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. These are 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, 8286, 8951, 9616, 10281, 10946, 11611, 12276, 12941, 13606, 14271, 14936, 15601, 31867, … (OEIS: A060528)

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