#minute physics

A staff is constructed by five parallel lines called ledger lines which represent the positioning of musical notes on or between them. Non-pitched percussive instruments are non-standard and will be mentioned further along in the series. As instruments have different pitch ranges (for example, our recorder running from 523.25 to 3160 Hz), these five lines certainly aren’t enough to represent the large number of musical notes which are performed. To suggest whether, in general, lower notes are needed versus higher notes: we introduce clefs.

There are three clefs most commonly used today, called the G, F, and C clefs. These are shown in the same order below, but note that each clef has several sub-uses and so how far up or down on the staff will depend on its usage. As a counterexample: nowadays, the G clef is always the treble clef. Way back in the day though, it used to be shifted down one ledger line to act as the “French violin” clef. In either case, the central swirl of the G clef encircles the ledger line which denotes a tone called (unsurprisingly) G.

Where do the notes even come from, though? Confidently, we said last week that the tone produced by our soprano recorder at 523.25 Hz was the note “C”. Who decided?

Boethius, in the sixth century, used the first fourteen letters of the Latin alphabet (excluding J, which hadn’t been... invented yet) to represent the most commonly used tones of the period. As the range extended, and mathematician/philosopher types uncovered some of the nice relationships between notes, musicians decided to repeat the first seven letters instead. Finally, the C-first ordering of the letters arose from the usage of major and minor scales. But the names aren’t really important. Different cultures use different letters and symbols and produce similar results! How do we know that 523.25 Hz is the right frequency? Why not 520 Hz? Or even simpler at 500 Hz?

It’s, again, anticlimactic but the specific frequencies we use are really arbitrary. Slightly prior to 1834, Johann Heinrich Scheibler trekked from region to region in Europe measuring tuning forks (which were all a little or a lot different) with a device called the tonometer. His rough average of all these measurements? That a specific A note should be standardized to 440 Hz. Several meetings in several countries later, A = 440 Hz is here to stay. What matters most are the frequency differences between the notes, or intervals, and that’s where Pythagoras of Samos comes in.

Apocryphally, Pythagoras was strolling by a blacksmith’s shop one fine day and heard the pleasant ringing of two different sized hammers striking anvils. He determined that he only needed two ratios (2:1 and 3:2) to build a tuning standard. From this spilled forth all of the first ever documented tuning system. Skepticism towards the Cult of Pythagoras aside, much of what we now know about consonance, or how pleasing two sounds combined are to the ear, comes from the original system of Pythagorean tuning.

Construction goes as follows: Take some base frequency and multiply by 3/2. This produces a new tone which is consonant with the first. Repeat five more times and record these frequencies. Now return to the base and divide by 3/2 five times. These twelve notes occupy a larger range than most instruments can even reach and so we can use the other ratio to reduce all of the notes into a smaller range between the base frequency and the note exactly twice its frequency. This special higher limit is called the octave. An example for 440 Hz is provided below.

The asterisks following note names represent a modifier to the standard tone called an accidental. Specifically, sharps and flats. More on this will come later but for now, keep in mind that flats (♭) are lower in frequency and sharps (♯) are higher.

This isn’t the end of the story due to an unfortunate side effect. No whole number of multiples of 3/2 can ever fit between octaves of the same note. The mathematical phrase would be that 2^n =/= 3^m for any integer n,m > 0. This suggests that instead of eventually looping around to the same note, the tuning method produces a spiral, and conflicts with itself. 

As such, the original ratios from the Pythagorean system were altered again and again to eventually produce different tuning methods called temperaments which included the famous equal and well temperaments. A quick explanation of modern piano tuning can be found in a fun Minute Physics video by Henry Reich: “Why It’s Impossible to Tune a Piano”.

As we now have separation of octaves into twelve equal pieces, sharps and flats are used to denote a “half step” above and below the modified note, respectively. Using the piano as our model, a half step is the frequency distance between adjacent keys (e.g. B to C) while a whole step is twice the size and has a separating key (e.g. A to B).

Note that this suggests E♯ and F are the same frequency and therefore are the same note or D♯ is the same as E♭.

So a staff indicates the notes to be played throughout the work and the frequencies of these notes are now (more or less) standardized in Western music. The ledger lines and the spaces between the lines hold notes which increase in name alphabetically and then loop around every seventh letter. As closing remarks, we can present the first piece of sheet music for this series.

Note that the squiggle is called a quarter note rest, named after what it requires the musician to do (...sort of... just no noise, okay?) and the rectangle is a half note rest. Until a deeper discussion of rhythm and timing: they’re the same. Consider all notes and rests to take the same amount of time and ignore the vertical bars.

This shows eleven notes being played. The treble clef identifies the first six notes are G. There is also a B-flat and E-flat, and an F played at the end. From the above discussion, B-flat is at a frequency of 463.5 Hz. Also, the depicted E-flat is an octave below the one calculated and should therefore be half the frequency at 313.25 Hz in the Pythagorean tuning system. Have you identified which song the sheet music is for?

For Thought:

1) C clefs are often used for vocals. Given that we can shift clefs up and down on staves, where might a C clef be positioned for a soprano singer relative to a tenor singer? Given four vocal parts: soprano, alto, tenor, bass, which vocal parts would most likely share the same clef positioning?

2) We generated frequencies from the standard A = 440 Hz tone. Would a Pythagorean tuning scheme from one of the other calculated frequencies give the original A tone back? Generate equal temperament tuned notes from A and see if the C = 523.25 Hz note from last lecture appears.

3) Instruments can easily span more than a single octave in their range. As such, one needs to be able to distinguish between notes of the same name. A = 880 Hz is of the same pitch class (note name) and so we can distinguish this by saying that A4 = 440 Hz, and A5 = 880 Hz. In other words, there are three A notes lower than 440 Hz commonly in use. What would C = 523.25 Hz be then? What about the G emphasized by the treble clef?

4) Calculate the Pythagorean and equal temperament frequencies of the treble clef G. Sketch the fingering of this note on the soprano recorder and graph the waveform of the fundamental and first two overtones (integer divisions) with accurate labeling. What are these overtones in musical notation?

5) Offer an informal proof as to why there are no integer solutions larger than zero for the equation 2^n = 3^m.