3-limit

Prime limit

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Pythagorean tuning

The 3-limit consists of all just intonation intervals whose numerators and denominators are both products of the primes 2 and 3. Some examples of 3-limit intervals are 3/2, 4/3, 9/8. All 3-limit intervals can be written as [math]\displaystyle{ 2^a \cdot 3^b }[/math], where a and b can be any (positive, negative or zero) integer. When octave-reduced, if b is non-zero, a and b are opposite signs. In other words, one number in the ratio is a power of 2 and the other number is a power of 3. Confining intervals to the 3-limit is known as Pythagorean tuning, and the Pythagorean tuning used in Europe during the Middle Ages is the seed out of which grew the common-practice tradition of Western music, as well as genres derived from it. The 3-limit can be considered a rank-2 temperament which tempers out no commas.

Terminology

A 3-limit interval is also known as a Pythagorean interval. Recently, composers Catherine Lamb and Marc Sabat have adopted tertial for intervals of HC3^{[citation needed]}, not to be confused with tertian which is the adjective associated with the third diatonic degree.

Edo approximation

Edos which do relatively well at approximating 3-limit intervals can be found as the denominators of the convergents and semiconvergents of the continued fraction for the logarithm base 2 of 3. The sequence is equivalent to the numbers of notes per octave in the mos scales generated by pure 3/2: 1, 2, 3, 5, 7, 12, 17, 29, 41, 53, 94, 147, 200, 253, 306, 359, 665, … (OEIS: A206788).

Another approach is to find edos which have more accurate approximation to 3 than all smaller edos. This results in 1, 2, 3, 5, 7, 12, 29, 41, 53, 200, 253, 306, 359, 665, … (OEIS: A060528). Bold ones are records of relative error. For a more comprehensive list, see Sequence of equal temperaments by error.

A stricter approach is to find edos with an increasingly stronger consistent circle of 3/2. These are 1, 12, 53, 665, 190537, … (with strengths 1, 2, 3, 11, 28, … respectively)

Table of intervals

3-limit intervals up to odd-limit 19683:

Ratio	Monzo	Size (¢)	Color Name		Diatonic Category
1/1	[0⟩	0.000	w1	wa unison	P1	C
2187/2048	[-11 7⟩	113.685	Lw1	lawa 1sn	A1	C#
256/243	[8 -5⟩	90.225	sw2	sawa 2nd	m2	Db
9/8	[-3 2⟩	203.910	w2	wa 2nd	M2	D
19683/16384	[-14 9⟩	317.595	Lw2	lawa 2nd	A2	D#
32/27	[5 -3⟩	294.135	w3	wa 3rd	m3	Eb
81/64	[-6 4⟩	407.820	Lw3	lawa 3rd	M3	E
8192/6561	[13 -8⟩	384.360	sw4	sawa 4th	d4	Fb
4/3	[2 -1⟩	498.045	w4	wa 4th	P4	F
729/512	[-9 6⟩	611.730	Lw4	lawa 4th	A4	F#
1024/729	[10 -6⟩	588.270	sw5	sawa 5th	d5	Gb
3/2	[-1 1⟩	701.955	w5	wa 5th	P5	G
6561/4096	[-12 8⟩	815.640	Lw5	lawa 5th	A5	G#
128/81	[7 -4⟩	792.180	sw6	sawa 6th	m6	Ab
27/16	[-4 3⟩	905.865	w6	wa 6th	M6	A
32768/19683	[15 -9⟩	882.405	sw7	sawa 7th	d7	Bbb
16/9	[4 -2⟩	996.090	w7	wa 7th	m7	Bb
243/128	[-7 5⟩	1109.775	Lw7	lawa 7th	M7	B
4096/2187	[12 -7⟩	1086.315	sw8	sawa 8ve	d8	Cb
2/1	[1⟩	1200.000	w8	wa 8ve	P8	C

Music

John Doe

• Building (A New Sun) (2017)

E8 Heterotic

• "Metal" from Elements (2019–2020)

ELSM123456

• Huan Xiang Wei Gong (2026)
• Huan Xiang Wei Gong (Traditional Chinese) (2026)

Zhea Erose

• Circles of Indigo - Dreamsura (2023)

Francium

• Pythagorean Metal (2023)

Charles Ives

• Johnny Reinhard's 2023 book, The Transcendental Tuning of Charles Ives, lays the foundation for AFMM's realizations of some of Ives' works, employing chains of up to 29 perfect fifths.
• String Quartet #2, The Unanswered Question, Three-Page Sonata, Universe Symphony
• Concord Sonata
• The Unanswered Question
• Universe Symphony

Peter 'Rush' Kosmorsky

• String Trio no. 2 (2013) – SoundCloud | play – in Pythagorean[17]

Bill Wesley

• Familly Across the Border (2023, performed on mbira tuned to 30-note extended Pythagorean)

See also

• Pythagorean tuning
• Harmonic limit
• 3-odd-limit
• Gallery of just intervals