In Part 1, we looked at equal temperament and just intonation, and saw how the former can be generalized to scales with more than 12 notes. Here, we will introduce a way to measure error, cover 19-tet, and show the "golden ratio" scale I mentioned at the beginning of Part 1.
Let's start by introducing a very small "unit of intervallic width." It is defined as one hundredth of a semitone in 12-tet, with a frequency ratio of 2^(1/1200). This unit is known as a cent. Therefore, there are 1200 cents in an octave, and the hypothetical 1200-tet would have intervals which are all multiples of a cent. A difference of one cent is imperceptible to humans, but it serves as a useful tool for measuring error between just intonation and equal temperament - after all, the goal of the latter is to approximate the former so that intervals sound harmonious and changing keys preserves said intervals.
First, let's revisit the 12-tet approximation of a fifth, just to demonstrate how cents are used. The error between the just fifth and the 12-tet fifth is just the interval between them, measured in cents. Since a 12-tet semitone is, by definition, 100 cents, a fifth is 7 of those, which is 700 cents. To find the number of cents a just fifth (3:2) spans, we need to remember the fact that multiplying ratios is the same as adding intervals, and dividing ratios is the same as subtracting intervals. This implies that pitch is measured on a logarithmic scale, and converting from ratios to portions of an octave requires taking the logarithm base 2. To get from octaves to cents, multiply by 1200, by the definition of cents. Putting it all together,
Let's start by introducing a very small "unit of intervallic width." It is defined as one hundredth of a semitone in 12-tet, with a frequency ratio of 2^(1/1200). This unit is known as a cent. Therefore, there are 1200 cents in an octave, and the hypothetical 1200-tet would have intervals which are all multiples of a cent. A difference of one cent is imperceptible to humans, but it serves as a useful tool for measuring error between just intonation and equal temperament - after all, the goal of the latter is to approximate the former so that intervals sound harmonious and changing keys preserves said intervals.
First, let's revisit the 12-tet approximation of a fifth, just to demonstrate how cents are used. The error between the just fifth and the 12-tet fifth is just the interval between them, measured in cents. Since a 12-tet semitone is, by definition, 100 cents, a fifth is 7 of those, which is 700 cents. To find the number of cents a just fifth (3:2) spans, we need to remember the fact that multiplying ratios is the same as adding intervals, and dividing ratios is the same as subtracting intervals. This implies that pitch is measured on a logarithmic scale, and converting from ratios to portions of an octave requires taking the logarithm base 2. To get from octaves to cents, multiply by 1200, by the definition of cents. Putting it all together,
log2(3/2)*1200 ≈ 701.955 cents.
The 12-tet 700-cent fifth is less than two cents flat of the just fifth (1.955 cents, to be more precise), which is good news for those who play music in 12-tet!
Before we get to 19-tet, we also need to review some music notation. The 12 notes of the "normal" Western musical scale are as follows:
C-Db-D-Eb-E-F-Gb-G-Ab-A-Bb-B
There are 7 letters used in the note names, which correspond to the 7 white keys on a piano. The "b" indicates a flat, which means "lowered by one semitone". There are also sharp notes, notated with a "#" sign and raising the note by a semitone. For example, D-sharp (D#) is the same note as E-flat (Eb) in the 12-tone scale because E is 2 semitones above D. However, this is not true for all pairs of adjacent white keys. E and F, as well as B and C, are separated by one semitone instead of two, so E# is F, Fb is E, B# is C, and Cb is B.
In 19-tet, this is not the case. A semitone is added in between each pair of adjacent white keys (e.g. D and E become separated by 3 semitones, B and C become separated by 2). This means that notes like D# and Eb are now different notes, and there is a new note between E and F, called E# or Fb, and another between B and C, called B# or Cb. This brings the total up to 19, with the order of notes being:
C-C#-Db-D-D#-Eb-E-E#-F-F#-Gb-G-G#-Ab-A-A#-Bb-B-B#
But why did I choose 19-tet for the subject of this page? The main reasons are:
1. It is better than most other equal-tempered scales at approximating the whole-number ratios of just intonation, like the 3:2 fifth, 5:4 major third, and 6:5 minor third (which has a particularly good approximation in 19tet, which is less than a fifth of a cent off).
2. It works perfectly with already-established Western musical notation, which is more than can be said of some other scales.
Before we get to 19-tet, we also need to review some music notation. The 12 notes of the "normal" Western musical scale are as follows:
C-Db-D-Eb-E-F-Gb-G-Ab-A-Bb-B
There are 7 letters used in the note names, which correspond to the 7 white keys on a piano. The "b" indicates a flat, which means "lowered by one semitone". There are also sharp notes, notated with a "#" sign and raising the note by a semitone. For example, D-sharp (D#) is the same note as E-flat (Eb) in the 12-tone scale because E is 2 semitones above D. However, this is not true for all pairs of adjacent white keys. E and F, as well as B and C, are separated by one semitone instead of two, so E# is F, Fb is E, B# is C, and Cb is B.
In 19-tet, this is not the case. A semitone is added in between each pair of adjacent white keys (e.g. D and E become separated by 3 semitones, B and C become separated by 2). This means that notes like D# and Eb are now different notes, and there is a new note between E and F, called E# or Fb, and another between B and C, called B# or Cb. This brings the total up to 19, with the order of notes being:
C-C#-Db-D-D#-Eb-E-E#-F-F#-Gb-G-G#-Ab-A-A#-Bb-B-B#
But why did I choose 19-tet for the subject of this page? The main reasons are:
1. It is better than most other equal-tempered scales at approximating the whole-number ratios of just intonation, like the 3:2 fifth, 5:4 major third, and 6:5 minor third (which has a particularly good approximation in 19tet, which is less than a fifth of a cent off).
2. It works perfectly with already-established Western musical notation, which is more than can be said of some other scales.
I found some Processing code which lets you play your keyboard like a piano and modified it to play 19-tet. Unfortunately, since it uses the minim sound library, it doesn't work in the browser. However, you can play the keyboard with the instructions in the document below (I didn't want to clutter up this page):
instructions.docx | |
File Size: | 39 kb |
File Type: | docx |
I recommend you follow the instructions and try this out if you already know the basics of the piano.