What should a temperament sound like? I've orderly tuned an equal temperament between F3 and F4. And I've given strong priority to the interval of the fifth.
Why? Because the interval of the fifth is the first and strongest harmonic interval of nature scale, the harmonic series. If we called F2 the first partial, here's the second. the third, and the fourth. So the first four notes of the harmonic series, three of these notes have the same letter name.
They're all the same note, they just occur in a different register. But the first interval with two letter names is the fifth. So again, the fifth is the first and the strongest harmonic interval.
Now, When I say that we're going to give it priority, it means that we want to tune the fifth in such a way that it's as clean and quiet as possible, and yet still keep the fifth tuned or tempered narrow. Now, there are only two intervals that are tuned narrow of perfect, the fifth and the minor third. All other intervals, the third, the sixth, the tenth, they're tuned wide of perfect. So let's go through the sequence that I used to tune this temperament.
I began with A4 using an aural pitch source. I set A to 440. Then I dropped down a tenth. And I tuned that interval to have 7 beats or a little bit faster than 7 beats per second.
From F3 I tuned A3. and I tuned A3 so that this third is slightly slower than this tenth. From F3, I tuned A flat 3, a minor third, and remember a minor third is tuned narrow of perfect. So let's listen to the comparison between the major third and the minor third. That's more of a difference than there is between this major third and the tenth.
So whereas there is only a slight difference between this third and this tenth, there is a distinct difference between this major third and the minor third. From A flat, I tune C. And the objective here is to tune this third slightly slower than this minor third. Minor third is faster.
Major third is slightly slower. Now, if we made this third the same speed as this minor third, the fifth would be perfect. And we don't want it perfect. but we want it very close to perfect. Therefore, the major third should be a bit slower than the minor third.
Having tuned this third, you'll notice we have this relationship. We have a fifth, the bottom note up, a third, and the top note down a third. Compare those two thirds.
Slow, fast. So again, we have a nice sounding fifth, major third, minor third, major third, major third. major third. Now we have two other intervals here. This minor third, the bottom minor third of the fifth, the top minor third of the fifth.
This is very useful. You'll notice that the fifth is defined by a major third and an adjacent minor third. If we were to expand this major third by raising A, we would increase the speed of the beats.
By raising A, Today, in that this is a narrow interval, we would also speed up the beats of this minor third. So if we lowered A to slow the beat rates down for this major third, at the same time, we would be slowing down the beat rate of this minor third. Now that's useful because it helps us to set the speed of the major third. If we made this too fast, that minor third would just become a blur.
If we made it too slow, maybe that minor third would be a little bit too easy to hear. At any rate, we have these four notes tuned. The next step is from the bottom note of the fifth, tune up a fourth, and from the top note of the fifth, tune down a fourth.
Now it's interesting, when we have this note tuned and this note tuned, those two notes form a minor third. That's very useful. Many people are familiar with taking an octave and dividing that octave into thirds by playing contiguous thirds. Well, you can do the same thing with the fifth, just using minor thirds. You know that these two fourths are tuned correctly when the two fourths seem to beat the same, and at the same time, this minor third fits between this minor third and that minor third so that you have an even stepped progression Now, that only leaves these two notes to have all of the notes tuned within the fifth.
So, how can we tune this? We'll begin with F sharp. How can we tune it?
Well, we can tune it so that this third is a little bit faster than this third. And it's interesting. This minor third and that minor third progress in the same way as the major thirds progress. A little faster, a little slower.
A little faster, a little slower. So we can quickly have F sharp in tune. That leaves B. We can do this. We can use this minor third to tune B, and we can also use this major third to tune B.
So we tune F sharp and B in such a way that we have an even progression of major and minor thirds. And that gives us the chromatic force, which seem all to be tempered about the same. Now, from there we need to complete the temperament.
So we can tune up using fifths. Tune C sharp. Use the major and minor third.
Tune a fifth. Major and minor thirds. Again go up. Whoops. Tune.
You might hear a little bit of activity in this fifth, and it's really not activity here. We have a false beat or a wild string here. Then up again.
Fourth fifth. Now let's listen to the sixth going up. Now there's another comparison, the outside 6 and the inside thirds. Now I want to comment about that. Traditionally, the major 6 and the major 3rd would be said to be equal in beating.
6th beat rate equals this 3rd beat rate. But when you give priority to the fifth, the sixth will beat a bit faster than the third. And the reason for that is that this fourth is a bit more expanded when you give priority to the fifth.
That makes this sixth beat a little bit faster. Now... Listen to the chord.
Very clean, very quiet. Settled. Has a bit of a gravitational pull to it. Very clean and quiet. The same thing.
Now this dominant seventh chord, let's just listen to the major chords and we'll hear the same thing. It's settled, quiet, clean.