For the past few weeks, I've been experimenting with musical scales, and I've found a scale that I find very mathematically beautiful. This scale goes beyond the notes of the piano and includes sounds you would not hear in "normal" Western music, with the golden ratio thrown into the bargain, and to top it all off, the first half-decent original image on this site! Before we get to that, though, we need to learn about the mathematics of harmony.
Musical notes as they appear on sheet music are usually read as a letter name with an optional sharp or flat symbol, like G or E-flat. However, what this really conveys is the frequency of a note. A, for example, is usually tuned to 440 hertz (Hz), which means that if you play an A on a stringed instrument like a guitar, the string will vibrate back and forth 440 times per second, and the sound wave that moves through the air will have the same frequency.
Harmonies are nothing but simple ratios between the frequencies of notes. An octave is just a 2:1 ratio, and a fifth is a 3:2 ratio. An octave is the span between any note and the note with the same name directly above it. An example of an octave would be the interval between the two notes in "Somewhere" in "Somewhere Over the Rainbow". A fifth is the interval between, for example, A and E. Think from one "Twinkle" to the next in "Twinkle, Twinkle, Little Star". If A is 440 Hz, the note an octave higher is also called A, but it would be tuned to 880 Hz, which is twice the frequency. The note a fifth higher is E, so it is tuned to 660 Hz, which is 3/2 times the frequency.
Or at least, that's how it seems it should be.
The scale used in Western music, the chromatic scale, is composed of 12 notes per octave (7 white keys and 5 black keys on a piano), and the interval between consecutive notes is known as a semitone. There are 12 semitones in an octave (necessarily) and 7 semitones in a fifth, so going up 12 fifths should be the same as going up 7 octaves (12*7 semitones = 7*12 semitones). Since going up an octave doubles the frequency, going up 7 octaves will double it 7 times, which is equivalent to multiplying it by 2^7, which equals 128. Likewise, since going up a fifth multiplies the frequency by 3/2, going up by 12 fifths is equivalent to multiplying by (3/2)^12. Punching that into a calculator, it gives us...
129.746337890625.
Yikes.
If we go up by 11 fifths, from A to E to B to F# and so on, all of the way to D, multiplying by 3/2 as we go, the final fifth from D back to A will not have a 3:2 ratio, because if it were, that A would be over 129 times the frequency of the initial A instead of the required 128. Therefore, the fifth between D and A will be too small and will sound flat and out of tune. This is called a wolf fifth, and I've put a sample of this interval for you to hear just how unbearable it is.
Musical notes as they appear on sheet music are usually read as a letter name with an optional sharp or flat symbol, like G or E-flat. However, what this really conveys is the frequency of a note. A, for example, is usually tuned to 440 hertz (Hz), which means that if you play an A on a stringed instrument like a guitar, the string will vibrate back and forth 440 times per second, and the sound wave that moves through the air will have the same frequency.
Harmonies are nothing but simple ratios between the frequencies of notes. An octave is just a 2:1 ratio, and a fifth is a 3:2 ratio. An octave is the span between any note and the note with the same name directly above it. An example of an octave would be the interval between the two notes in "Somewhere" in "Somewhere Over the Rainbow". A fifth is the interval between, for example, A and E. Think from one "Twinkle" to the next in "Twinkle, Twinkle, Little Star". If A is 440 Hz, the note an octave higher is also called A, but it would be tuned to 880 Hz, which is twice the frequency. The note a fifth higher is E, so it is tuned to 660 Hz, which is 3/2 times the frequency.
Or at least, that's how it seems it should be.
The scale used in Western music, the chromatic scale, is composed of 12 notes per octave (7 white keys and 5 black keys on a piano), and the interval between consecutive notes is known as a semitone. There are 12 semitones in an octave (necessarily) and 7 semitones in a fifth, so going up 12 fifths should be the same as going up 7 octaves (12*7 semitones = 7*12 semitones). Since going up an octave doubles the frequency, going up 7 octaves will double it 7 times, which is equivalent to multiplying it by 2^7, which equals 128. Likewise, since going up a fifth multiplies the frequency by 3/2, going up by 12 fifths is equivalent to multiplying by (3/2)^12. Punching that into a calculator, it gives us...
129.746337890625.
Yikes.
If we go up by 11 fifths, from A to E to B to F# and so on, all of the way to D, multiplying by 3/2 as we go, the final fifth from D back to A will not have a 3:2 ratio, because if it were, that A would be over 129 times the frequency of the initial A instead of the required 128. Therefore, the fifth between D and A will be too small and will sound flat and out of tune. This is called a wolf fifth, and I've put a sample of this interval for you to hear just how unbearable it is.
wolf_fifth_on_c.mid | |
File Size: | 0 kb |
File Type: | mid |
(Source: Wikimedia Commons)
The tuning system we have just described is called Pythagorean tuning, and it preserves all but one fifth perfectly at the cost of that one being very out of tune. Can we do better? Maybe there would be some way to spread out the error so that every fifth is only slightly out of tune. In fact, there is such a way, and it's known as equal temperament.
Since the octave is divided into 12 semitones, going up by 12 semitones should be the same as going up an octave (but that may go without saying!) Call the frequency ratio between two notes one semitone apart x. The interval between two notes 12 semitones apart should therefore be x^12, and since that's the same as an octave,
x^12 = 2
Solving for x,
x = 2^(1/12), or the twelfth root of 2.
The ratio between two notes a fifth apart would be x^7 (7 semitones), which is 2^(7/12), which is about 1.4983. Not exactly 1.5, but the human ear can barely tell the difference. The important thing is that every fifth sounds the same, so playing in different keys will change the pitch of the notes, but all intervals will remain the same. This comes at a price, though: no interval besides the octave sounds perfectly harmonious in equal temperament because the twelfth root of 2 is an irrational number, so raising it to any integer power besides a multiple of 12 will also yield an irrational number. Sorry.
Here are a few more intervals with whole-number ratios and their equal-tempered approximations.
The tuning system we have just described is called Pythagorean tuning, and it preserves all but one fifth perfectly at the cost of that one being very out of tune. Can we do better? Maybe there would be some way to spread out the error so that every fifth is only slightly out of tune. In fact, there is such a way, and it's known as equal temperament.
Since the octave is divided into 12 semitones, going up by 12 semitones should be the same as going up an octave (but that may go without saying!) Call the frequency ratio between two notes one semitone apart x. The interval between two notes 12 semitones apart should therefore be x^12, and since that's the same as an octave,
x^12 = 2
Solving for x,
x = 2^(1/12), or the twelfth root of 2.
The ratio between two notes a fifth apart would be x^7 (7 semitones), which is 2^(7/12), which is about 1.4983. Not exactly 1.5, but the human ear can barely tell the difference. The important thing is that every fifth sounds the same, so playing in different keys will change the pitch of the notes, but all intervals will remain the same. This comes at a price, though: no interval besides the octave sounds perfectly harmonious in equal temperament because the twelfth root of 2 is an irrational number, so raising it to any integer power besides a multiple of 12 will also yield an irrational number. Sorry.
Here are a few more intervals with whole-number ratios and their equal-tempered approximations.
# Semitones |
Ratio |
Approximation |
Name |
2 |
9:8 = 1.125 |
~1.122 |
Major second, whole tone |
3 |
6:5 = 1.2 |
~1.189 |
Minor third |
4 |
5:4 = 1.25 |
~1.260 |
Major third |
5 |
4:3 ≈ 1.333 |
~1.335 |
(Perfect) fourth |
7 |
3:2 = 1.5 |
~1.498 |
(Perfect) fifth |
8 |
8:5 = 1.6 |
~1.587 |
Minor sixth |
9 |
5:3 ≈ 1.667 |
~1.682 |
Major sixth |
10 |
9:5 = 1.8 |
~1.782 |
Minor seventh |
11 |
15:8 = 1.875 |
~1.888 |
Major seventh |
It might seem like I just pulled all of these whole-number ratios out of thin air, but we can work through it and derive all of these ourselves.
First, the interval of 5 semitones, a perfect fourth, is an octave minus a fifth (12-7=5), and in order to subtract intervals, we must divide the ratios. 2/(3/2) = 4/3, so the perfect fourth has a 4:3 frequency ratio. Next, 2 semitones, or a whole tone, is a fifth minus a fourth, or (3/2)/(4/3) = 9/8, so the whole tone has a 9:8 ratio.
Everything else in the table is based on just one more assumption, which is that the interval of 4 semitones - a major third - has a ratio of 5:4, which is almost true (2^(4/12) ≈ 1.260). Adding it to the fourth and fifth gives us (5/4)*(4/3) = 5/3 for the 9-semitone major sixth and (5/4)*(3/2) = 15/8 for the 11-semitone major seventh, respectively. 3 semitones, a minor third, is a fifth minus a major third, which is (3/2)/(5/4) = 6/5. If you add the minor third to the fourth and fifth in the same way, the 8-semitone minor sixth turns out to have a ratio of 8:5 and the 10-semitone minor seventh has a ratio of 9:5.
This covers everything except the semitone itself and the 6-semitone interval, which is known as a tritone because it is equal to three stacked whole tones (3*2=6). The semitone's ratio is close to 1, because a 1:1 frequency ratio means that the two notes are the same, and a semitone deviates relatively little from that. Any close approximation must thus involve large whole numbers (think (n+1):n, where n is around 20), and only ratios of small numbers sound consonant and pleasing. The equal-tempered tritone has a frequency ratio of 2^(6/12), or just 2^(1/2), which is the square root of 2 (about 1.414). It's somewhat close to a 7:5 or 10:7 ratio, and these are small numbers, so why didn't I include it? I didn't include it because of the 7. All of the other ratios have only 2, 3, and 5 as prime factors. If some note is tuned to 7/5 of the frequency of the bottom note (or tonic), none of the intervals including that note will be preserved because the 7 can't be canceled out. Restricting the prime factors of the numbers involved to those at or below some prime p is called p-limit tuning. The just intonation of the 12-tone scale I derived here is a 5-limit tuning, for example, because no primes above 5 were used.
Tuning a scale with whole-number ratios like this is called "just intonation". The reason it isn't commonly used either, despite so many perfect intervals, is that it only sounds great in one key, similar to how Pythagorean tuning sounds terrible in one key because of the wolf fifth and mediocre in other keys because most intervals are defined as ratios between very large whole numbers. For example, the major third, whose ratio is close to 5:4, is approximated in Pythagorean tuning by four stacked fifths minus two octaves: ((3/2)^4)/(2^2) = 81/64 ≈ 1.266.
What if you don't want 12-tone equal temperament, though? Maybe you want more notes in the scale. That's easy! A scale step is the 12th root of 2 in the 12-tone scale, so it would be, say, the 15th root of 2 in 15-tone equal temperament, or the 22nd root of 2 in 22-tone equal temperament, or the 283rd root of 2 in 283-tone equal temperament - intervals smaller than the semitone which bring forth a whole range of new possibilities! Welcome to the world of microtonal music, and we'll explore one scale of particular interest - 19-tone equal temperament - in Part 2!
First, the interval of 5 semitones, a perfect fourth, is an octave minus a fifth (12-7=5), and in order to subtract intervals, we must divide the ratios. 2/(3/2) = 4/3, so the perfect fourth has a 4:3 frequency ratio. Next, 2 semitones, or a whole tone, is a fifth minus a fourth, or (3/2)/(4/3) = 9/8, so the whole tone has a 9:8 ratio.
Everything else in the table is based on just one more assumption, which is that the interval of 4 semitones - a major third - has a ratio of 5:4, which is almost true (2^(4/12) ≈ 1.260). Adding it to the fourth and fifth gives us (5/4)*(4/3) = 5/3 for the 9-semitone major sixth and (5/4)*(3/2) = 15/8 for the 11-semitone major seventh, respectively. 3 semitones, a minor third, is a fifth minus a major third, which is (3/2)/(5/4) = 6/5. If you add the minor third to the fourth and fifth in the same way, the 8-semitone minor sixth turns out to have a ratio of 8:5 and the 10-semitone minor seventh has a ratio of 9:5.
This covers everything except the semitone itself and the 6-semitone interval, which is known as a tritone because it is equal to three stacked whole tones (3*2=6). The semitone's ratio is close to 1, because a 1:1 frequency ratio means that the two notes are the same, and a semitone deviates relatively little from that. Any close approximation must thus involve large whole numbers (think (n+1):n, where n is around 20), and only ratios of small numbers sound consonant and pleasing. The equal-tempered tritone has a frequency ratio of 2^(6/12), or just 2^(1/2), which is the square root of 2 (about 1.414). It's somewhat close to a 7:5 or 10:7 ratio, and these are small numbers, so why didn't I include it? I didn't include it because of the 7. All of the other ratios have only 2, 3, and 5 as prime factors. If some note is tuned to 7/5 of the frequency of the bottom note (or tonic), none of the intervals including that note will be preserved because the 7 can't be canceled out. Restricting the prime factors of the numbers involved to those at or below some prime p is called p-limit tuning. The just intonation of the 12-tone scale I derived here is a 5-limit tuning, for example, because no primes above 5 were used.
Tuning a scale with whole-number ratios like this is called "just intonation". The reason it isn't commonly used either, despite so many perfect intervals, is that it only sounds great in one key, similar to how Pythagorean tuning sounds terrible in one key because of the wolf fifth and mediocre in other keys because most intervals are defined as ratios between very large whole numbers. For example, the major third, whose ratio is close to 5:4, is approximated in Pythagorean tuning by four stacked fifths minus two octaves: ((3/2)^4)/(2^2) = 81/64 ≈ 1.266.
What if you don't want 12-tone equal temperament, though? Maybe you want more notes in the scale. That's easy! A scale step is the 12th root of 2 in the 12-tone scale, so it would be, say, the 15th root of 2 in 15-tone equal temperament, or the 22nd root of 2 in 22-tone equal temperament, or the 283rd root of 2 in 283-tone equal temperament - intervals smaller than the semitone which bring forth a whole range of new possibilities! Welcome to the world of microtonal music, and we'll explore one scale of particular interest - 19-tone equal temperament - in Part 2!