the second over C is a major one. Starting at E - the third scale degree of C-Major - on the other hand brings us to F, only one semitone ahead - the second over E is a minor one.
However, fourths and fifths are irregular as they are both next to the symmetric middle and the same from most scale degrees of the natural tonality and in this case said to be natural. They are respectively augmented or diminished, written ♯4 or ♭5, respectively depending on whether we're coming from the fourth or the seventh scale degree. They both describe the distance of 6 semitones, also called a tritone, the most dissonant interval, opposite to its starting point on the circle of fifths.
As frequently mentioned above, notes repeat in octaves, so for every interval there is an inversion that leads to the same destination in opposite direction.
| Distance | Interval | Inversion | Natural Scale Degrees |
|---|---|---|---|
| 1 | ♭2 | ♯7 | 3 7 |
| 2 | ♯2 | ♭7 | 1 2 4 5 6 |
| 3 | ♭3 | ♯6 | 2 3 6 7 |
| 4 | ♯3 | ♭6 | 1 4 5 |
| 5 | 4 | 5 | 1 2 3 5 6 7 |
| 6 | ♯4 | ♭5 | 4 7 |
| 7 | 5 | 4 | 1 2 3 4 5 6 |
| 8 | ♭6 | ♯3 | 3 6 7 |
| 9 | ♯6 | ♭3 | 1 2 4 5 |
| 10 | ♭7 | ♯2 | 2 3 5 6 7 |
| 11 | ♯7 | ♭2 | 1 4 |
Chords, Triads, Extensions, Inversions and Voicings
When playing more than just an interval at the same time we call that a chord - or a triad in the special case of three notes. The most typical triads consist of a major or minor third, followed by a fifth (which happens to lie a respective minor or major third above the first third). The third decides whether we call it a major or minor chord. If the third is not a real third but diminished or augmented (that is, they technically the distance of a second or fourth, see above for details on musical intervals) it is called a suspended chord. If the chord is said to be diminished or augmented it means its fifth is (e.g. two consecutive minor thirds yield a diminished minor triad, two major ones an augmented major one).
In cases when the root is not the note with the lowest pitch, we speak of inversions of a chord, however, there are often multiple ambiguous ways to notate chords, and for chords more complex than simple traids the inversions can become difficult to enumerate. In cases like that it is often easier to notate them in terms of their voicings, that is spell out which note belongs into which octave where it matters:
When a chord name is followed by a slash and a note name, and the chord contains that note, it is usually to select the inversion based on that note, put that note into the bass or hold it steady (see below), if the note is not present, it is supposed to be added.
If multiple consecutive chords are notated with a slash followed by the same note, it usually means that it's a pedal note, one that is to be held steady while others change over the chord progression (those were historically to be played via pedal on an organ).
For chord extension, that is adding harmonic content with additional notes, the intervals are often counted into the next octave:
| Interval > 8 | to same note < 8 |
|---|---|
| 9 | 2 |
| 10 | 3 |
| 11 | 4 |
| 12 | 5 |
| 13 | 6 |
| 14 | 7 |
Modes
The mode describes the gravitational center of harmony and meldoy of a piece or part thereof. It is very often also the note / chord root of its beginning and its end. Each mode gives a particular feel and cultural prevalence may differ. For example, playing the white piano keys centered around C, this way playing in major also called Ionian, is often said to sound happier than centering around A, in which case we're playing in minor also called Aeolian. Logically, you can use any scale degree as the center point. Playing E to E is sometimes said to sound Spanish, and as all the modes have names, that one is called Phrygian. Knowing these names may (or may not) help you memorize the shapes of the corresponding scale degrees and provides a means to refer to them without depending on a fixed starting point for counting.
| Scale Degree | Mode Name | Natural Intervals |
|---|---|---|
| 1 | Ionian | ♯2 ♯3 4 5 ♯6 ♯7 |
| 2 | Dorian | ♯2 ♭3 4 5 ♯6 ♭7 |
| 3 | Phrygian | ♭2 ♭3 4 5 ♭6 ♭7 |
| 4 | Lydian | ♯2 ♯3 ♯4 5 ♯6 ♯7 |
| 5 | Mixolydian | ♯2 ♯3 4 5 ♯6 ♭7 |
| 6 | Aeolian | ♯2 ♭3 4 5 ♭6 ♭7 |
| 7 | Locrian | ♭2 ♭3 4 ♭5 ♭6 ♭7 |
In fact, as any mode can be used as the tonic (tonal center), it makes some sense to count that mode as the first degree. Ionian (a.k.a. Major) and Aeolian (a.k.a. Natural Minor) are the most common, conventionally notated as upper and lower case roman numerals (but since that is rather limiting and potentially reduces readability I won't do that here). The following table lets you translate scale degrees in respect to any mode to refer to Ionian. For example, if your piece is in Minor, read column six to translate its scale degrees corresponding to respective rows to ones referring to Major (compatible to the counting used in the interval table above):
| 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|---|---|---|---|---|---|---|
| 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| 2 | 3 | 4 | 5 | 6 | 7 | 1 |
| 3 | 4 | 5 | 6 | 7 | 1 | 2 |
| 4 | 5 | 6 | 7 | 1 | 2 | 3 |
| 5 | 6 | 7 | 1 | 2 | 3 | 4 |
| 6 | 7 | 1 | 2 | 3 | 4 | 5 |
| 7 | 1 | 2 | 3 | 4 | 5 | 6 |
The following table can be used to translate back. For example, read column six to translate scale degrees corresponding to respetive rows referring to Major (compatible to the counting used in the interval table above) to ones referring to Minor:
| 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|---|---|---|---|---|---|---|
| 1 | 7 | 6 | 5 | 4 | 3 | 2 |
| 2 | 1 | 7 | 6 | 5 | 4 | 3 |
| 3 | 2 | 1 | 7 | 6 | 5 | 4 |
| 4 | 3 | 2 | 1 | 7 | 6 | 5 |
| 5 | 4 | 3 | 2 | 1 | 7 | 6 |
| 6 | 5 | 4 | 3 | 2 | 1 | 7 |
| 7 | 6 | 5 | 4 | 3 | 2 | 1 |