Prelude for Thought

Intervals

Any two pitches make up a dyad which contains an interval. This interval can be classified in two parts: 1) the generic name and 2) the quality.

To generic interval is simply the number of notes separating two pitches. Drop all accidentals and count letters between B♭ and F♯... B C D E F. The interval is a fifth. For another example, what is the generic interval between G♯ and A? It’s a second.

How about identifying intervals on sheet music?

If two notes are both on the same staff component (e.g. both spaces or both lines), the interval will be odd. The third dyad in the above image has notes both on lines with one line between them. This interval is a fifth. Count the note names... G A B C D, and this agrees. The last interval has two lines between the pitches and therefore is a seventh. When the components the notes sit on are distinct, the interval will be even. The first dyad has a separating space and line, so this interval is a fourth. Count the names: G A B C. The second interval is the same because accidentals don’t matter in generic intervals.

Consider the rectangular accidental on the second interval. This is the natural sign and we need it to distinguish from G♭ in the first dyad. Accidentals will persist through the entire measure (segments of the staff separated by the vertical bars). Measures will be discussed thoroughly in the next topic. This means that the G in the next dyad is still a natural G, and if C was notated again after the second interval without the natural sign, musicians would play C♯.

Understandably, playing G and D together is very different from G♭ and D♯ as one sounds nice and one... less so. The quality is different. There are five most common qualities one can impart upon an interval: Perfect, Major, minor, Augmented, and diminished.

Perfect intervals include unison, the fourth, the fifth, and the octave. Major and minor modifiers are used for seconds, thirds, sixths, and sevenths. Augmented intervals come from increasing any perfect or major pitch distances by a half-step. Diminishing an interval is lowering any perfect or minor pitch distance.

The first measure is the same as above. Try determining the complete interval names and find the whole thing for each dyad in the new measure.

We know G and C together make a fourth, but this G is flat, and therefore augments the perfect fourth. In the second dyad, G♭ is now raised to a natural G, but C has also been raised to a sharp and therefore the interval remains the same. The third interval is a perfect fifth, which is why G and D are adjacent in the circle. The final dyad shows a seventh, but we need to know the notes of the root (lowest) pitch’s scales to identify the quality.

In a major scale, the final note before the octave is a half-step lower. Therefore, F is in the major scale and it’d be called a major seventh interval. This trick doesn’t help with some of the more central tones in the scales, so we can also do this the systematic way.

G♭ is as far away from C as possible and the scale has six flats (or six sharps depending on your viewpoint - see question 4 in section 1.2) which appear in the order: B♭ E♭ A♭ D♭ G♭ C♭ in a key signature. This means that F is a tone in the G♭Maj scale and so the interval is a major seventh!

Inversion

One final method for generating any pitch distance is by utilizing the cyclical nature of note names in an octave. We can invert the interval by moving either of the members of the dyad past the other by jumping octaves.For example, the dyad A4 + C4 (which is a minor third) is inverted to either C3 + A4 (a major sixth) or C4 + A5 (a major sixth). Note that the inverted interval is the same no matter the direction.

The rules are simple and fairly intuitive. This interval plus the inverted interval value will always add to nine (check for yourself with the above example). Major intervals will become minor and augmented intervals will become diminished. Perfects stay perfect.

How can we use this trick to our advantage? We want to generate a major seventh above G♭. Inverting our desired interval means we want a minor second below the root, or just a half step away. Again, this is F.

Okay, enough of that example. Let’s try to create a diminished sixth below A♯! Inverting the sought after distance gives an augmented third above our pitch. A major third is two whole-steps above... C, D... then augment it: D♯.

Triads

A third note can be added to the intervals above to make chords. In western music, the most common types of chords are tertian, or built from thirds. Take the root note and add a major third above it. Then add a minor third above that. Using just third intervals, we now have a root, major third, and perfect fifth playing simultaneously. If we instead add a minor third and then a major third, we have a root, minor third, and again a perfect fifth. These are major and minor triads. We can also move the position of the fifth. In a major triad, raising the fifth by one semitone retains that all intervals are thirds (specifically major thirds) and the chord is augmented. Lowering the minor third would give a diminished third, but this gives an interval of only one whole-step, which can be quite dissonant. Likewise, lowering the fifth in a minor triad gives a diminished chord.

When all three are stacked directly on top of one another, they are said to be closed. When there are octave gaps, the chord is open. These octave gaps often arise because the chords, like the dyads above, have been inverted. Raising the root up an octave most the chord from root position to first inversion. Raising both the root and third, so that the perfect fifth is now in the bass is called second inversion.

Of the following chords, identify the root note, the quality, and whether or not it is inverted.

We see that the first chord is C, E, and G. As E is a major third (two whole-steps) away from the root and G is a perfect fifth (adjacent to C in the Circle), this is a Cmaj triad. The second chord is Cmaj in second inversion, as the fifth is in the bass position. For the third chord, we see D in the base with F (the minor third) above that and G♯ (the diminished fifth) stacked on top. This is a diminished triad, written as Dm♭5 for the minor third and a flat fifth, or D°.  Lastly, the chord F♯, B♭, D, F♯ is still a triad as the base note is doubled. After some inspection, one might be able to see that D is a major third above B♭ and F♯ another major third above D. This is an open, augmented B♭ chord in second inversion. Our notation would be B♭+.

Seventh Chords

We can continue to stack another third on top of our four possible triads to give seven possible seventh chords: diminished, half-diminished, minor, minor major, major minor, major, and augmented major. The names make less sense now that each chord contains so much information, so we rely on chord symbols more extensively as additional notes are added.

To add another third to any triad, we can count three semitones or four. The distance from C to B is called a major seventh and spans eleven semitones. The minor seventh spans ten. Adding the major seventh to the major triad gives the major seventh chord: CM7. Adding the minor seventh results in the minor major, or dominant, seventh: C7.

Adding the major third of our perfect fifth (the major seventh interval) gives the minor major seventh: CmM7, or the “opposite” of our major minor seventh. The minor seventh interval atop the minor triad gives a minor seventh chord: Cm7.

With the minor seventh interval added to the above triad, we form a half-diminished seventh chord: Cm7♭5 or Cø7, but with a diminished seventh (ten half-steps) on a diminished triad, we have the diminished seventh chord: Co7.

Lastly, there is only one real option here. Adding a minor seventh breaks the informal rule of stacking thirds and counting four semitones forward merely doubles the root. Adding a major seventh interval (a minor third above the augmented fifth) gives an augmented major seventh chord: C+M7.

As one might imagine, we can continue to stack thirds all the way to thirteenth chords before we have every note in the scale and the rules outlined above give a systematic way of generating each one. As complexity increases, the symbols stay consistent. Desiring an eleventh chord with an augmented triad, minor seventh, major ninth (fourteen semitones), and perfect eleventh (as it’s essentially a fourth above the root, plus an octave) gives a dominant eleventh chord: C11, seen below. For nontraditional chords, using sharps and flats before the superscript intervals is all any musician should really need.

What might a Sokovia chord sound like? The fictional country is modeled around the inland autonomies of southeastern Europe, like Slovakia/Moldova/Hungary. One of the most famous Hungarian composers of all times was Franz Liszt, the same man from the War of the Romantics! His piece Hungarian Rhapsody No. 2 opens with a single C♯4, fleshed out by an F3 and G♯3. He doubles both the C♯ and G♯ in the lower register. 

This is a C♯maj triad in first inversion, with the doubling. Quite a step up from the triads above. Is it distinctly southeastern European? Liszt allegedly based his rhapsodies off of local folk music, so perhaps.

Half a minute later though, as the opening call slows, he restates the beginning now with an E instead of F in the chord. This would be C♯m, still in first inversion. Maybe this is more distinct. In fact, the whole intro (called the Lassan) is in the key of  C♯ minor, and it is fairly... moody. Which most people associate with minor keys, flawed thinking or not. It might not be until the Friska (dance) that anything stereotypically Hungarian folk emerges. More on that a little later.

For Thought:

1) Construct the G♭ harmonic minor scale. Now, on each note, build a chord on top using the third and fifth relative to the given root. Be sure to remain in the key. Identify each triad using the generic interval and quality as above. On a general level, do certain scale degrees or positions in the scale tend to have the same quality no matter the tonic? Are there smaller cases to break the trends up? Could someone do the same thing with seventh chords?

2) Generate ninth, eleventh, and thirteenth chords using the same systematic method of adding major and minor thirds onto seventh chords without breaking any established rules. Label them using superscript sharps and flats on the necessary intervals. Often times, these extended chords will emit middling notes as they do not do much to change the overall sound.

3) Suspended chords can come in two flavors: suspended second and suspended fourth. Instead of a third, these chords include seconds or fourths, respectively. Create sus2 and sus4 chords for any six pitch classes and write their inversions. As an example, A♭sus2 would include A♭, B♭, and E♭; while A♭sus4 is A♭, D♭, and E♭. Do any patterns emerge?

4) Construct a proof (visual or otherwise) for the rules of inversion: 1) The interval and inverted interval distances sum to nine, 2) Perfect intervals stay perfect, 3) Major intervals become minor and vice versa, 4) Augmented intervals become diminished and vice versa. Be sure the argument includes the extended chords mentioned in the second question.

5) Below are definitions for three more types of chords. They might have different musical function, but the tones and their order are identical to a chord already defined above. Find at least one alternative definition previously mentioned for each. What is the unifying idea?

Slash Chords - These are written as Chord/Pitch where the /Pitch describes what note the chord should have in the base. 

Modern Sixth Chords - These are chords with an added sixth above the root.

Polychords - Notated as a fraction instead of the slash above, they feature two chords that are to be played simultaneously. Take as an example Cm7 over F. 

Encouragement abound for the interested reader to continue digging! The combatants, musical pieces, critiques, newspaper headlines, etc. are all worth their own discussion. Today though, the thirteenth variation of Paganini’s twenty-fourth caprice deserves special mention. Within the ledger lines of the sheet music is a very special motion called a glissando. As it takes place on a piano, it could even be called discrete glissando, meaning that there is audible gliding between two notes, with distinct tones heard between the glide’s endpoints.

This is clearly from Disney’s Peter Pan.

Picture the above photo, except less pirate-y... and more glissando. Piano glissando can be: all the white keys, or all the black keys, or with impressive timing, every key in order. During these moments in the variation, the pianist cares less about what notes are being used in the overall piece. The same cannot be said for everywhere else in the music. There are some notes during the melody which when played would sound very off indeed.

Why is this happening? In effect, the piece of music has been restricted to a set of acceptable notes to limit dissonance. In Western music (our system of choice), the set of acceptable notes is a seven-pitch, or heptatonic, scale. So five of the constructed notes discussed last time are thrown out. These notes are chosen based on the special rules of the scale and there are a lot of them. Seven very popular sets of rules are called Ionian, Lydian, Mixolydian, Dorian, Phrygian, Aeolian, and Locrian. These are also called modes, and some theorists would be quick to note that the relationship between modes and scales are more subtle than described above. Let’s ignore that until a bit later.

The two modes to be discussed initially are the Ionian and Aeolian modes, more commonly referred to as the major and natural minor scales.

The Major Scale

We can start from C4, or middle C, at 440*(1/2)^(3/4) Hz, or 261.6 Hz and move up two whole steps, then a half step, then three more whole steps, and finally another half step. This brings us to the next octave in our initial pitch class, C5.

So the notes in the C major scale are: C, D, E, F, G, A, and B. Super easy! No accidentals required. We could also put the twelve semitones into a ring and draw connecting lines as follows.

Doing the same for every major scale makes... a mess... but it’s fun to visualize.

The Natural Minor Scale

This time, our progression moves through a whole step, then a half step, then two whole steps, then another half step, then two final whole steps. Using any of the diagrams above, we see the set of pitches in the A natural minor scale are A, B, C, D, E, F, and G! Again, really easy with no accidentals!

This means though that C major and A minor are actually the same set of notes, and only the ordering is different. Another phrasing would be that A is C’s relative minor. Conveniently, we can pair up all relative scales by rotating the above circle. G♯ sits to the left of A, and is therefore the relative minor of B major, which sits to the left of C. This can also be reversed for relative majors!

Alternatively, G major and G minor would be considered parallel, because they have the same base tone, called the tonic.

Writing the two scale progressions with whole (w) and half (h) tones, we say the major scale follows w,w,h,w,w,w,h and the natural minor scale is w,h,w,w,h,w,w. Note again a rotational relationship. Moving the first two letters of the natural minor scale to the end produces the major scale!

Other Minor Scales

There are also the harmonic and melodic minor scales which raise the seventh note or the sixth and seventh notes by a half step. So the harmonic A minor (hereafter Am) is A, B, C, D, E, F, G♯ and the melodic Am has an additional sharp with F♯ instead of F.

The Rest of the Modes

So where do the other five modes come in?

Consider again the C major scale. When it starts on C, it is the Ionian mode. Now shift the scale to D, but keep all the original tones: D, E, F, G, A, B, C. This is D Dorian. It is essentially the minor (Aeolian) scale but with a raised sixth note. Continue shifting to E. This is E Phrygian. Then F Lydian, G Mixolydian, A Aeolian, and finally B Locrian. Not so tricky after all. The order of these modes can be remembered by a mnemonic device such as I Don’t Particularly Like Modes A Lot.

The Circle of Fifths

So now we fast forward 155 years and the modern musician reads from left to right and sees two things: a clef and a series of accidentals. Odds are it’s a treble clef but the accidentals are relatively unpredictable. These describe the key of the piece, or movement, or section. Take for example a piece with no written accidentals following the clef:

Can you guess the name of this one? We know because of the key signature, which in this case is blank, that this song is either in the key of C major or A minor. To determine between the two, one would need to look at which tone is being positioned as the tonic through phrasing and other analysis. As a freebie, the excerpt of Ode to Joy above is in C.

Not all key signatures are so simple however. Generating a Dm scale gives the set of notes: D, E, F, G, A, B♭, C. The relative major, B, has five sharps!

In the first two bars of Scarborough Fair, the pianist never even gets to play a B♭, but the signature is more about the overall key and tonic center no matter how frequently the flat accidental is added.

The specifics of each key and how many flats are present is all too much for the average musician to remember right away. One of the most important tools of introductory music theory is called the Circle of Fifths. This tool has a myriad of interesting properties to be discussed over the next several lectures which relate keys and chords and progressions and so on. Leave space for each of the twelve semitones to fit inside and outside of a circle. Start with the key of C at the top. The next fifth is the fifth note in the current scale progression. So the fifth to the right of C is the fifth note in C major: G. Then, jumping by fifths in the same way as in Pythagorean tuning, we wind around the circle until we return to C. On the inside, we can plot the relative minors using the rotation trick again.

To remember the ordering of the fifths, students often implement other mnemonic devices. In this case, clockwise from the key of F might read: Father Charles Goes Down And Ends Battle. Likewise, counterclockwise from B♭ reads: Battle Ends And Down Goes Charles. This works for the minors also.

So how might this help remembering the accidentals in each key? Consider that C has neither, while G has... G, A, B, C, D, E, F♯... one sharp. From writing out Dm earlier, we know the relative major, F, contains one flat! Likewise, D has two sharps: F♯ and C♯ and B♭ has two flats: B♭ and E♭. Incredibly, the pattern continues all the way around the circle.

Can you guess how the War of the Romantics ended? If the fight between classical and more modern music is unsurprising to you, its because notions of the war are still present even today. Sooner than later, proponents of House and Alternative styles might be competing with a newer form of music not yet invented.

For Thought:

1) Glissando and portamento are two words which both essentially mean a slide between notes. Many individuals however would be quick to point out subtle differences. Try to construct as accurate of definitions for each as possible, or argue towards the idea that they are in fact the same.

2) Identify one system of music which does not use heptatonic scales and describe their alternative.

3) Construct any natural, melodic, and harmonic minor scales that have not yet been discussed and calculate the frequency differences between the sixth and seventh tone for each scale. Generate the D♭ Locrian mode from the appropriate major scale and indicate which major scale was used.

4) Determine the order that flats and sharps appear as one moves around the circle of fifths. Does this line up with the order they appear in key signatures? Do the overlapping areas at the bottom of the circle cause any issues?

5) Eduard Hanslick once said of Wagner’s Götterdämmerung, “One can listen to this incoherent ardour amidst the fluctuations of deafening and nerve-racking orchestral effects for only a short time without relaxation.” Which side of the war was Hanslick likely on? What specific type of music was he against?

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.

Without much forethought, we know that the recorder needs to be cylindrical, produce a tone, and have keyholes and a mouth piece. This can be represented best by a plastic pipe with one closed end as the mouthpiece is essentially a small opening on a solid base. This allows for sound waves to reflect off of its inner surface.

When the user blows into the mouth piece, a continuous wave of pressure moves from the mouthpiece and into the main chamber. Fast moving air particles push on their slower neighbors and propagate the wave front away from the source. If that source of pressure is actually a vibration, then the tube of fluid compresses and expands in horizontal fashion, creating a sound wave.

Created by wonderful Dan Russell from Penn State Acoustics:

The wave has a length, a frequency, and an amplitude as all other waves (sine, light, crowd) do. In musical instruments, this source of vibration is not simply the column of air being blown into a tube. For reed instruments, the flexible wood does the vibrating. Brass players buzz their lips. For our recorder, there is an additional opening in the mouth piece which is seen in the cross section below. The column of air switches between flowing above and below the (unfortunately named) labium and this provides sufficient oscillation for sound.

As the wave moves to the end of the pipe, the surroundings become more freeing and now the air can move in all directions. This is a change in impedance to the wave, and it causes some of the wave to reflect back into the instrument. The two waves will pass through one another and will either reinforce or cancel out the amplitude in various regions. Once the longitudinal sound waves stabilize, an unchanging result forms called a standing wave.

If these standing waves fit within the length of the pipe, then they sound with a certain frequency, called a tone. Moreover, the specifics of the length, shape, and material of the instrument will affect the final tones heard. For any given breath, multiple standing waves which satisfy the fitting condition with distinct wavelengths can exist. In this way, the final sound is a combination of many related frequencies. 

For now, we can take the word “fit” to mean: satisfies our physical intuition. Considering that the closed end of the recorder will experience no vibration while the open end will experience the largest vibration possible (a node and antinode, respectively), the largest wavelength standing wave which fits is one quarter of the complete wavelength. Note the figure below plots a standing displacement wave, whereas a pressure wave would appear flipped horizontally. This means air has greatest displacement at the open end and the closed end experiences none.

The longest wavelength has the lowest resulting frequency and is therefore called the fundamental frequency. Shorter wavelengths which have antinodes at the pipe opening must specifically be integer multiples of that fundamental frequency. This collection of frequencies (n*Freq for n an integer) forms the harmonic series.

The last piece of the puzzle is the placement of finger holes on the body, or rather, the fingering . When all of the holes are covered, the recorder acts as a simple pipe. When the holes become uncovered higher up on the body, shorter wavelengths will fit within the tube and higher notes (as a general trend) will be produced. A fairly complex fingering chart is shown below. The note names and their positioning lack much meaning right now. Just know that the lowest note (from the longest wavelength and smallest frequency) is called C and it sounds at 523.25 Hz. The highest playable note is called G at 3160 Hz.

Image provided by Barnaby Walters:

The positioning of the finger holes might feel very precise and in a way, that assumption is correct, but the recorder has also been around for almost three quarters of a millennium! Trial and error were essential for the modern recorder’s construction. Even today there is still discussion to be had about the merits of German versus Baroque fingering styles.

As no small feat, in this introductory lesson we’ve learned about the acoustics of simple wind instruments and given cursory details on how individuals go about creating the soprano recorder.

For Thought:

1) What might standing waves on a taut membrane (like a drum) look like? How do standing waves appear on a stringed instrument? Are the waves on these more tactile surfaces also longitudinal?

2) A simple formula in physics relates the speed of a wave to its frequency and wavelength. Specifically, speed is the product of the two. If the speed of sound is roughly 332 m/s, what are the wavelengths of the two extremes of the recorder’s range in meters? In feet?

3) Given the wavelength of the lowest possible note on a recorder, and considering the diagram above of one fourth of the wavelength fitting within the tube, what is the minimum length a recorder can be to play the low range? Compare this to the actual standard length of 32 cm and consider what that means about our approximations.

4) Brainstorm reasons for why there is also a higher end to the instrument’s range.

5) The harmonic series also has a mathematical analog as the sum of inverse integers: 1 + 2^-1 + 3^-1 + ... Why was this series named as such?

Topics to be Covered:

1) Scales, Chords, Modes

Also, the physics of waves and anatomy of the inner ear.

2) Counterpoint

Includes notions of musical set theory and Schenkerian analysis.

3) Form, Genre, Sound Coding

Discusses how modern musicians interface with computer software.

Recommended Prerequisites:

A love for music and science. Also... probably trigonometry? High school biology?

Those who have musical backgrounds will likely find initial topics largo and those with more mathematical leanings might find analysis of the wave equation strung along. Nevertheless, we might all finish this series with a deeper love for and understanding of music. 

Feedback is both welcomed and enjoyed!