Capstone 2: Design Your Own Scale — A Mathematical Composition Challenge

Overview

Western music uses twelve equal semitones per octave. But this is not a law of nature. It is one solution — historically contingent, mathematically elegant in some respects and imperfect in others — to the problem of dividing the octave into intervals useful for melody and harmony. Other cultures have solved this problem differently. The Arabic maqam system uses quarter-tones. The Indian classical tradition uses 22 shrutis. Indonesian gamelan tunings vary from island to island and ensemble to ensemble, deliberately rejecting standardization. And throughout Western history, before equal temperament, theorists proposed dozens of alternative tuning systems — meantone, Pythagorean, just intonation, and many more — each with distinct mathematical structures and distinct aesthetic characters.

This project asks you to do what those theorists did: design a complete musical scale from scratch.

Your scale must meet four requirements. First, it must be derivable from a physical or mathematical rationale — not arbitrary, but grounded in the acoustic or mathematical principles developed throughout this textbook. Second, it must have between 5 and 12 notes per octave — enough to make melody and at least rudimentary harmony possible, not so many as to be unperformable. Third, it must have at least one uniquely interesting interval property — something that distinguishes it from the standard Western scales in a musically meaningful way. Fourth, it must be accompanied by at least one original piece of music composed within it.

The point of this project is not to produce a "better" scale than the one Western music uses. It is to experience the mathematical structure of scales from the inside — to discover what it feels like to choose a tuning rationale and discover its consequences, to confront the comma problem firsthand, to realize that every scale is a trade-off, and to compose music that cannot exist in any other tuning system because it depends on intervals that no other system contains.

Learning Objectives

By the end of this project, students will be able to:

1. Derive a complete set of musical pitch ratios from a chosen physical or mathematical foundation, and express those ratios as both frequency ratios and cent values.
2. Explain the mathematical relationship between a tuning system's generator interval and the structure of its pitch space, including the conditions under which the system closes on itself (or fails to).
3. Calculate the consonance profile of a scale by identifying which intervals approximate small integer ratios and quantifying their deviation from just intonation using the cent as a unit.
4. Construct a consonance table for all intervals within a scale and use it to develop a systematic framework for harmonic tension and resolution within that scale's theory.
5. Compose original music in a non-standard scale, making documented compositional decisions about melody, voice-leading, and harmonic movement based on the scale's physical properties.
6. Explain what information is preserved and what is lost when comparing any non-standard scale to both 12-TET and just intonation.
7. Identify which known world music traditions most closely approximate the interval structure of the designed scale, and explain why that resemblance arises.
8. Articulate the first principles of a music theory for the designed scale, including rules governing consonance, the analog of the "dominant" function, and the possibilities for transposition.

Background Reading

Review the following chapters before beginning:

• Chapter 6 (The Harmonic Series): The physical basis of consonance — why integer frequency ratios produce smooth, beating-free sound, and how the harmonic series generates the just intervals.
• Chapter 11 (Scales and Tuning Systems): A survey of historical tuning systems, from Pythagorean to just intonation to meantone to 12-TET, with the mathematical rationale for each.
• Chapter 12 (The Geometry of Intervals): The cent as a unit of interval measurement; interval arithmetic; the circle of fifths as a geometric structure.
• Chapter 13 (Equal Temperament and Its Discontents): The specific trade-offs of 12-TET, what it gains (transposability) and what it sacrifices (pure fifths and thirds).
• Chapter 14 (Microtonality and Alternative Tuning Systems): Survey of non-standard Western tuning experiments and non-Western tuning systems.
• Chapter 16 (Consonance and Dissonance): The physics of consonance — roughness, beating, and the harmonic overlap model.
• Chapter 20 (Symmetry in Music and Physics): Musical symmetry groups and why transposability is a symmetry property.

Phase 1: Define Your Physical/Mathematical Foundation

Estimated time: 2–4 hours

Your scale must be derivable from a principle, not assembled arbitrarily. Choose one of the four derivation strategies below. Each strategy produces a different class of scale and a different set of analytical problems.

Strategy A: Harmonic Series Derivation

Choose a set of partials from the harmonic series and use their frequency ratios (relative to the fundamental) as your scale steps. For example, the first eight partials of the harmonic series have ratios 1:1, 2:1, 3:2, 4:3, 5:4, 6:5, 7:6, 8:7 (normalizing each to lie within a single octave). The septimal scale uses partials up to the 7th harmonic; the 11-limit scale includes the 11th partial (with ratio 11:8, an interval that sounds startlingly neither major nor minor to Western ears).

What to document: List exactly which partials you chose and why. Calculate the frequency ratio and cent value of each scale step. Explain any choices you made about octave reduction (bringing ratios > 2:1 back into a single octave range by halving).

Strategy B: Generator Interval Derivation

Choose a single generating interval (not the octave) and stack it repeatedly, reducing each result to the range [1:1, 2:1] as needed. This is how the Pythagorean scale is derived (generator = 3:2, the perfect fifth) and how many historical and non-Western scales are structured.

Interesting generator intervals to consider: - 7:6 (septimal minor third) — generates a scale with very different structure from Pythagorean - 5:4 (major third) — generates a scale with characteristic tritone-like saturations - 11:8 (undecimal augmented fourth) — generates a scale with no interval resembling the perfect fifth - Any equal-tempered interval of the form 2^(k/n) for chosen integers k and n

What to document: State your generator interval and the number of stacking steps (this determines the number of notes). Show all stacking operations. Calculate the "comma" — the small interval by which N stacked generator intervals fail to return exactly to the octave.

Strategy C: Mathematical Sequence Derivation

Use a mathematical sequence to determine your frequency ratios. This produces scales with unusual mathematical properties that may or may not correspond to acoustic simplicity.

Possibilities: - Fibonacci ratios: The Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, 21...) has the property that consecutive ratios converge to φ (the golden ratio ≈ 1.618). Construct a scale using ratios of consecutive Fibonacci numbers as intervals. - Prime number scale: Use the first N prime numbers as frequency ratios, reduced to a single octave. - Equal-step n-TET for non-standard n: Divide the octave into 7, 17, 19, 31, or 53 equal steps. Note that 19-TET and 31-TET are historically documented as providing better approximations to just intonation than 12-TET.

What to document: State the mathematical sequence and the derivation rule. Calculate the ratios. Identify any coincidental approximations to familiar intervals.

Strategy D: Consonance/Dissonance Design

Work backward from a desired consonance structure. Decide in advance which intervals in your scale should be "most consonant" and which "most dissonant," then find a set of frequency ratios that achieves that structure.

For example: design a scale in which the most consonant interval is not the perfect fifth but a different ratio (say, 7:5, the septimal tritone), and all other intervals are arranged around it.

What to document: State the consonance structure you were aiming for. Show how you chose the frequency ratios to achieve it. Calculate the resulting intervals and assess how well they meet your design goals.

Phase 1 Required Calculations

Regardless of strategy, produce the following for your completed scale:

Conversion formula: Cents = 1200 × log₂(ratio)

Phase 1 Exercise

Calculate and compare your scale's intervals to both 12-TET and just intonation: - Which of your scale's intervals most closely approximate just intervals (small integer ratios)? - Which of your scale's intervals are furthest from any just interval? - Which 12-TET semitone is each of your scale's steps nearest to?

Write a paragraph explaining how your scale's interval structure relates to the tuning systems covered in Chapters 11–14. Does your scale resemble any known historical or non-Western system?

Phase 2: Analyze the Physical Properties

Estimated time: 2–3 hours

Consonance Analysis

For every interval in your scale (every pair of notes), calculate a consonance rating using the harmonic overlap model from Chapter 16. The simplest version: find the simplest integer ratio approximation to each interval, and rate consonance inversely to the product of the numerator and denominator.

Symmetry Analysis

A scale is transposable to N levels if you can start on N different notes within the scale and produce a scale with the same interval pattern. 12-TET is transposable to all 12 levels; just intonation is transposable to only 1 (the key it was tuned for). Where does your scale fall?

To check: take your interval sequence (e.g., [204¢, 182¢, 204¢, 204¢, 182¢, 204¢, 204¢] for a Pythagorean major scale) and test whether it is cyclically symmetric. If the same pattern appears starting on a different step, that step is a transposable level.

The Comma Problem

Stack your generator interval (or the smallest step of your scale) repeatedly and track how far you deviate from a whole number of octaves:

After N steps: deviation = N × generator - M × 1200¢ (where M is the nearest integer number of octaves)

Make a table showing this deviation for N = 1, 2, 3, ..., up to the point where either (a) the deviation returns near zero (your scale "closes") or (b) you have gone far enough to conclude it never closes within a manageable number of notes.

If your scale closes, it has a finite, well-defined structure. If it does not close, it is an "open" scale that can in principle generate infinitely many distinct pitches — like Pythagorean tuning.

Phase 2 Exercise

Construct a "consonance table" — a grid showing every interval between every pair of scale degrees, with your consonance rating. Identify: - The three most consonant non-unison intervals in your scale - The three most dissonant intervals - Any interval that is surprisingly consonant or dissonant relative to your intuitions

Write 3–4 sentences interpreting this table for a composer: what are the "stable" harmonic destinations in this scale? What interval would serve the role that the perfect fifth serves in Western harmony?

Phase 3: Design a Consonance/Dissonance Framework

Estimated time: 1–2 hours

This phase turns your physical analysis into a usable compositional theory.

Building Your Harmonic Grammar

Using your consonance table from Phase 2, establish the following:

Stable tones: Which notes in your scale, when held, create a sense of rest or arrival? (These will typically be notes connected to the root by your most consonant intervals.)

Active tones: Which notes in your scale create a sense of motion, incompletion, or tension? (These will typically be notes connected by your most dissonant intervals.)

Resolution rules: Define at least three resolution rules — statements of the form "interval X tends to resolve to interval Y" — based on your consonance analysis. In Western harmony, the classic rule is that the tritone (highly dissonant) resolves by contrary motion to a major third or minor sixth (highly consonant). What is the analogous rule in your scale?

The "dominant" analog: In Western functional harmony, the dominant chord (built on the fifth scale degree) has a strong tendency to resolve to the tonic. Does your scale have an equivalent? Find the scale degree whose most dissonant interval against the root can resolve most smoothly to consonance.

Documenting Your Framework

Write a one-page "Theory Sheet" for your scale that a composer could use: - The name of your scale (invent one) - Its interval structure (list of intervals in cents, ascending) - The three most consonant intervals and their notation - The three most dissonant intervals and their notation - Three resolution rules - The "dominant analog" and how it resolves

Phase 4: Compose a Short Piece

Estimated time: 3–5 hours

Requirements

Your composition must be 2–4 minutes in duration and must contain all of the following:

A melody: A single-voice melodic line that uses the characteristic intervals of your scale. The melody should exploit at least one interval that does not exist in 12-TET — this is what makes composition in your scale different from ordinary composition with a slightly odd key signature.

Harmonic movement: At least two voices (or one voice with accompaniment) must sound simultaneously for some portion of the piece. The harmonic movement must use your consonance/dissonance framework: there must be moments of relative stability (consonant intervals) and moments of tension (dissonant intervals).

A defined beginning and ending: The piece must start from and return to a clear tonal center (the root of your scale, or whatever pitch functions as "home"). The ending must be marked by a cadence — a harmonic gesture that signals conclusion, using your resolution rules.

Tension and resolution: There must be at least one clear moment of tension (use of your scale's most dissonant interval, or a departure from the tonal center) followed by resolution (return to consonance and the tonal center). This is the equivalent of a dominant-tonic cadence in your scale's grammar.

Technical Approaches

You may use any of the following approaches to realize your composition:

By ear / by hand: Compute the frequencies of your scale degrees (f_n = f_0 × ratio_n) and use software that allows arbitrary frequency input (Max/MSP, SuperCollider, Pure Data, Csound, or a MIDI instrument with microtonal pitch-bend). Notate your composition in any way that allows it to be reproduced — staff notation with annotations, tablature, a frequency-time graph, or written instructions.

Software synthesis: Many digital audio workstations support microtonal tuning via Scala (.scl) format. Create a Scala file for your scale and use it with a software synthesizer. Scala format is simply a text file listing cent values; see Chapter 14's further reading for resources.

Python synthesis: Generate your composition as a WAV file by computing sine waves (or simple additive synthesis) at your scale's frequencies and concatenating them. This is the most transparent approach — you will directly hear the consequences of every number you computed.

Documenting Compositional Choices

Keep a compositional journal during Phase 4. For each structural decision, write one sentence explaining it: - Why did you choose this opening melodic gesture? - Which interval forms the backbone of your melody, and what is its physical character? - Where did you place the tension moment, and which dissonant interval creates it? - How did you resolve the tension, and which consonance provides the arrival? - Was there any moment where the scale's unusual intervals did something unexpected — created a surprise, a color, or an effect you did not anticipate?

The journal will become part of your Reflection (Phase 5) and should be at least 200 words.

Phase 5: Reflection and Analysis

Estimated time: 1–2 hours

Comparing Your Scale to the World

To Western major/minor: What do you gain by using your scale instead of the Western major or natural minor scale? What do you lose? Be specific: which intervals in your scale are more consonant than their nearest Western equivalents? Which are less consonant? What kinds of melodies are easier to write in your scale? Which are harder?

To world music: Search the ethnomusicological literature (Chapter 30's further reading is a starting point) for tuning systems that resemble your scale. The Scala archive (a database of hundreds of historical and non-Western tuning systems) is freely available online. Which known tuning system most closely approximates your scale, and why might that resemblance exist? Is it mathematical coincidence, or did you and that tradition converge on similar interval relationships for similar acoustic reasons?

Developing a full music theory: If you were going to write a comprehensive music theory textbook for your scale — equivalent to a Western counterpoint manual — what would the first three rules be? Not preferences, but rules: things that must be done and things that must be avoided, grounded in the physical analysis you have done. Write these as formal rules, with brief physical justifications.

Deliverables and Grading Rubric

1. Scale Documentation with Frequency Ratios (25 points) A complete table of your scale's frequency ratios and cent values, the Phase 1 comparison table, and the Phase 1 written paragraph. Must be precise (ratios as fractions or decimals to at least 4 decimal places; cent values to the nearest cent).

2. Physical and Mathematical Analysis (25 points) The consonance table, symmetry analysis, comma analysis, and Phase 2 written interpretation.

3. Original Composition (30 points) The composition itself (audio file, score, or both), the Theory Sheet, and the compositional journal.

4. Reflection Essay (20 points) A minimum 400-word essay responding to all three reflection questions in Phase 5.

Extension Challenges (Optional)

Develop a Scale Pair Many musical traditions use not one but a family of related scales — major and minor, Dorian and Phrygian, the raga family. Develop a "dark" variant of your scale that uses the same generator or rationale but shifts the balance of intervals toward more dissonant combinations, and compose a second short piece in it. What is the affective relationship between your two scales?

Build an Instrument Using physical materials (a stringed instrument, a set of resonant tubes, a percussion instrument), construct an instrument tuned to your scale. The physical construction forces you to confront the scale's ratios as physical measurements — string lengths, tube lengths, or mass ratios. Document the construction and the tuning process.

Teach Your Scale Present your scale to a classmate or group who has not seen your work. Without playing them the composition you wrote, ask them to improvise a short melody in your scale. Then play your composition. What did they discover about the scale that you had missed? What did you discover that they missed? Write a reflection on what this exercise reveals about the relationship between a scale's mathematical structure and a listener's intuitive response to it.