It Just Makes Cents -- A Deeper Dive into Music and Tuning!

        In the 7th grade, my band directors would tell us, the little amateur musicians in the middle school band that we were, that the individual chords that we were playing in our daily chorale had to be adjusted to make the chord resonant. Mr. Raney, one of my band directors, drew on the whiteboard the notes of the chord that we were playing. If you were playing the 5th, then you had to be playing the note +2 cents sharp. The 3rd had to be -13 cents flat. The root of the chord was the only note that could be in tune, per say. What did this all mean? Seventh-grade me was freaking out. Everybody else in the room looked confused as well. Mr. Raney wrote down "Pythagorian Tuning" on the board yet I had no idea how that correlated to what I was doing. "It's simple math," he said, looking at all of us with a look that said, "How can you not know this?" . Math? In music? How could that be? The questions I had over this topic consumed me.
        Fast forward a year later, when eighth-grade me was assigned a year-long project over ANY topic that would have to be thoroughly researched and well-presented by April. When I was asked to write down the topic of my project, of course the words "Pythagorian" and "Tuning" were what blotched out of my pen. Sure, I had heard of Pythagoras. The a^2+b^2=c^2 equation was what we had just gone over in the geometry unit of my mass class was aptly called the Pythagorean Theorem. But what did he have to do with tuning? Thus, the commencement of my intriguing yet complicated project began. I had no idea what I what I was getting myself into.
        I had to start with the basics. And so, I will tell you the basics because I am assuming you, as I did, did not know where to start in understanding music theory.

        The discovery of music can be traced down tens of thousands of years back. Fragments of bone flutes have been found at Neanderthal sites. Early instruments prove that humans have produced pitched sound for a long time. As different musical styles began to emerge in different regions, the history of tuning progressed as new discoveries in the physics and mathematics behind the music were being made. If you were to go back 1000 years in time and hear a group of musicians play, the sounds of the music would not sound like the music played today. Why? The reason for this is that different systems of intonation, or tuning, were changed into being the standard of the status quo as time went on. To know and come to appreciate tuning systems, the physics behind sound has to first be understood. So, let's start with the simplest question.

What is sound? 

        Sound is a type of energy that is caused by mechanical vibrations in the air: When an object vibrates it causes movement in the air particles surrounding it, and then those particles bump into other particles that are close to them, and those into more particles, and so on, that vibrate until they run out of energy. This energy can be defined as sound waves or periodic oscillations. Irregular repeating vibrations sound harsh to our ears, which can be described as “noise”, while regular repeating vibrations can be described as musical notes. When the vibrations are fast, a high note is produced, and when the vibrations are slow, a low note is produced. Two characteristics define sound: frequency, which is the rate at which a wave repeats itself in a given period of time, and amplitude, which measure the intensity of the wave. Frequency can be measured in Hertz (Hz), which is equal to 1 cycle per second, while amplitude is measured in decibels (dB). Pitch is essentially the frequency of a note. It is the quality of a sound determined by the rate of vibrations producing it; it is the degree in which the tone is high or low. The Wind and brass instruments produce sound by vibrating air inside a tube. By pressing down keys that open or close holes in the tube, the air column inside becomes longer or shorter, determining the pitch of the desired sound. With string instruments, fingers or a bar press down on strings that caused the strings’ length to change and vibrate at different pitches. That being said, where is the math in sound?

Pythagorean tuning and other systems of tuning:

        Although Pythagoras may be best known for the Pythagorean theorem in geometry, there are other things that Pythagoras is credited for that you may not know about. The Pythagorean discovery that “all things known have number – for without this, nothing could be thought of or known” – was indeed experimented first through music. Musicians by the time of Pythagoras had already been tuning instruments for centuries, so people were aware that sometimes a lyre would make pleasing sounds and sometimes it would not. According to legend, during the sixth century B.C. the Ancient Greek philosopher Pythagoras observed blacksmiths at work and noticed that some of the hammers made consonant sounds when they were struck at the same time. He realized that these altered pitches came from the differentiating weights of the hammers: When one hammer weighing six pounds struck together with a hammer of 12 pounds, a 2:1 simple ratio was produced. Pythagoras found that simple fractions, such as 3:2 or 2:1, distinguish pleasing intervals. He discovered the important principle of harmony. Thus, the Pythagorian tuning system was invented based on the constant intervals 1:1 (unison), 2:1 (the octave), 3:2 (the perfect fifth), and 4:3, (the perfect fourth). To find the next degree in a scale (a set of musical notes ordered by fundamental frequency of pitch) you would have to start with the root note (fundamental pitch) and multiply it by 1.5, or 3/2. The problem with Pythagorean tuning is that even when the system was created all the way around through a circle of twelve pitches, a Pythagorean comma (two different pitches exist for the same note) is formed, which keeps essentially any system built on an interval from being a perfect circular system. Octaves spread farther and farther apart as the pitches form a spiral. Also, since the system is based off of open fifths (a fifth without the third), thirds are theoretical dissonances, the major third being 408 cents wide rather than the in tune 386 cents, so they were avoided at final cadences. At the end of the 15^th century, composers got weary of the medieval strictness of open fifths, so they began to use a different tuning system that engulfed the sweetness of the major third. So, different tuning methods were invented by musicians. 

The one we use today and has been the standard for the past 200 years is called equal temperament and consists of a compromise -- it sacrifices all the pitches being in tune to obtain the possibility of changing from one key to any other given key. This means that all the notes in the equal-tempered 12-tone chromatic scale are the same distance in frequency apart. 

The "natural" system of tuning is called just intonation. Just intonation is a tuning system that contemplates pitches in a chord rather than an arbitrary frequency. It is based on the naturally occurring phenomenon of the overtone series, or harmonic series. Overtones are the ‘building blocks’ of sound, meaning they make up the sound we hear - sort of like the DNA in living organisms. A singular sound that we think we hear isn’t only one sound – there are many quieter subsidiary sounds supporting that one sound that we recognize, called the fundamental pitch. To find overtones of a sound, the fundamental pitch has to be multiplied by multiples of itself expressed as f₀, 2 f­₀, 3 f₀, 4 f₀, 5 f₀, etc., where f₀ is the fundamental frequency. For example, the series of frequencies 1000, 2000, 3000, 4000, 5000, etc. given in Hertz is a harmonic series. This phenomenon can be demonstrated on a string: When the string is plucked, it not only vibrates over its entire length producing the “ground note” or fundamental pitch, but also in fractional divisions of its length such as 1/2, 1/3, 1/4, 1/5, 1/6, etc. These fractional divisions are overtones. Theoretically, there are infinite overtones in just one note, but the overtones get softer in amplitude as the pitch series ascends, so it is very hard to hear the higher partials. Musical instruments and human voices emphasize, or “bring out” certain overtones more than others because of the intricate differences in the way their structures resonate and amplify sound. This creates a variety of sounds and is the reason why a trumpet ends up sounding like a trumpet and a clarinet sounding like a clarinet. 

A visual at how the vibrations of the fundamental pitch (top) is divided into overtones. 

What does it mean to be "in tune"?
        
        Regardless of the tuning method you use (I have been using equal temperament because that is what is set on most digital tuning devices), being in tune means to play the certain pitch that the note is set in. When you're out of tune, you will sound either "flat" or "sharp". As referred to in my last post, being flat is shown by the symbol ♭ and being sharp is shown by the symbol #. Being flat means that the pitch you're playing is lower than what it should be, and being sharp means that the pitch you're playing is higher than what it should be. Standard pitch in the United States is A440, which means that the musical tone of the note 'A' is set at 440 Hz. So, if you were to be playing this specific A at 435 Hz, you'd be 5 cents flat. If it was 445 Hz, you'd be 5 cents sharp.Why is it important to be in tune? If you're in tune, you will sound harmonious. If not, it will be dissonant. Professional musicians are constantly adjusting their pitch.