Overview — what we will learn
You will learn three things step by step:
- How a monochord (or octochord) shows harmonic intervals by changing string length.
- How to make a 12‑note Pythagorean scale by a folding/marking exercise (the idea behind McClain’s paper technique) and by stacking perfect fifths on a monochord.
- What goes wrong with pure (Pythagorean) tuning and why musicians invented temperaments such as equal temperament.
1) What a monochord/octochord is and how harmonics give intervals
A monochord is a single string stretched over a board with a movable bridge so you can change string length. A vibrating string’s pitch (frequency) is inversely proportional to its length: if you make the vibrating part half as long, the pitch goes up one octave. If you make the vibrating part two thirds as long, the sound is a perfect fifth above the original.
Important simple ratios (lengths give lower pitch when longer):
- Octave: length ratio 1 : 1/2 (or frequency 1 : 2)
- Perfect fifth: length ratio 1 : 2/3 (frequency 1 : 3/2)
- Perfect fourth: length ratio 1 : 3/4 (frequency 1 : 4/3)
- Major third (Pythagorean type): 1 : 4/5 is not Pythagorean; Pythagorean major third is produced from stacking fifths and equals ratio 81:64 (we'll explain below).
Try this on a monochord
Materials: a stringed monochord (or a long elastic string across a board), a movable bridge, ruler.
- Tune the string to a note with full length L (call this note C for example).
- Move the bridge to the midpoint (L/2) and pluck: that is one octave above the original.
- Move the bridge so the vibrating length is two-thirds of L (L × 2/3) and pluck: that is a perfect fifth above the original.
Hearing the octave and fifth directly helps you understand how simple number ratios make pleasant intervals.
2) How to generate a 12‑note Pythagorean scale (two ways)
A. Stacking perfect fifths on the monochord (practical numbers)
Start with a base string length L for a note (C). Each perfect fifth above corresponds to a new vibrating length multiplied by 2/3. To keep the new note inside the same octave (so lengths stay between L and L/2), multiply or divide by 2 as needed.
Example using L = 60 cm as the base (C) and producing 12 notes by successive fifths:
- C: 60.000 cm
- G (one 5th up): 60 × 2/3 = 40.000 cm
- D (two 5ths up, adjusted into the octave): 40 × 2/3 = 26.667 → ×2 = 53.333 cm
- A: 53.333 × 2/3 = 35.556 cm
- E: 35.556 × 2/3 = 23.704 → ×2 = 47.407 cm
- B: 47.407 × 2/3 = 31.605 cm
- F#: 31.605 × 2/3 = 21.070 → ×2 = 42.140 cm
- C#: 42.140 × 2/3 = 28.093 → ×2 = 56.186 cm
- G#: 56.186 × 2/3 = 37.458 cm
- D#: 37.458 × 2/3 = 24.972 → ×2 = 49.944 cm
- A#: 49.944 × 2/3 = 33.296 cm
- F: 33.296 × 2/3 = 22.197 → ×2 = 44.394 cm
If you tune your monochord bridges to these 12 lengths, you get the Pythagorean 12‑note set (each step derived by pure 3:2 fifths, adjusted by octaves). Listen: some intervals (especially the thirds) will sound different from what you're used to on a piano.
B. Paper folding idea (McClain’s Pythagorean paper folding — a simple classroom version)
The idea of McClain’s paper method is to use repeated, consistent folding and marking to create positions that correspond to the ratios you get by stacking fifths and octaves. Here is a hands‑on classroom version you can try that captures the idea:
Materials
- A long paper strip (about 1 m long, 3–4 cm wide).
- Ruler and pencil.
- Scissors (optional).
Steps
- Mark one end as 0 and the other end as 1 (this whole strip represents one octave — the low C at 1 and the high C at 1/2 in length, but you can just think of it as the base length).
- Fold the strip in half and crease it. Unfold and mark the half point — this shows the octave midpoint (1/2).
- To get points corresponding to a perfect fifth (which is 2/3 of the string length), fold so that one end reaches two thirds along the strip: a simple way is to fold the strip back on itself and then slide the layers until two-thirds lines up — mark that crease. (If exact folding is hard, measure 2/3 with a ruler and mark it.)
- Now, starting from your base point (0 or 1), repeatedly mark new points by moving 2/3 along from the last marked point and whenever a mark falls outside the main interval (past the ends), fold or halve it back into the main length by marking the equivalent position an octave up or down (multiply or divide by 2) so all marks remain between 0 and 1.
- After 12 such steps (or sooner) you will have about 12 distinct marks — these are the positions of the 12‑note Pythagorean set. The marks will not be evenly spaced; they are governed by powers of 3 and 2.
Note: the exact folding trick McClain describes can involve several neat folding shortcuts to generate the needed powers of 2 and 3, but the simple ruler/fold method above reproduces the main idea: repeated 3:2 steps moved into the same octave give the 12 notes.
3) Why tuning becomes a problem: the Pythagorean comma and the wolf
When you use pure perfect fifths (ratio 3/2) to build a 12‑note scale, you might expect that after stacking twelve fifths you would get exactly 7 octaves higher. But the math shows they do not match exactly.
Compute the difference:
- Twelve pure fifths multiply frequencies by (3/2)^12.
- Seven octaves multiply frequencies by 2^7.
- The ratio (3/2)^12 ÷ 2^7 = 531441 ÷ 524288 ≈ 1.013643. That’s about 1.36% higher in frequency.
- In musical terms that tiny difference equals about 23.46 cents (a cent is 1/100 of a semitone). This small mismatch is called the Pythagorean comma.
Because the twelve pure fifths don’t close to exactly seven octaves, you cannot make all twelve fifths perfectly pure and also have everything line up. If you try to tune twelve pure fifths around the circle of fifths, one of the fifths will end up very bad — that is called the wolf interval (a very dissonant interval). To avoid that, tuners spread the small mismatch across many intervals — that spreading is called temperament.
Equal temperament — the modern solution
Equal temperament divides the octave into 12 equal semitones. Each fifth in equal temperament is slightly narrower than a pure 3:2 fifth:
- Pure fifth (3:2) is ~701.955 cents.
- Equal‑tempered fifth is exactly 700 cents.
So each equal‑tempered fifth is about 1.955 cents flat compared with a pure fifth. Spread over 12 fifths, the small differences add up to the Pythagorean comma and the circle closes neatly — no wolf interval, and every key is usable though slightly impure compared with pure intervals.
Simple experiment to hear the differences (classroom activity)
- Make two 12‑note sets: one using the stacked‑fifths lengths you calculated on the monochord (Pythagorean set), and one where you equally divide the octave into 12 equal steps (you can use a tuner app set to equal temperament to place the bridge positions).
- Play a few common chords (C major, G major, D major, etc.) using each tuning. Notice how the thirds and some chords sound more or less sweet in the Pythagorean vs equal tempered set.
- Find and compare one of the fifths that sounds noticeably worse in Pythagorean tuning (this might be the wolf if you’ve tuned a chain of fifths). Notice how equal temperament avoids that extreme bad fifth but changes every fifth slightly.
Quick summary (what to remember)
- Monochords show that simple number ratios (2:1, 3:2, 4:3, ...) give pleasant musical intervals.
- Pythagorean scales are built by stacking pure 3:2 fifths and moving notes by octaves; paper folding can be used to mark these positions physically (McClain’s idea) and you can also compute the lengths to tune a monochord.
- Pure fifths don’t close neatly after 12 steps (Pythagorean comma ~23.46 cents), so tuners must choose a temperament. Equal temperament spreads the small error across all intervals so each key is usable.
If you want, I can give you a printable worksheet with the exact fold sequence, a drawing of how to mark the paper strip step‑by‑step, or a numbered monochord tuning chart you can use in class. Which would you like?