INTERVALS & INTONATION

I would like to share with you something that is part of my research in music.

It is the beauty of pure intervals. Intervals can be melodic or harmonic. What I want to show you is most obviously perceived when we hear the notes together, in the harmonic way.

The simpler the relationship between two frequencies, the more we experience harmony.

The first interval is the unison with a ratio of 1:1. Perfectly consonant, it’s not very exciting, because nothing is happening.

The second one is the octave, the ratio is 1:2. Still very stable but now something opens up.

Then comes the perfect fifth 2:3. It’s stable and radiant and at the same time there is an interesting interaction between the two frequencies. The fifth can be a perfect end chord because the ground note is the tonic and it doesn’t need a resolution.

The fourth has a ratio of 3:4. This time the lowest note is not the tonic but the dominant and it asks for a resolution. It’s not considered very consonant for this reason.

Thirds represent the feeling element with their minor and major colors. The major third has a ratio of 4:5 and the minor third has a ratio of 5:6. Their complements are the minor sixth 5:8 and the major sixth 3:5.

Now you may ask: ”What’s the point?”

The point is that the pure fifths don’t fit exactly in the pure octaves. On the cello we can play perfect intervals because there is freedom of intonation. On an instrument with fixed tuning, like the piano, you have to find a way to handle the imperfection differently because you cannot change the tuning of a note while playing.

It’s fascinating that music is very simple and incredibly complex at the same time.

It’s like a puzzle with perfect pieces. You can put some of them perfectly together but you can never put them all into the frame at once. To have perfection in one place, you have to sacrifice perfection in another place. Musicians have always struggled with this problem and have always looked for the best compromise. There is no solution, it’s a sort of mystery.

Today, we tune the piano in the equal temperament, reducing slightly all the perfect fifths. All the notes are the same distance apart, to make it possible to play in all the different keys, but none of the intervals is really pure or perfect anymore.

What I am looking for is to play with perfect intervals. Why ? Because I love the sound and feeling of it.

How can we do that on a cello ?

We have the four open strings that have a fixed pitch. All the notes we produce with our fingers though, can be adjusted.

We will have to talk about notes and frequencies. It’s the physics and mathematics, that are part of music. I will show you the frequencies in the 432 Hz tuning, which can be a subject on its own for a next video. The wonderful thing is that we can divide 432 by almost every number and so we get nice divisions with whole numbers.

The A string of the cello is an octave lower than the A that defines the pitch: 432 : 2 = 216 Hz. The D is 2/3 of 216 = 144 Hz. The G is 2/3 of 144 = 96 Hz and the C is 2/3 of 96 = 64 Hz.

Let’s start with the first finger in the first position on the D string, giving an E.

For the E, a whole tone above the D we find the frequency being 144 x 9/8 = 162Hz. That corresponds exactly with the fourth in relation to the open A string 162 x 4/3 = 216Hz. So far, so good.
Now we want to play the E against the open G string, making a major sixth.

From the G 96 Hz we find the E being 96 x 5/3 = 160Hz.

Two beats less than the E found before. Do we hear the difference? Yes, we do. So, depending on the context we have to adjust our finger position to get a perfect interval.

You remember what we said about harmonics in a previous video ?

We talked about the C string and the A on the G string producing the same harmonic.

Playing the C we can hear the harmonic E which is vibrating five times faster than the C. 64 x 5 = 320 Hz. If we play the A as a whole note above the G, that makes 96 x 9/8 = 108 Hz. The harmonic E will be 108 x 3 = 324 Hz. Slightly different from the harmonic of the C. But if we play the A as a major sixth above the C it will be slightly lower making the same harmonic of 320 Hz, vibrating in sympathy with the other harmonic.

You see how we can adjust a note to get the perfect consonant or resonance. 

It’s a bit of a puzzle sometimes, but it’s rewarding.

Where it becomes even more interesting, is with the major and minor thirds. Thirds represent feelings and emotions. They play an important role in music, defining major and minor keys. We will explore the difference between an equal temperament tuning and really consonant thirds.

On the D string, the F natural is 171 Hz in equal tuning. When we play it as the minor third on the D it is 144 x 6/5 = 172,8 Hz, almost two beats higher.

Same thing with the F#, which is 181,6 Hz in equal tuning. Played as the major third on the D is gives 144 x 5/4 = 180 Hz, slightly lower.

In general, flats are higher, sharps are lower. When we play a major third or sixth, we have to make it a bit smaller and when we play a minor third or sixth, we should make it bigger. 

To know what to do, we have to take into account the function of the notes in the harmonic context. Because the same note can be the tonic or the third and will have a slightly different intonation.

Small differences, but if you play these intervals with this intonation in mind, the sound gets another meaning.

I love to play these pure intervals. Why are they perceived as harmonious? Because the two frequencies interact and join all the time. Perfect intervals are calming, harmonizing and aligning our energy.

I’m curious to know if this inspires you to experiment. Please let me know and enjoy.