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Topic: Pythagorean comma

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	Syntonic comma
		The syntonic comma, also known as the comma of Didymus, is a small interval between two musical notes, equal to the frequency ratio 81:80, or around 21.51 cents.
		Pythagorean tuning tunes the fifths as exact 3:2s, but uses the relatively complex ratio of 81:64 for major thirds.
		In Quarter comma meantone, the major and minor tones are made equal to the square root of 5:4.
	www.ebroadcast.com.au /lookup/encyclopedia/sy/Syntonic_comma.html (306 words)

	NationMaster - Encyclopedia: Comma (music)
		In music theory, a comma is a small or very small interval between two enharmonic notes tuned in different ways.
		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.
		The schisma, also spelled skhisma, is the ratio between a Pythagorean comma and a syntonic comma and equals 32805/32768, which is 1.
	www.nationmaster.com /encyclopedia/Comma-%28music%29 (545 words)

	Pythagorean Tuning and Medieval Polyphony - Table of Contents
		The unsuitability of medieval Pythagorean intonation for Renaissance music should not be seen as a "flaw," any more than Renaissance meantone tuning is "flawed" because it is hardly suitable for the works of Wagner or Max Reger.
		Readers interested in the practical details of Pythagorean tuning are encouraged to jump directly from Section 2 to Section 4.
		Section 3, on stylistic considerations, is linked in many ways to a companion article on 13th-century polyphony, and owes a special debt of gratitude to studies by Vincent Corrigan on the Notre Dame conductus repertory, and by Mark Lindley on the later 13th and 14th centuries, although any flaws or infelicities are of course mine.
	www.medieval.org /emfaq/harmony/pyth.html (766 words)

	[No title] (Site not responding. Last check: 2007-10-18)
		The Pythagorean Comma is the difference of the interval of Ab to C# compared to the 3:2 interval ratio of the pure fifth.
		The Pythagorean Comma is a considerable portion of a semitone, since it is greater than the just noticicable difference(jnd) cutoff of 1.01.
		Pythagorean Intonation developed into a preference after the 15th century to various just intonation forms where beatless thirds were added to the perfect consonances of Pythagorean fifths and octaves.
	people.cs.uchicago.edu /~dtshoda/CS295_files/working_dir/draft.html (2800 words)

	pythagorean - 3-limit just intonation musical tuning system
		Because the comma vanishes, so does the notational distinction between the sharps and flats: thus, A# = Bb, C# = Db, etc. Many modern composers and theorists, whose use of 12-edo does not reference its older tonal connotations, prefer to use the digits 9 - 11 to represent the degrees of the tuning.
		Assuming Mercator's comma to be below the "margin of error" in interval perception (usually given as ~5 cents), 53edo thus provides an approximation which is audibly indistinguishable from a 53-tone "pure" Pythagorean chain; note that 53-edo actually is a closed tuning and thus gives a circle of 5ths rather than an open-ended chain.
		In 53-edo it is 1 degree, identical to the Pythagorean comma; in 306-edo it is 5 degrees, one less than the 6 degrees which compose the Pythagorean comma.
	tonalsoft.com /enc/p/pythagorean.aspx (1559 words)

	Onelist Tuning Digest # 483 message 26, (c)2000 by Joe Monzo
		Basically, there are only two independent commas that come into play when trying to analyze or render music of the Western conservatory tradition in just intonation.
		Paul is using 'comma' here in a very generic sense to mean a small interval, which in many cases will be used as a unison vector to delimit the boundaries of a periodicity-block.
		The other 'commas' can be used for bridging in other, "invented" musical systems, motivating certain corresponding tuning systems as shown in the "Equal Temperament" entry.
	sonic-arts.org /td/monzo/o483-26new5limitnames.htm (2611 words)

	[No title] (Site not responding. Last check: 2007-10-18)
		[pythagorean comma] The frequency for an initial pitch tuning by fifths is 3/2^(12) = 129.75.
		The Pythagorean Comma is a considerable portion of a semitone, since it is greater than the just noticicable difference cutoff of 1.01(see jnd section) as all octaves and fifths cannot be tuned exactly in terms of the harmonic series interval ratios.
		[pythagorean] It is not possible to tune all octaves and fifths perfectly to the natural harmonic series since it is impossible for all fifths to be true to both the ratio of the fifth and the ratio of the octave.
	people.cs.uchicago.edu /~dtshoda/CS295_files/working_dir/rough_2.html (4302 words)

	Contact
		The pitch discrepancy between the two (just a little over 1 percent) is the Pythagorean comma, which is the amount by which the sequence of perfect fifths misses being "closed" after the sequence is carried out past twelve pitches.
		If the perfect fifth were to be replaced by another slightly smaller interval, called the tempered fifth then the discrepancy of the Pythagorean comma could be spread equally among twelve intervals, each of which would be very close indeed to the perfect fifth.
		Because the Pythagorean comma is small, we expect the discrepancy between the tempered fifth (T5) and the perfect fifth (P5) to be even smaller.
	tomscarff.tripod.com /music/equal_temperament.htm (252 words)

	Tonalsoft Encyclopaedia of Tuning -- Septimal schisma as xenharmonic bridge?, (c)1998 by Margo Schulter
		Pythagorean tuning is not only a system of just intonation with a very long history, but also an amazingly successful system not only ideally suited to much Gothic music but inviting "Neo-Gothic extensions" approximating intervals in 5-based and 7-based systems.
		It is the difference between the Pythagorean comma (531441:524288, about 23.46 cents) and the syntonic comma (81:80, about 21.51 cents) which separates a number of basic 3-based intervals from their 5-based counterparts.
		It is the difference between the Pythagorean comma and the septimal comma (64:63, about 27.26 cents) which separates basic Pythagorean intervals from their 7-based counterparts.
	sonic-arts.org /td/schulter/septimal.htm (1195 words)

	[No title] (Site not responding. Last check: 2007-10-18)
		The syntonic comma was seen as beneficial in medieval music where 3rds were not meant to be consonant.
		Kirnberger III uses the 1/4 comma of the previous example, but only the 5.5 cent flattening to 4 of the 5ths.
		Its partner, the syntonic comma, started as a 'feature' but quickly became a 'bug' (as the computer programmers would have it) and the history of tunings is very much about the reduction of the syntonic comma to acceptable levels, with the removal of the Pythagorean comma being a useful by-product, if possible,
	www.j2b.co.uk /tuning/commas.htm (524 words)

	tunetemp \| David Schulenberg
		Pythagorean intonation is a system of just intonation traditionally ascribed to the mythical ancient Greek mathemetician Pythagoras in which the most common intervals are defined as follows:
		There is another comma as well: the so-called Pythagorean comma, which is the difference between six major whole tones and an octave.
		Nine Pythagorean commas would equal 1.12970812218183304558660833854287, which is somewhat closer to a major whole tone than are nine syntonic commas.
	www.wagner.edu /faculty/dschulenberg/tunetemp.html (4097 words)

	[No title]
		His system is based in part on the established ratios of Pythagorean just intonation: the octave at a pure 2:1, the fifth at 3:2, the fourth at 4:3, and the major second or whole-tone at 9:8 (the difference between the pure consonances of the fifth and fourth).
		The Pythagorean contrast between complex and "beatful" thirds and sixths, and pure fifths and fourths, accentuates the musical tension of cadences; the smoother, more "streamlined" effect of an unstable sonority such as 7:9:12 might have a striking effect for either 14th-century or 21st-century ears.
		This scheme is very similar to a 24-note Pythagorean tuning with a chain of 23 pure fifths, except that the distance between the two 12-note manuals is slightly increased from a Pythagorean comma (531441:524288, ~23.46 cents) to a septimal comma (64:63, ~27.26 cents).
	veenet.value.net /~mschulter/marchetmf.txt (4863 words)

	What is Pythagorean Comma
		There are several ways to explain the Pythagorean comma.
		Start at middle C and tune perfect 4ths and 5ths in both directions.
		Stay in the FF octave around Middle C. Leave the pythagorean Comma between G# and Eb.
	www.music.indiana.edu /som/piano_repair/temperaments/pythagorean_comma.html (210 words)

	Temperaments and the circle of fifths (Site not responding. Last check: 2007-10-18)
		An advantage of using the Syntonic Comma is that by dividing it by four, and then tempering four consecutive 5ths by a quarter of a comma each produces a pure major 3rd.
		One of the obvious ways of creating a circulating temperament, was to begin as with 1/4 comma mean tone: flatten the four ascending 5ths from C to E. In doing so, the syntonic comma has been accounted for, and there remains only the schisma: the two cent difference between the Syntonic and Pythagorean commas.
		This entails flattening one of the remaining 5ths by 1/12 (2 cents) of a Pythagorean comma.
	www.kirnberger.fsnet.co.uk /TempsII.htm (2353 words)

	[No title]
		The Pythagorean comma, a natural unit to use in describing a 12-note circulating temperament, is the amount by which 12 fifths at 3:2 exceed seven pure octaves at 2:1, a difference of 531441:524288 or about 23.46 cents.
		The syntonic comma, a typical unit for measuring temperament in regular meantone tunings, is the difference between the rather active and complex Pythgorean major third formed from four pure 3:2 fifths (e.g.
		In this system, a Pythagorean comma is taken as having 720 TU, and a syntonic comma as having 660 TU, a rounding which reflects their actual sizes with the latter almost exactly 11/12 of the former.
	www.bestii.com /~mschulter/Lehman_Bach_Neidhardt.txt (2129 words)

	Musical Scales and Frequencies (Site not responding. Last check: 2007-10-18)
		As with "just intonation," Pythagorean tuning sounds pretty good near the home key but the 3rds sound slightly too high or low and get worse the farther the key moves from home.
		There are various schemes to adjust certain intervals slightly to make the Pythagorean scale sound closer to the "just" scale; these form the basis of various "meantone" tuning schemes.
		The most famous of these is to spread the Pythagorean comma evenly around all 12 intervals in the chromatic scale in equal proportions, effectively making every interval just slightly sour but none worse than any other.
	www.swarthmore.edu /NatSci/ceverba1/Class/e5/MusicalScales.html (446 words)

	Natasha Mostert (Site not responding. Last check: 2007-10-18)
		The riddle of the Pythagorean Comma is one of the oldest mysteries in the science of sound and forms the cornerstone of Natasha Mostert's suspense novel, The Other Side of Silence.
		The problem of the Pythagorean Comma is a complex one and any discussion of this topic is usually highly esoteric.
		The reason lies in a strange-sounding phenomenon, The Pythagorean Comma: a mathematical blip, an imperfection in musical intervals, and the reason why musicians have no choice but to make use of equal temperament and a flawed musical scale.
	www.natashamostert.com /novel2e.html (678 words)

	Stichting Huygens-Fokker: Logarithmic Interval Measures
		The chromatic and diatonic semitones are 4 and 5 commas.
		In regular meantone temperaments where the tempering of the fifth is expressed in a fraction of a comma, the comma is the syntonic comma.
		The syntonic comma is 11.008 schismas, the Pythagorean comma 12.008, and the minor diesis 21.016 schismas, so practically 11, 12 and 21.
	www.xs4all.nl /~huygensf/doc/measures.html (2184 words)

	How to calculate the cents values for a mean tone scale from the size of the comma
		Since we want the interval c to e to be a pure 5/4, all the fifths in C' G' D A e need to be flattened slightly in order to achieve 5/1 for the interval from C' to e.
		A pure fifth is 701.955 cents, and the syntonic comma is 21.5063 cents.
		Of couse, the quarter comma meantone isn't the only meantone scale of interest, and one will want to be able to find the cents values for any meantone scale.
	tunesmithy.netfirms.com /japplets/mean_tone_in_cents.htm (949 words)

	Zeke Hoskin: Non-Equal-Tempered Tunings
		In the Pythagorean scale, the interval from C to E is 81/64.
		Theorists have given names to two places where the Pythagorean scale runs into trouble: the amount by which a Pythagorean third exceeds a good-sounding 5/4 is called the Syntonic Comma, and the amount by which 12 Pythagorean fifths miss coming back to the starting note (seven octaves up) is the Pythagorean Comma.
		As long as you take more than an eighth of a comma from the fifths, the error in the third is less than half a comma.
	www.openaccess.org /~zeke/tuning.htm (2245 words)

	Pythagorean comma - Wikipedia, the free encyclopedia
		Put more succinctly, twelve perfect fifths are not exactly equal to seven perfect octaves, and the Pythagorean comma is the amount of the discrepancy.
		This interval has serious implications for the various tuning schemes of the chromatic scale, because in Western music, 12 perfect fifths and seven octaves are treated as the same interval.
		Equal temperament, today the most common tuning system used in the West, accomplished this by flattening each fifth by a twelfth of a Pythagorean comma (2 cents), thus giving perfect octaves.
	en.wikipedia.org /wiki/Pythagorean_comma (207 words)

	Das Wohltemperirte Clavier - Pitch, Tuning and Temperament Design [by Charles Francis]
		In this regard, Lindley proposes[11] that the elaborate theoretical tuning models expounded by the likes of Neidhardt and Sorge, were unable to capture the subtle nuances that Bach customarily achieved in tuning[12].
		He, suggests, moreover, that the cause of the theoretical deficiency was that Sorge and Neidhardt would never split their basic unit of measurement for tempering, namely 1/12 of a Pythagorean comma.
		The Pythagorean third (~408 cents) is excluded, so that a sequence of four pure fifths does not occur, so satisfying Lindley’s second condition.
	www.bach-cantatas.com /Articles/Das_Wohltemperirte_Clavier.htm (2304 words)

	[No title]
		This implies that the Pythagorean comma is distributed somewhat evenly along the circle of fifths.
		Alterations of 1/4-syntonic comma would require an additional bisection (the minimum to perform Renaissance meantone, a wonderful tuning for all triadic diatonic music), although there are those who like Renaissance music just fine in JI or even 12-tet.
		But for the purists (like me), the final result is 288-tet, or steps of just over 4 cents, which is the usual jnd or "just-noticeable difference." It's too bad we're stuck with the lousy decimal system, as it's mentally and physically easier to approximate bisections and a trisection than the quintisections required by decimals.
	members.tripod.com /~tuning_archive/Mills/html/s___4/msg_3800-3999.html (6720 words)

	Just Intonation the Harmonic Piano: Michael Harrison's Work and Tunings
		The minute difference between the tuning of this note B# and the original C is the “Pythagorean” comma (approximately 1/8 of a tone).
		In equal temperament these fifths are each tuned 1/12 of a “Pythagorean” comma flat so that the B# and C are equalized to become the same pitch.
		The comma is thus freed from its restricted status as an "out-of-tune" dissonance that, until recently, was disguised, avoided or obliterated by tempered tunings, compositional styles, performance practices and instrument designs.
	www.michaelharrison.com /harmonic-tunings.html (2583 words)

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