John Grieve: Diapason: Pythagorean just interval    
 Diapason: Pythagorean just interval6 comments
6 Jun 2010 @ 17:38, by John Grieve

very interesting dictionary item


Pythagorean interval
Names Ratio Cents
comma 531441/524288 23.46
limma
minor semitone 256/243 90.225
apotome
major semitone 2187/2048 113.685
tone 9/8 203.91
semiditone 32/27 294.135
ditone 81/64 407.82
perfect fourth
diatessaron
sesquitertium 4/3 498.045
tritone 729/512 611.7
perfect fifth
diapente
sesquialterum 3/2 701.955
octave
diapason 2/1 1200.0
The intervals of Pythagorean tuning are just intervals involving only powers of two and three.

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, 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.


See alsoWhole-tone scale
List of meantone intervals

External linksNeo-Gothic usage by Margo Schulter
Cent (music)

The cent is a logarithmic unit of measure used for musical intervals. Typically cents are used to measure extremely small intervals, or to compare the sizes of comparable intervals in different tuning systems, and in fact the interval of
..... Click the link for more information. Pythagorean tuning

Pythagorean tuning is a system of musical tuning in which the frequency relationships of all intervals are based on the ratio 3:2.
..... Click the link for more information. Just intonation

In music, just intonation is any musical tuning in which the frequencies of notes are related by ratios of whole numbers. Any interval tuned in this way is called a just interval
..... Click the link for more information. Superparticular number

Superparticular numbers, also called epimoric ratios, are improper vulgar fractions of the form

..... Click the link for more information. Ancient Greek

Note: This article contains special characters.

Ancient Greek
αρχαία ελληνικά

..... Click the link for more information. Perfect fifth
perfect fifth
Inverse perfect fourth
Name
Other names diapente
Abbreviation P5
Size
Semitones 7
Interval class 5
Just interval 3:2
Cents
Equal temperament 700
..... Click the link for more information. Perfect fourth
perfect fourth
Inverse perfect fifth
Name
Other names diatessaron
Abbreviation P4
Size
Semitones 5
Interval class 5
Just interval 4:3
Cents
Equal temperament
..... Click the link for more information. Consonance and dissonance

In music, a consonance (Latin consonare, "sounding together") is a harmony, chord, or interval considered stable, as opposed to a dissonance — considered unstable (or temporary, transitional).
..... Click the link for more information. Major second
major second
Inverse minor seventh
Name
Other names whole tone
Abbreviation M2
Size
Semitones 2
Interval class 2
Just interval 9:8 or 10:9
Cents
Equal temperament
..... Click the link for more information. Størmer's theorem

In number theory, Størmer's theorem, named after Carl Størmer, gives a finite bound on the number of consecutive pairs of smooth numbers that exist, for a given degree of smoothness, and provides a method for
..... Click the link for more information. Major third
Major third
Inverse Minor sixth
Name
Other names -
Abbreviation M3
Size
Semitones 4
Interval class 4
Just interval 5:4
Cents
Equal temperament 400

..... Click the link for more information. Syntonic comma

In music theory, the syntonic comma, also known as the comma of Didymus or Ptolemaic comma, is a small interval between two musical notes, equal to the frequency ratio 81:80, or around 21.51 cents.
..... Click the link for more information. Minor third
Minor Third
Inverse Major Sixth
Name
Other names -
Abbreviation m3
Size
Semitones 3
Interval class 3
Just interval 6:5
Cents
Equal temperament 300

..... Click the link for more information. Meantone temperament

Meantone temperament is a musical temperament, which is a system of musical tuning. In general, a meantone is constructed the same way as Pythagorean tuning, as a chain of perfect fifths, but in a meantone, each fifth is narrowed
..... Click the link for more information. Equal temperament

Equal temperament is a musical temperament, or a system of tuning in which every pair of adjacent notes has an identical frequency ratio.
..... Click the link for more information. Pythagorean comma

In music, when ascending from an initial (low) pitch by a cycle of justly tuned perfect fifths (ratio 3:2), leapfrogging twelve times, one eventually reaches a pitch approximately seven whole octaves above the starting pitch.
..... Click the link for more information. Whole tone scale
Whole tone scale
Qualities
# of pitch classes: 6
Maximal evenness
Degenerate well-formed collection

In music, a whole tone scale
..... Click the link for more information. List of meantone intervals

The following is a list of intervals of meantone temperament. These intervals constitute the standard vocabulary of intervals for the Western common practice era.
..... Click the link for more information.



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6 comments

7 Jun 2010 @ 02:37 by mortimer : Diapason - Pythagorean MP3
Instrument - Tuning;

Acoustic Bass - Pythagorean minor third
Flute - minor seventh
Rhodes Piano - perfect fourth diatessaron sesquiter
Vibraphone - minor second limma minor semitone

Strings progression - | root | octave diapason | fourth diatessaron | minor seventh |

The MP3 - [link]
 



8 Jun 2010 @ 18:46 by mortimer : Oops
The Rhodes was overriding the pitch bends and playing Equal Temperament,
I fixed the track and uploaded new MP3  



10 Jun 2010 @ 07:08 by mortimer : Some Notes
with Hz...

12-tone Pythagorean scale

2187/2048
9/8
32/27
81/64
4/3
729/512
3/2
6561/4096
27/16
16/9
243/128
2/1


880.0000 Hz
835.3125 Hz
782.2222 Hz
742.5000 Hz
704.7949 Hz
660.0000 Hz
626.4844 Hz
586.6667 Hz
556.8750 Hz
521.4815 Hz
495.0000 Hz
469.8633 Hz
440.0000 Hz
417.6563 Hz
391.1111 Hz
371.2500 Hz
352.3975 Hz
330.0000 Hz
313.2422 Hz
293.3333 Hz
278.4375 Hz
260.7407 Hz
247.5000 Hz
234.9316 Hz
220.0000 Hz  



10 Jun 2010 @ 07:17 by mortimer : Pythagorean - Ritual Bath

New Age / Grime genre, with 12 tone Pythagorean scale - [link]  



10 Jun 2010 @ 12:31 by johnjoseph : pythagorean music
How remarkable. Immersing oneself in this music, bathing oneself in it is obviously a cleansing/purifying process, and it also reveals to us our destiny  


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