Tonality diamond

Key takeaways

Tonality diamond belongs to science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Tonality diamond to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Tonality diamond from memory before moving on to harder problems.

Reference excerpt

In music theory and tuning, a tonality diamond is a diamond-shaped diagram of ratios comprising overlapping tonalities in which the major tonalities orient along one diagonal and the minor tonalities along the other. An n-limit tonality diamond ("limit" here is in the sense of odd limit, not prime limit) is an arrangement in diamond-shape of the set of rational numbers r, 1 ≤ r < 2 {\displaystyle 1\leq r<2} , such that the odd part of both the numerator and the denominator of r, when reduced to lowest terms, is less than or equal to the fixed odd number n. Equivalently, the diamond may be considered as a set of pitch classes, where a pitch class is an equivalence class of pitches under octave equivalence. The tonality diamond is often regarded as comprising the set of consonances of the n-limit. Although originally invented by Max Friedrich Meyer, the tonality diamond is now most associated with Harry Partch ("Many theorists of just intonation consider the tonality diamond Partch's greatest contribution to microtonal theory.").

The diamond arrangement Partch arranged the elements of the tonality diamond in the shape of a rhombus subdivided into (n+1)2/4 smaller rhombuses, where n is the odd limit (the highest odd number used in the ratios defining the pitch classes of the tonality diamond). In the rhombuses along the upper left edge of the diamond are placed ratios whose overnumbers (numerators) are the odd numbers from 1 to n. The undernumber (denominator) of each is the minimum integer power of 2 such that 1 ≤ r < 2 {\displaystyle 1\leq r<2} , where r {\displaystyle r} is the value or quotient of the ratio. For example, 1/1 3/2 5/4. The ratios are then sorted in ascending order (1/1 5/4 3/2) within those rhombuses. In the rhombuses along the lower left edge are placed the corresponding reciprocal ratios with the undernumbers containing the odd numbers 1 to n and the overnumbers taking the powers of 2 such that 1 ≤ r < 2 {\displaystyle 1\leq r<2} . For example, 1/1 8/5 4/3. In all other rhombuses are placed the products of the diagonally upper-left and lower-left ratios also satisfying 1 ≤ r < 2 {\displaystyle 1\leq r<2} . That defines all the elements of the tonality diamond, with some repetition. Diagonals sloping in one direction form Otonalities and diagonals in the other direction form Utonalities. One of Partch's instruments, the diamond marimba, is arranged according to the tonality diamond.

Numerary nexus A numerary nexus is an identity shared by two or more interval ratios in their numerator or denominator, with different identities in the other. For example, in the following Otonality, the denominator is always 1, thus 1 is the numerary nexus:

1 1 2 1 3 1 4 1 5 1 e t c . ( 3 2 ) ( 5 4 ) {\displaystyle {\begin{array}{cccccc}{\frac {1}{1}}&{\frac {2}{1}}&{\frac {3}{1}}&{\frac {4}{1}}&{\frac {5}{1}}&\mathrm {etc.} \\&&({\frac {3}{2}})&&({\frac {5}{4}})\end{array}}}

In the following Utonality, the numerator is always 1 and the numerary nexus is thus also 1:

1 1 1 2 1 3 1 4 1 5 e t c . ( 4 3 ) ( 8 5 ) {\displaystyle {\begin{array}{cccccc}{\frac {1}{1}}&{\frac {1}{2}}&{\frac {1}{3}}&{\frac {1}{4}}&{\frac {1}{5}}&\mathrm {etc.} \\&&({\frac {4}{3}})&&({\frac {8}{5}})\end{array}}}

… excerpt ends here. Continue reading the full article.

Illustrations

Worked examples

Example 1 — a first encounter with Tonality diamond

Start with the simplest possible case. Write down what Tonality diamond claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Tonality diamond before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Tonality diamond ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Tonality diamond

In research: Tonality diamond appears in science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Tonality diamond in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Tonality diamond is common in secondary-school and first-year university syllabi. It links to neighbouring topics Harry Partch, Pitch space, so understanding it makes those chapters shorter.
In everyday life: Look for Tonality diamond outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Tonality diamond in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Tonality diamond means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Tonality diamond out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Tonality diamond in simple terms?

In music theory and tuning, a tonality diamond is a diamond-shaped diagram of ratios comprising overlapping tonalities in which the major tonalities orient along one diagonal and the minor tonalities along the other. An n-limit tonality diamond ("limit" here is in the sense of odd limit, not prime…

Why does Tonality diamond matter?

Because it connects several science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Tonality diamond?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Tonality diamond.