Pythagorean interval

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 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 [http://commons.wikimedia.org/wiki/File:Interval_ratios_in_D-based_symmetric_Pythagorean_tuning_(powers_of_2_%26_3).PNG here] and [http://commons.wikimedia.org/wiki/File:Interval_ratios_in_D-based_symmetric_Pythagorean_tuning.PNG here].]]

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

File:Perfect fourth on C.png

File:Major second on C.svg

File:Lesser just minor seventh on C.png

File:Pythagorean major sixth on C.png

File:Semiditone on C.png on C (1/1 - 32/27) {{audio|Pythagorean minor third on C.mid|Play}}, three Pythagorean perfect fifths inverted.]]

File:Ditone on C.png

File:Pythagorean minor sixth on C.png

File:Pythagorean major seventh on C.png

File:Pythagorean augmented fourth on C.png

File:Diminished fifth tritone on C.png

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.

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

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

{{reflist}}

• [http://www.medieval.org/emfaq/harmony/marchetmf.html Neo-Gothic usage by Margo Schulter]

{{Intervals|state=expanded}}

{{DEFAULTSORT:Pythagorean Interval}}

Category:3-limit tuning and intervals

cs:Ditón

de:Ditonus

es:Ditono

eo:Ditono

nl:Ditonus

ru:Дитон