Index
Introduction
Ever wondered why some musical sounds feel happy, sad, or just right? It's all about intervals – the distance between two notes. These tiny gaps of sound have a fascinating history, stretching back centuries! And believe it or not, there's some cool math behind it all!
Let's start with the basics. Imagine the musical alphabet: A, B, C, D, E, F, G. When we play two of these notes together, we create an interval. But not all intervals are created equal! Some sound harmonious, while others clash. Why?
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Ancient Discoveries: Pythagoras and the Perfect Sounds (with Ratios!)
Way back in ancient Greece, the philosopher and mathematician Pythagoras discovered something amazing about music. He found that simple ratios between the lengths of vibrating strings created beautiful, harmonious sounds. These special intervals, like the perfect fifth and perfect fourth, became the foundation of Western music.
There is an interesting tale of Pythagoras walking past a blacksmith's shop when he heard the intervals of the fifth from the blacksmith's hammer. Handel wrote music based on the story. Pythagoras noticed that hammers of different weights made different sounds as they hit the metal. He realised that some combinations of the notes were more pleasing to the ear than others. He discovered that the secret to the harmonious sounds is the ratio between the length of two vibrating strings. If the ratio is a simple whole number you hear a pleasing harmonious sound. This discovery in 500 BCE still affects Western music today.
There is an interesting tale of Pythagoras walking past a blacksmith's shop when he heard the intervals of the fifth from the blacksmith's hammer. Handel wrote music based on the story. There is a TQ Level 3 Achieve piece that is based on this song, listen for the perfect octaves, fifths and fourths, you can follow in the score.
Free Download for Piano
A perfect fifth, like the interval between C and G, is created by a string length ratio of 3:2. That means if one string is 3 units long, the other string that produces the perfect fifth will be 2 units long. A perfect fourth (like C to F) has a ratio of 4:3. These simple whole number ratios are what Pythagoras considered consonant in sound.
Think of the first two notes of Twinkle, Twinkle, Little Star – that's a perfect fifth! The perfect fourth can be heard at the beginning of the hymn Amazing Grace
Examples Perfect Fourths, Fifths and Octaves
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Building on the Basics: Major and Minor (and More Ratios!)
Centuries later, during the Renaissance, music started to get more complex. Composers wanted to express a wider range of emotions. That's where major and minor intervals came in. Gioseffo Zarlino, an important music theorist from the 16th century, helped solidify the understanding of major thirds and minor thirds.
A major third (like C to E) has a ratio of 5:4, while a minor third (C to Eb) has a ratio of 6:5. Notice how these ratios are a little more complex than the perfect intervals. These more complex ratios are what give major and minor intervals their distinctive color. Major intervals generally sound bright and happy, while minor intervals tend to sound darker and more melancholic. Think of the difference between a major and a minor chord – it's all about the third!
Examples Major and Minor Thirds
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Examples Major and Minor Sixths
A major sixth has a ratio of 5:3, (like C to A) and a minor sixth has the ratio 8:5 (C to Ab) these are very similar to the minor 3rd,
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Expanding the Family: Major and Minor Scales
Heinrich Glarean, another influential theorist, added even more to our understanding of scales and modes. He formally described the major scale (also known as the Ionian mode) and the minor scale (Aeolian mode). These are the two important scales of Western music. The major scale is bright and cheerful and the minor scale is seen as dark and melancholic. These scales, built from specific patterns of major and minor intervals, became the basis for much of the music we hear today.
The major scale follows a pattern of whole and half steps, or tones and semitones; it has a specific sequence of intervals with the major seventh. This gives it the characteristic sound we know so well.
The minor scale has a different pattern of tones and semitones, it has a minor third and a minor seventh giving it a more melancholic character.
The scales are, in essence, a carefully chosen sequence of these specific ratios. There are different scales all over the world that have their own interval structures and sonic characteristics. These scales help you decipher the emotional content of the music.
What About the 2nd and 7th? (And Their Ratios!)
You're right to ask about the 2nd and 7th! These intervals are also crucial. The major second (like C to D) has a ratio of 9:8, while the major seventh (C to B) has a ratio of 15:8.
Examples Major and Minor Seconds
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Examples Major and Minor Sevenths
The minor second and minor seventh have slightly different ratios, creating their characteristic sound.
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The Augmented and Diminished: Adding Spice (and Even More Ratios!)
Sometimes, composers want to create tension or surprise. That's where augmented and diminished intervals come in. These are created by slightly widening or narrowing a perfect or major interval, respectively. This changes the ratio slightly, creating a dissonant sound. They add a bit of "spice" to the music, making it more interesting. Calculating the exact ratios for augmented and diminished intervals can get a bit more complex, often involving irrational numbers, but the principle remains the same: a slight change in the ratio creates a noticeable change in the sound.
Putting It All Together
The Circle of Fifths is a visual representation that shows the relationship between different musical keys. Imagine a circular clock face. Every point on the circle represents a different musical key. Each note on that circle is a perfect 5th away from the next one.
If you start at F and go clockwise you will get F, C, G, D, A, E, B etc . It shows how harmony works in Western music. Keys that are closer together share more notes, they sound more harmonious when you play them and can be used together. Keys that are further apart have fewer shared notes and they create a larger contrast. The interplay between intervals and scales gives music its structure, its emotion, its power to move the person listening or playing it. The augmented and diminished intervals add to this sonic tapestry by bringing instability and spice to the sound, like adding spices to your favourite recipe. They introduce different instabilities, tensions, spooky sounds and yearning into the music. They build suspense, unexpected twists and turns and surprises. Just a semitone can change the mood of a piece of music, a small change can equal a big change in the emotional character of the music.
So next time you listen to music, pay attention to the intervals. They're the secret ingredients – and the math behind them is pretty cool too! From the ancient Greeks to modern musicians, the exploration of intervals has shaped the music we love. It’s a fascinating journey through sound, history, and emotion, and if you can decode the power of music you have a superpower that can change the world.
Enjoy your sonic adventures!
Worksheets
Major and Minor Intervals Worksheet
Quiz
Define a musical interval and explain how it is determined using the musical alphabet.
Who was Pythagoras and what was his significant discovery related to music?
Describe the mathematical ratio associated with a perfect fifth and give a song example where you can hear this interval.
What is the ratio of a major third? How does the sound of a major third differ from that of a minor third?
Who was Gioseffo Zarlino and what contribution did he make to music theory?
Explain the difference in sound quality between consonant and dissonant intervals.
Provide an example of a consonant interval and a dissonant interval.
Who was Heinrich Glarean and what scales did he formally describe?
How are augmented and diminished intervals created, and what effect do they have on the overall sound?
Explain why dissonance is important in music.
Essay Questions
Consider the following essay questions to further deepen your understanding.
Discuss the historical significance of Pythagoras' discoveries in relation to musical intervals and their impact on Western music.
Explain the differences between major and minor intervals, including their mathematical ratios and the emotional qualities they evoke.
Analyze the role of consonance and dissonance in music composition, providing specific examples of how these elements are used to create emotional impact.
Trace the evolution of musical understanding of consonance and dissonance from the time of Pythagoras to today.
Discuss the significance of augmented and diminished intervals in creating musical tension and providing harmonic colour.
Further Exploration:
* Look up the circle of fifths and how it relates to intervals and ratios.
* Explore online resources that demonstrate the ratios of intervals using vibrating strings or other visual aids.
* Try playing with intervals on an instrument or using online music tools to hear the differences in their ratios.
Glossary
Interval: The distance between two musical notes, measured by the number of letter names spanned between the notes.
Consonance: A combination of notes that sounds pleasing, stable, and resolved.
Dissonance: A combination of notes that sounds tense, unstable, and unresolved.
Perfect Interval: A type of interval (e.g., perfect fourth, perfect fifth, perfect octave) characterized by simple mathematical ratios and a stable, consonant sound.
Major Interval: A type of interval (e.g., major third, major sixth) that generally sounds bright and happy.
Minor Interval: A type of interval (e.g., minor third, minor sixth) that generally sounds darker and more melancholic.
Augmented Interval: An interval created by widening a perfect or major interval by a semitone, creating a dissonant sound.
Diminished Interval: An interval created by narrowing a perfect or minor interval by a semitone, creating a dissonant sound.
Ratio: The mathematical relationship between the lengths of vibrating strings that produce specific musical intervals.
Scale: An ordered sequence of notes, typically based on a specific pattern of intervals.
Mode: A type of musical scale characterized by a specific pattern of intervals and a distinct melodic character.
Further Reading
Chi-Chung Tang, A. (2011). Pythagoras at the Smithy. [online] Texas Scholar Works. Available at: https://repositories.lib.utexas.edu/bitstream/handle/2152/27195/TANG-MASTERSREPORT-2012.pdf [Accessed 19 Aug. 2022].
Evdokimoff, T. (n.d.). Recognize Intervals with Consonance and Dissonance. [online] Musical U. Available at: https://www.musical-u.com/learn/recognize-intervals-with-consonance-and-dissonance/ [Accessed 2 Aug. 2023].
Rivera, B.V. (1995). Theory Ruled by Practice: Zarlino’s Reversal of the Classical System of Proportions. Indiana Theory Review, [online] 16, pp.145–170. Available at: https://www.jstor.org/stable/24044523 [Accessed 13 Jun. 2022]. Read Online.
Senyshyn, T.L. (2003). The Theory and Practice of a-modes in Glarean’s ‘Dodecachordon’, 1547. [online] open.library.ubc.ca. Available at: https://open.library.ubc.ca/soa/cIRcle/collections/ubctheses/831/items/1.0091065 [Accessed 13 Jun. 2022].
Wold, E. and Tenney, J. (1989). A History of Consonance and Dissonance. Computer Music Journal, [online] 13(3), p.94. doi:https://doi.org/10.2307/3680020.