Harmonic Selection
Here, we work directly with the harmonic series. We take some number of harmonics, and then we derive the ratios between each harmonic and all of the others. In doing so we find every rational interval within that series. Then we remove the duplicates, or intervals that can be reduced down to other existing intervals. While this might be mathematically interesting in some way, it’s not a standard method in any way that I’m aware of. It’s just for fun and curiosity.
Here, I choose to go with seven harmonics. Although the way we go about reducing the number of intervals makes it so that the end result is the same as if I had chosen eight harmonics.
| 1/1 | 2/1 | 3/1 | 4/1 | 5/1 | 6/1 | 7/1 |
| 1/2 | 2/2 | 3/2 | 4/2 | 5/2 | 6/2 | 7/2 |
| 1/3 | 2/3 | 3/3 | 4/3 | 5/3 | 6/3 | 7/3 |
| 1/4 | 2/4 | 3/4 | 4/4 | 5/4 | 6/4 | 7/4 |
| 1/5 | 2/5 | 3/5 | 4/5 | 5/5 | 6/5 | 7/5 |
| 1/6 | 2/6 | 3/6 | 4/6 | 5/6 | 6/6 | 7/6 |
| 1/7 | 2/7 | 3/7 | 4/7 | 5/7 | 6/7 | 7/7 |
To get started on eliminating the duplicates, from here we can remove any cell that has a power of 2 in the numerator or the denominator (2, 4, 8, 16, etc). We know that multiplying or dividing an interval by 2 is effectively the same interval, just shifted by an octave. If we were to start on a note C, for example (and I’m mixing concepts by speaking in note names here, but it’s fine), and I go to its third harmonic which is G, within the same octave, then that’s an interval of 3/2. If I go to 3/1, then that’s still G, just an octave further up. If I go to 3/4, that’s still G, just an octave down. So from here, we remove the duplicates (or empty out their cells)
| 1/1 | 3/1 | 5/1 | 6/1 | 7/1 | ||
| 1/3 | 3/3 | 5/3 | 6/3 | 7/3 | ||
| 1/5 | 3/5 | 5/5 | 6/5 | 7/5 | ||
| 1/6 | 3/6 | 5/6 | 6/6 | 7/6 | ||
| 1/7 | 3/7 | 5/7 | 6/7 | 7/7 |
Continuing with the same logic, I remove anything that has a multiple of 2 in either the numerator or the denominator. Using the C to G example again. If I start with the interval 3/2, and then go up by an octave, we said it’s also 3/1 above, but another way of looking at it is that it’s also the interval 6/2 (which reduces to 3/1). Either way, it’s still G. So we can remove multiples of 2 (not just powers of 2), knowing that equivalent class intervals are already accounted for in other cells
| 1/1 | 3/1 | 5/1 | 7/1 | |||
| 1/3 | 3/3 | 5/3 | 7/3 | |||
| 1/5 | 3/5 | 5/5 | 7/5 | |||
| 1/7 | 3/7 | 5/7 | 7/7 |
What’s left now is to remove the remaining reducible intervals. If I move up (multiplication-wise) by 15 and down by 5, it’s the same as going up by 3 and “going down” by 1 (where going down by 1 multiplication-wise represents staying in place). A simpler example would be if there’s an interval 15/15, it’s the same as the interval 1/1. So in that way, we go about removing the reducible intervals
| 1/1 | 3/1 | 5/1 | 7/1 | |||
| 1/3 | 5/3 | 7/3 | ||||
| 1/5 | 3/5 | 7/5 | ||||
| 1/7 | 3/7 | 5/7 |
And now we’re left with 13 intervals. What’s left to do is to convert these intervals so that they each sit between 1.00 and 2.00. We do this by multiplying the interval by some power of two, or some power of 1/2 (which is also a power of 2) until it’s within that range. This gives us a functionally equivalent interval, shifted by octave(s). So for example, 3/1 would become 3/2, 5/7 would become 10/7, etc. In the case of our interval group above, we get the following result:
| 1/1 | 3/2 | 5/4 | 7/4 | 4/3 | 5/3 | 7/6 | 8/5 | 6/5 | 7/5 | 8/7 | 12/7 | 10/7 |
| 1.0000 | 1.5000 | 1.2500 | 1.7500 | 1.3333 | 1.6667 | 1.1667 | 1.6000 | 1.2000 | 1.4000 | 1.1429 | 1.7143 | 1.4286 |
Rearranging the group, and putting it all in increasing order, we get
| 1/1 | 8/7 | 7/6 | 6/5 | 5/4 | 4/3 | 7/5 | 10/7 | 3/2 | 8/5 | 5/3 | 12/7 | 7/4 |
| 1.0000 | 1.1429 | 1.1667 | 1.2000 | 1.2500 | 1.3333 | 1.4000 | 1.4286 | 1.5000 | 1.6000 | 1.6667 | 1.7143 | 1.7500 |
As a reference to compare against, here are the 12-TET and Pythagorean tables
12-TET
| 2^(0/12) | 2^(1/12) | 2^(2/12) | 2^(3/12) | 2^(4/12) | 2^(5/12) | 2^(6/12) | 2^(7/12) | 2^(8/12) | 2^(9/12) | 2^(10/12) | 2^(11/12) |
| 1.0000 | 1.0595 | 1.1225 | 1.1892 | 1.2600 | 1.3348 | 1.4142 | 1.4983 | 1.5874 | 1.6818 | 1.7818 | 1.8877 |
Pythagorean
| 1/1 | 2187/ 2048 | 9/8 | 19683/ 16384 | 81/64 | 177147/ 131072 | 729/ 512 | 3/2 | 6561/ 4096 | 27/16 | 59049/ 32768 | 243/ 128 |
| 1.0000 | 1.0679 | 1.1250 | 1.2014 | 1.2656 | 1.3515 | 1.4238 | 1.5000 | 1.6018 | 1.6875 | 1.8020 | 1.8984 |
The 12-TET and Pythagorean track each other closely enough that you can use just one or the other to compare with.
Comparing our 13 intervals to the 12-TET variations:
The 8/7 (1.1429) and 7/6 (1.1667) intervals both sit somewhere between the third and fourth intervals in the 12-TET table (i.e., between 1.1225 and 1.1892).
The 6/5 (1.2000) interval is close to the 1.1892 12-TET interval.
The 5/4 (1.2500) interval tracks close to the 1.2600 in 12-TET.
The 4/3 (1.3333) interval is very close to the 1.3348 in 12-TET.
The 7/5 and 10/7 intervals are both close to the 1.4142 (square root of 2) interval in 12-TET. Interestingly, if you take the geometric mean of 7/5 and 10/7, you get this square root of 2 value. In other words, multiply 7/5 and 10/7 by each other, and then take the square root of the result, and you get the square root of 2. So it’s as though you can collapse those two values into sqrt(2).
Next, the 3/2 (1.5000) interval is close to the 1.4983 12-TET interval.
The 8/5 (1.6000) interval is close to the 1.5874 interval in 12-TET.
The 5/3 (1.6667) interval is close to the 1.6818 12-TET interval.
And then the remaining intervals, 12/7 (1.7143) and 7/4 (1.7500) sit between the 1.6818 and 1.7818 intervals in 12-TET.
To summarize, the 6/5, 5/4, 4/3, 3/2, 8/5, and 5/3 intervals from our 13-interval table derived from the 7 harmonics, all have analogs in the 12-TET (and by extension in the Pythagorean) tables. Musically, these would be the minor third, major third, perfect fourth, perfect fifth, minor sixth, and major sixth intervals. Missing therefore are the minor and major second, and the minor and major seventh; and approximated twice is the tritone.
As for the other intervals in the 13-interval table, the 8/7 and 7/6 intervals would sit somewhere between the third and fourth intervals in the other tables (i.e., between the major second and minor third in music terms). The 12/7 and 7/4 intervals would sit somewhere between the tenth and eleventh intervals in the other tables (i.e., between the major sixth and minor seventh in music terms). And the 7/5 and 10/7 intervals as mentioned could be combined geometrically to produce the sqrt(2) interval in the 12-TET table (analogous to the 729/512 interval in the Pythagorean table; i.e., the tritone in music terms).
Just Intonation
Just intonation is the tuning system that’s often implied when talking about “pure” tones, and “pure” intervals. It uses three prime numbers to derive all of its intervals. The first two, 2 and 3, we’re already familiar with from Pythagorean tuning. The third prime is 5. All intervals are some combination of the powers of 2, powers of 3, and powers of 5.
As we did with Pythagorean tuning, at the beginning, we reduce the powers of 2 all the way down to 1, since they just move us along different octaves.
To set things up, we use a table. On one axis of the table, we have just a few powers of 3, symmetrically arranged around 3^0 = 1/1. On the other axis, we have just a few powers of 5, symmetrically arranged around 5^0 = 1/1.
| 1/9 | 1/3 | 1/1 | 3/1 | 9/1 | |
| 1/5 | |||||
| 1/1 | |||||
| 5/1 |
From here, the values in the inner cells are derived by multiplying the current row interval by the current column interval. So for example, the top left inner value would be the result of multiplying 1/5, the row interval, by 1/9, the column interval. Next, staying in the same row and moving over to the next cell, that value would be the result of multiplying 1/5, the row interval, by 1/3, the column interval. The next cell would be the result of multiplying 1/5, the row interval, by 1/1, the column interval, and so on for that row, and then we do the same for the other two rows. Filling in the rest of the values, we get the following table:
| 1/9 | 1/3 | 1/1 | 3/1 | 9/1 | |
| 1/5 | 1/45 | 1/15 | 1/5 | 3/5 | 9/5 |
| 1/1 | 1/9 | 1/3 | 1/1 | 3/1 | 9/1 |
| 5/1 | 5/9 | 5/3 | 5/1 | 15/1 | 45/1 |
As with Pythagorean tuning, it’s at this point that we apply our powers of two, to bring each of these intervals to be within the range 1.00 to 2.00. We multiply each interval by some power of two, raising or lowering it by some number of octaves, until the interval is within that range. Doing that, we now end up with the following values:
| 1/9 | 1/3 | 1/1 | 3/1 | 9/1 | |
| 1/5 | 64/45 | 16/15 | 8/5 | 6/5 | 9/5 |
| 1/1 | 16/9 | 4/3 | 1/1 | 3/2 | 9/8 |
| 5/1 | 10/9 | 5/3 | 5/4 | 15/8 | 45/32 |
This gives us 15 intervals. Putting them in order, we have:
| 1/1 | 16/15 | 10/9 | 9/8 | 6/5 | 5/4 | 4/3 | 45/32 | 64/45 | 3/2 | 8/5 | 5/3 | 16/9 | 9/5 | 15/8 |
Using the 12-TET or Pythagorean tables as a reference again
12-TET
| 2^(0/12) | 2^(1/12) | 2^(2/12) | 2^(3/12) | 2^(4/12) | 2^(5/12) | 2^(6/12) | 2^(7/12) | 2^(8/12) | 2^(9/12) | 2^(10/12) | 2^(11/12) |
| 1.0000 | 1.0595 | 1.1225 | 1.1892 | 1.2600 | 1.3348 | 1.4142 | 1.4983 | 1.5874 | 1.6818 | 1.7818 | 1.8877 |
Pythagorean
| 1/1 | 2187/ 2048 | 9/8 | 19683/ 16384 | 81/64 | 177147/ 131072 | 729/ 512 | 3/2 | 6561/ 4096 | 27/16 | 59049/ 32768 | 243/ 128 |
| 1.0000 | 1.0679 | 1.1250 | 1.2014 | 1.2656 | 1.3515 | 1.4238 | 1.5000 | 1.6018 | 1.6875 | 1.8020 | 1.8984 |
In this case, unlike the harmonic-based table created earlier, the Just Intonation intervals do a good job of mapping to the intervals contained in these tables, with extra intervals to spare.
As was the case with the harmonic-based table of intervals, the Just Intonation method also produces two different intervals for the tritone. Here it’s 45/32 and 64/45. And as was the case before, if you take the geometric mean of these two intervals, thereby combining them, the interval you get is the square root of 2, the tritone value used in 12-TET.
In practice, what you would actually do is select one or the other, let’s say 45/32, the smaller of the two. That leaves you with 14 intervals. From there, it again comes down to choice. There are two intervals generated by the Just Intonation method (the 10/9 and 9/8 intervals) that could be used to correspond to a minor second. Likewise, there are two intervals generated (the 16/9 and 9/5 intervals) that could be used to correspond to a major seventh. 9/8 is usually chosen for the minor second, and 16/9 is usually chosen for the major seventh. That gets rid of the two extra intervals, 10/9 and 9/5, and leaves us with the 12 intervals that can map directly onto the other two reference tables.
Here they are, all together:
Pythagorean
| 1/1 | 2187/ 2048 | 9/8 | 19683/ 16384 | 81/64 | 177147/ 131072 | 729/ 512 | 3/2 | 6561/ 4096 | 27/16 | 59049/ 32768 | 243/ 128 |
| 1.0000 | 1.0679 | 1.1250 | 1.2014 | 1.2656 | 1.3515 | 1.4238 | 1.5000 | 1.6018 | 1.6875 | 1.8020 | 1.8984 |
Just Intonation
| 1/1 | 16/15 | 9/8 | 6/5 | 5/4 | 4/3 | 45/32 | 3/2 | 8/5 | 5/3 | 16/9 | 15/8 |
| 1.0000 | 1.0667 | 1.1250 | 1.2000 | 1.2500 | 1.3333 | 1.4063 | 1.5000 | 1.6000 | 1.6667 | 1.7778 | 1.8750 |
12-TET
| 2^(0/12) | 2^(1/12) | 2^(2/12) | 2^(3/12) | 2^(4/12) | 2^(5/12) | 2^(6/12) | 2^(7/12) | 2^(8/12) | 2^(9/12) | 2^(10/12) | 2^(11/12) |
| 1.0000 | 1.0595 | 1.1225 | 1.1892 | 1.2600 | 1.3348 | 1.4142 | 1.4983 | 1.5874 | 1.6818 | 1.7818 | 1.8877 |