EPTPart 1: Introduction and the major scale
Part 2: More about intervals, and the minor scale
Part 3: The history of modal scales
To know what I am talking about, I recommend to read the first parts first, if you haven’t done it yet. This is the longest blog yet, it concludes the series about intervals and scales.
Let's recapitulate the last blog: Major and Minor are just two out of seven scales modes. While the general note position is determined by the scale, the actual pitch tuning of scales changed over time to improve certain intervals while compromising others as little as possible. Modal scales include the older and simpler pentatonic (five-note) scales. Most modal scales are also a variant of either major or minor.
Whole Tone And Semitone
A whole tone, from the first step to the second step of a scale, is called the second. There are different ways to calculate the pitch ratio, the most common one is to use the difference from a fourth to a fifth. It gets us the ratio of 8 : 9, meaning the pitch of the second tone is 1/8 higher than the root note. Other definitions exist, but are not needed for this blog.
There is a minor second interval, too. As there is no black key between notes E and F, and B and C, we have a minor second between those notes. We can calculate the pitch ratio of the minor seconds as the difference from a major third to a perfect fourth, and get 15:16.
The smaller step appear in their position as the result of how modal scales are created. The same smaller steps at the same locations also appear if we string three triads together, which is another way to get to the major or minor scale. That is why the efforts of the first two blogs where made: To clarify that the appearance and the distribution of the steps smaller than a whole tone is not arbitrary.
Effects Of Small Steps
The minor second can be considered a semitone because it is narrower than a whole tone. Its pitch ratio of 15:16 looks complicated, and it is. As a result, this interval causes much dissonance. B and C played at the same time create a very harsh, dissonant sound. However, B is still part of the C-major scale. The crass dissonance of B in relationship to C makes us begging for a resolution to C.
![[image loading]](http://i.imgur.com/5kV1RrN.png)
More About Intervals
Let's analyze every interval which appears in a major scale. The example notes are for C-major.
C = prime: The first note, the root of the scale, acting as tonal center, the gravity point for the melody and the chords.
D = major second: A whole tone above the prime. A dissonance, seeks resolution to the prime.
E = major third: Sounds very good if played together with the prime and gives this scale its 'major' characteristics.
F = perfect fourth: Consonant or dissonant depending on the context, today often used as instable consonance.
G = perfect fifth: Very consonant sound, seeks resolution to the prime, the dominant interval.
A = major sixth. Complementary interval of the minor third, creates a minor-like sound within the major scale.
B = major seventh. Most dissonant interval in this scale, seeks resolution to the tonic.
C = octave. Like the prime, but doubled in pitch. Acts almost as the prime itself.
Please note that effect of an interval changes greatly depending on the musical context. For example, jazz music often uses the major seventh in a way which requires no resolution.
The octave complement of a major interval is a minor interval. Since the major scale includes any major interval, it also includes minor complement intervals.
![[image loading]](http://i.imgur.com/xxa43b5.png)
Expanding Our Intervals
The first idea which comes to our mind is the interval of the ninth, because why shouldn't we use intervals bigger than an octave?
This is of course possible. The ninth shares its characteristic with the second, as the same note is played, just an octave higher. The second creates dissonance. But the second it is a narrow interval, which adds to the tension. This is because the combined sounds heavily depends on the overtones. Two tones with quite similar, yet not exactly the same pitches create a very complicated overtone spectrum which is not close to a natural overtone spectrum of a single sound. We perceive this complex overtone spectrum as an unnatural and dissonant.
The ninth however is a very wide interval. This results in a colorful, yet not very dissonant sound because the interval wideness relaxes the tension. This is because the overtones of the higher-pitch note are more stretched out, spreading over a wider range, which results in less complication.
In just intonation, the ratio of the major ninth is 4 : 9. That is no natural interval, but still simple enough to be somewhat harmonic. Another way to get to a ninth is to stack two fifths on top of each other. If we play those three notes – root, its fifth, and the fifth of that fifth – it sounds almost consonant. We now hear the short string of fifths and identify the ninth as the fifth of the fifth.
However, if we play a minor ninth, the wideness of the interval doesn't help much to cover the dissonance caused by the very complex pitch ratio of 15:32.
Intervals Should Not Be Too Wide
The ninth, in both versions, sometimes appears in today's music. The tenth interval is too wide for some piano players as they don't have hands that big, though compositions using a tenth do exist. Even larger intervals are used in chord theory, but also appear in sheets as they can be played on certain instruments.
Our brain uses similarities to assume a connection. A narrow interval implies a close connection of the two notes, as they are similar in pitch. On the other hand, if intervals are getting too wide, they hard to interpret for our brain, because now we tend to interpret both notes as independent events.
Changing Our Intervals
Why only play the pure interval, why not sharpen or flatten it? To create a sharper sound, we need to slightly increase the pitch. To flatten it, we need to slightly decrease it. This deviation from the pure interval changes its tone, its color. That means, we are using chromaticism. Chroma is the Greek word for color.
The most commonly used chromatic step, also called chromatic semitone, should be tuned as 24:25. That is the ratio difference of a minor to a major third. The chromatic semitone is therefore smaller than the diatonic semitone of 15:16.
There are actually even more semitone variants. Do we really need all them? As it turns out, we can make our life simple by using only one half-tone step, tuned in a ratio between the chromatic and the diatonic one. The context of the music allows us to recognize the intended meaning.
Another Example Regarding Tuning Issues
Over the ages, many scales and tuning were proposed, all with a particular goal in mind. Let us – again! – consider a string of fifths which we shift back into the same octave to construct a scale. From the prime we get the fifth by using its 2:3 ratio. As the next note, we calculate the fifth of the previous fifth and get 4 : 9. Because that already exceeds the octave, we shift it down one octave and get 8 : 9.
So far, so good. Let us calculate the next fifth. The fifth of 8 : 9 has a very complicated ratio of 16:27. If we tune it as 3:5 instead, we are reasonably close to the strict tuning by fifths, but avoid a very complex interval. Do we want to use the simpler interval or continue with stacking actually pure fifths? There is no solution which includes only advantages and has no drawbacks.
Our Goal: To Do Almost Anything
Not only we have to ask ourselves about the intricacies of the actual instrument tuning, we also would like to play the major scale from any other note instead of having to use C as the first degree. We even want to play any diatonic scale beginning from any note. And we want to be able to use chromaticism, that means we want the option to sharpen (slightly increase) or flatten (slightly decrease) any note.
It looks like we need a lot of steps. A whole lot of steps. But we are lucky and a nice mathematical happenstance makes our life much easier.
![[image loading]](http://i.imgur.com/Au99fkS.png)
From A Scale To A Circle
So far, any scale we examined could be constructed by a series of fifths. What happens when we continue to stack fifths after we got the seven notes for our scale? It turns out that the thirteenth note is the same as the first one, just some octaves higher.
Well, it is almost the same note.
What we do in today’s music, is to slightly adjust – to temper – the fifth, in a way that the thirteenth note in our string of fifths comes out to exactly the first, leaving us twelve different steps when we don't take the octaves into account. A modern piano has 88 keys, spanning 7 1/4 octaves, but offers twelve steps per octave. You now know, why! Many instruments in a modern orchestra are able to play these twelve notes per octave as well.
As we still tune the scale by using the concert pitch for the note labeled A, we can calculate the pitch of every other note. It is important that every instrument in the orchestra is in tune.
Enharmonic Equivalents
There is another way to get to this kind of tuning. A whole tone step is the interval of 8 : 9. Six of such whole tones are just slightly exceeding an octave. We can temper the whole tone, and calculate a semitone step which cuts a tempered whole tone in two equal parts.
This tempered semitone step is smaller than a diatonic semitone, but larger than a chromatic semitone. That means, we can use our single semitone step for both semitone types. This tuning is called the equal temperament because the octave is divided into twelve equal steps. Any other interval is made out of those semitone steps.
Within this scale, the note B♯ sounds exactly like C, as both notes are played with the same piano key. They are not the same note, but enharmonic equivalent and sound as if they were the same note. This allows to play any western scale – while having only twelve steps per octave.
The scale of twelve degrees per octave is also called the chromatic scale, because if we project a seven-note scale on the chromatic scale, we can sharpen or flatten each note by a semitone, therefore use chromaticism.
Sometimes, a sharpened or flattened white-key note is played with another white key. For example, E♯ is played with the F key and F♭ with the E key.
Using Chromaticism
The minor scale is often used with one or two chromatic steps: The seventh, and sometimes the sixth gets sharpened, turning them from minor to their major version. This has harmonic reasons as the modern ear expects the leading tone (the major seventh) which is not a part of the natural minor scale.
With the exception of some simple songs, almost any piece of music uses more or less chromaticism, A diatonic scale does provide useful intervals, but is too limited for longer musical scores.
Endless Flexibility
We have seven white-key notes. Each one can be played in three versions: pure, sharpened or flatted. So we have seven times three = 21 different notes which we project on the twelve-step chromatic scale. But since a note like D♭(D-flat) is a perfectly valid note, we could flatten it, too, and get D-doubleflat. On a piano, we play D-doubleflat with the C key.
The glyph of a double flat looks like two ♭ close to each other. Double sharp even has its own glyph, looking similar to a small-letter x. Double accidentals do appear in some sheets. Still, with enharmonic equivalents thanks to our equally tempered chromatic scale, we can play such notes while still having only twelve steps per octave.
Other scales were proposed, using more steps. But they either don't provide an almost perfectly tuned fifth, or they need an impractical number of steps. The common chromatic scale with only twelve degrees has an extremely good ratio of flexibility versus complexity.
Atonal Music
Instead of using the chromatic scale as note material to project other scales on it, musicians began to use the entire chromatic scale as actual scale for music. That means, the stable diatonic intervals are no longer dominantly present. Instead, any chromatic interval appears as often as any other. This implies that no tonal center can be recognized, resulting in atonal music. Other scales for atonal music are possible.
Conclusion
The fifth is the dominant interval, ruling over any consideration regarding harmony. Creating a scale means to create a selection of harmonic relationships. These include other intervals than the fifth, as the harmonic series implies. If we allow for some small tuning errors, twelve equally tempered steps per octave are sufficient to model any harmonic interval in the series of 1:2:3:5:6. Also the whole tone step of 8 : 9 is available. The different semitone steps are consolidated into a single one.
The enharmonic equivalents notes and intervals are used as a resource today, which is often considered more important than having pure intervals. While intervals create the meaning, a slight temperament is all which is needed to allow for a staggering, marvelous harmonic order created by the circle of twelve fifths and the chromatic scale of twelve steps per octave.
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