Pythagorean interval

File:Perfect fifth on C.png

Audio file "Just perfect fifth on C.mid" not found}}Template:Category handler: C-G (3/2 ÷ 1/1 = 3/2).

In musical tuning theory, a Pythagorean interval is a musical interval with a frequency ratio equal to a power of two divided by a power of three, or vice versa. For instance, the perfect fifth with ratio 3/2 (equivalent to 3¹/ 2¹) and the perfect fourth with ratio 4/3 (equivalent to 2²/ 3¹) are Pythagorean intervals.

All the intervals between the notes of a scale are Pythagorean if they are tuned using the Pythagorean tuning system. However, some Pythagorean intervals are also used in other tuning systems. For instance, the above-mentioned Pythagorean perfect fifth and fourth are also used in just intonation.

Interval table

Name	Short	Other name(s)	Ratio	Factors	Derivation	Cents	ET Cents	MIDI file	Fifths
diminished second	d2		524288/531441	219/312		−23.460	0	{{errorTemplate:Main other|Audio file "Pythagorean comma on C.mid" not found}}Template:Category handler	−12
(perfect) unison	P1		1/1	30/20	1/1	0.000	0	{{errorTemplate:Main other|Audio file "Unison on C.mid" not found}}Template:Category handler	0
Pythagorean comma			531441/524288	312/219		23.460	0	{{errorTemplate:Main other|Audio file "Pythagorean comma on C.mid" not found}}Template:Category handler	12
minor second	m2	limma, diatonic semitone, minor semitone	256/243	28/35		90.225	100	{{errorTemplate:Main other|Audio file "Pythagorean minor semitone on C.mid" not found}}Template:Category handler	−5
augmented unison	A1	apotome, chromatic semitone, major semitone	2187/2048	37/211		113.685	100	{{errorTemplate:Main other|Audio file "Pythagorean apotome on C.mid" not found}}Template:Category handler	7
diminished third	d3	tone, whole tone, whole step	65536/59049	216/310		180.450	200	{{errorTemplate:Main other|Audio file "Minor tone on C.mid" not found}}Template:Category handler	−10
major second	M2		9/8	32/23	3·3/2·2	203.910	200	{{errorTemplate:Main other|Audio file "Major tone on C.mid" not found}}Template:Category handler	2
semiditone	m3	(Pythagorean minor third)	32/27	25/33		294.135	300	{{errorTemplate:Main other|Audio file "Pythagorean minor third on C.mid" not found}}Template:Category handler	−3
augmented second	A2		19683/16384	39/214		317.595	300	{{errorTemplate:Main other|Audio file "Pythagorean augmented second on C.mid" not found}}Template:Category handler	9
diminished fourth	d4		8192/6561	213/38		384.360	400	{{errorTemplate:Main other|Audio file "Pythagorean diminished fourth on C.mid" not found}}Template:Category handler	−8
ditone	M3	(Pythagorean major third)	81/64	34/26	27·3/32·2	407.820	400	{{errorTemplate:Main other|Audio file "Pythagorean major third on C.mid" not found}}Template:Category handler	4
perfect fourth	P4	diatessaron, sesquitertium	4/3	22/3	2·2/3	498.045	500	{{errorTemplate:Main other|Audio file "Just perfect fourth on C.mid" not found}}Template:Category handler	−1
augmented third	A3		177147/131072	311/217		521.505	500	{{errorTemplate:Main other|Audio file "Pythagorean augmented third on C.mid" not found}}Template:Category handler	11
diminished fifth	d5	tritone	1024/729	210/36		588.270	600	{{errorTemplate:Main other|Audio file "Diminished fifth tritone on C.mid" not found}}Template:Category handler	−6
augmented fourth	A4		729/512	36/29		611.730	600	{{errorTemplate:Main other|Audio file "Pythagorean augmented fourth on C.mid" not found}}Template:Category handler	6
diminished sixth	d6		262144/177147	218/311		678.495	700	{{errorTemplate:Main other|Audio file "Pythagorean diminished sixth on C.mid" not found}}Template:Category handler	−11
perfect fifth	P5	diapente, sesquialterum	3/2	31/21	3/2	701.955	700	{{errorTemplate:Main other|Audio file "Just perfect fifth on C.mid" not found}}Template:Category handler	1
minor sixth	m6		128/81	27/34		792.180	800	{{errorTemplate:Main other|Audio file "Pythagorean minor sixth on C.mid" not found}}Template:Category handler	−4
augmented fifth	A5		6561/4096	38/212		815.640	800	{{errorTemplate:Main other|Audio file "Pythagorean augmented fifth on C.mid" not found}}Template:Category handler	8
diminished seventh	d7		32768/19683	215/39		882.405	900	{{errorTemplate:Main other|Audio file "Pythagorean diminished seventh on C.mid" not found}}Template:Category handler	−9
major sixth	M6		27/16	33/24	9·3/8·2	905.865	900	{{errorTemplate:Main other|Audio file "Pythagorean major sixth on C.mid" not found}}Template:Category handler	3
minor seventh	m7		16/9	24/32		996.090	1000	{{errorTemplate:Main other|Audio file "Lesser just minor seventh on C.mid" not found}}Template:Category handler	−2
augmented sixth	A6		59049/32768	310/215		1019.550	1000	{{errorTemplate:Main other|Audio file "Pythagorean augmented sixth on C.mid" not found}}Template:Category handler	10
diminished octave	d8		4096/2187	212/37		1086.315	1100	{{errorTemplate:Main other|Audio file "Pythagorean diminished octave on C.mid" not found}}Template:Category handler	−7
major seventh	M7		243/128	35/27	81·3/64·2	1109.775	1100	{{errorTemplate:Main other|Audio file "Pythagorean major seventh on C.mid" not found}}Template:Category handler	5
diminished ninth	d9	(octave − comma)	1048576/531441	220/312		1176.540	1200	{{errorTemplate:Main other|Audio file "Unison on C.mid" not found}}Template:Category handler	−12
(perfect) octave	P8	diapason	2/1		2/1	1200.000	1200	{{errorTemplate:Main other|Audio file "Perfect octave on C.mid" not found}}Template:Category handler	0
augmented seventh	A7	(octave + comma)	531441/262144	312/218		1223.460	1200	{{errorTemplate:Main other|Audio file "Pythagorean comma on C.mid" not found}}Template:Category handler	12

Notice that the terms ditone and semiditone are specific for Pythagorean tuning, while tone and tritone are used generically for all tuning systems. Despite its name, a semiditone (3 semitones, or about 300 cents) can hardly be viewed as half of a ditone (4 semitones, or about 400 cents).

File:Interval ratios in D-based symmetric Pythagorean tuning (powers for large numbers).PNG

Frequency ratio of the 144 intervals in D-based Pythagorean tuning. Interval names are given in their shortened form. Pure intervals are shown in bold font. Wolf intervals are highlighted in red. Numbers larger than 999 are shown as powers of 2 or 3. Other versions of this table are provided here and here.

12-tone Pythagorean scale

The table shows from which notes some of the above listed intervals can be played on an instrument using a repeated-octave 12-tone scale (such as a piano) tuned with D-based symmetric Pythagorean tuning. Further details about this table can be found in Size of Pythagorean intervals.

File:Just perfect fifth on D.png

Audio file "Just perfect fifth on D.mid" not found}}Template:Category handler: D-A+ (27/16 ÷ 9/8 = 3/2).

File:Perfect fourth on C.png

Audio file "Just perfect fourth on C.mid" not found}}Template:Category handler, one perfect fifth inverted (4/3 ÷ 1/1 = 4/3).

File:Major second on C.svg

Audio file "Major tone on C.mid" not found}}Template:Category handler: C-D (9/8 ÷ 3/2 = 3/2), two Pythagorean perfect fifths.

File:Lesser just minor seventh on C.png

Audio file "Lesser just minor seventh on C.mid" not found}}Template:Category handler, two perfect fifths inverted.

File:Pythagorean major sixth on C.png

Audio file "Pythagorean major sixth on C.mid" not found}}Template:Category handler, three Pythagorean perfect fifths.

File:Semiditone on C.png

Audio file "Pythagorean minor third on C.mid" not found}}Template:Category handler, three Pythagorean perfect fifths inverted.

File:Ditone on C.png

Audio file "Pythagorean major third on C.mid" not found}}Template:Category handler, four Pythagorean perfect fifths.

File:Pythagorean minor sixth on C.png

Audio file "Pythagorean minor sixth on C.mid" not found}}Template:Category handler, four Pythagorean perfect fifths inverted.

File:Pythagorean major seventh on C.png

Audio file "Pythagorean major seventh on C.mid" not found}}Template:Category handler, five Pythagorean perfect fifths.

File:Pythagorean augmented fourth on C.png

Audio file "Pythagorean augmented fourth on C.mid" not found}}Template:Category handler, six Pythagorean perfect fifths.

File:Diminished fifth tritone on C.png

Audio file "Diminished fifth tritone on C.mid" not found}}Template:Category handler, six Pythagorean perfect fifths inverted.

Fundamental intervals

The fundamental intervals are the superparticular ratios 2/1, 3/2, and 4/3. 2/1 is the octave or diapason (Greek for "across all"). 3/2 is the perfect fifth, diapente ("across five"), or sesquialterum. 4/3 is the perfect fourth, diatessaron ("across four"), or sesquitertium. These three intervals and their octave equivalents, such as the perfect eleventh and twelfth, are the only absolute consonances of the Pythagorean system. All other intervals have varying degrees of dissonance, ranging from smooth to rough.

The difference between the perfect fourth and the perfect fifth is the tone or major second. This has the ratio 9/8, also known as epogdoon and it is the only other superparticular ratio of Pythagorean tuning, as shown by Størmer's theorem.

Two tones make a ditone, a dissonantly wide major third, ratio 81/64. The ditone differs from the just major third (5/4) by the syntonic comma (81/80). Likewise, the difference between the tone and the perfect fourth is the semiditone, a narrow minor third, 32/27, which differs from 6/5 by the syntonic comma. These differences are "tempered out" or eliminated by using compromises in meantone temperament.

The difference between the minor third and the tone is the minor semitone or limma of 256/243. The difference between the tone and the limma is the major semitone or apotome ("part cut off") of 2187/2048. Although the limma and the apotome are both represented by one step of 12-pitch equal temperament, they are not equal in Pythagorean tuning, and their difference, 531441/524288, is known as the Pythagorean comma.

Contrast with modern nomenclature

There is a one-to-one correspondence between interval names (number of scale steps + quality) and frequency ratios. This contrasts with equal temperament, in which intervals with the same frequency ratio can have different names (e.g., the diminished fifth and the augmented fourth); and with other forms of just intonation, in which intervals with the same name can have different frequency ratios (e.g., 9/8 for the major second from C to D, but 10/9 for the major second from D to E).

File:Pythagorean diatonic scale on C.png

Audio file "Pythagorean diatonic scale on C.mid" not found}}Template:Category handler.

See also

• Generated collection
• Just intonation
• List of meantone intervals
• List of intervals in 5-limit just intonation
• Shí-èr-lǜ
• Whole-tone scale

External links

• Neo-Gothic usage by Margo Schulter