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Topic: Quarter comma meantone

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	Meantone temperament
		The most common form of meantone temperament tunes all the major thirds to the just ratio of 5:4 (so, for instance, if G is tuned to 400Hz, B is tuned to 500Hz).
		These systems, like quarter comma meantone, are also named after the amount which the size of the perfect fifths is flatter than a 3:2.
		Quarter comma meantone is particularly suitable for music which treats the third as a consonant interval.
	ebroadcast.com.au /lookup/encyclopedia/qu/Quarter_comma_meantone.html (311 words)

	How to convert a ratio to cents
		Quarter here means a quarter of the value of the comma in cents, not a quarter of its value as a ratio.
		So the effect is that in quarter comma meantone, one of the twelve scales can't be used in the normal way at all.
		Quarter comma meantone continued to be used for church organs for a long time.
	members.tripod.com /~tuning_archive/on_site_tree/robertwalker/mos.html (1260 words)

	Syntonic comma
		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.
		Quarter comma meantone, on the other hand, uses exact 5:4s for major thirds, but flattens each of the fifths by a quarter of a syntonic comma.
		In Quarter comma meantone, the major and minor tones are made equal to the square root of 5:4.
	www.gamelow.com /Music-S/Syntonic_comma.php (309 words)

	Meantone - "Original Graveyard Blues"
		Meantone is "Original Graveyard Blues." A duo comprised of Meantone, fretless, slide guitar and throat and Young Brett, drums and percussion.
		Another unique aspect of Meantone's fretless guitar and songs is that they are in a tuning that was popular in Europe from the 15th to 18th centuries called quarter comma meantone.
		Meantone chooses to use 31 notes to the octave from this tuning as opposed to 12 notes on normal guitars.
	www.meantoneblues.com /biography.html (576 words)

	1/7th comma meantone
		While "meantone" is supposed to refer to the ratio of the tone between the tonic and the major 3rd, there is little in the theory(s) of meantone to suggest this was actually used to construct the scales.
		The important words in the name "quarter comma meantone" are "comma" Here comma refers to the comma of Didymus or the syntonic comma.
		So when one encounters the term "1/7 meantone" that means the 5th has been tempered or narrowed by 1/7 of the syntonic comma.
	www.ptg.org /pipermail/caut/2001-May/004094.html (585 words)

	Meantone temperament
		Meantone tuning involves flattening the fifth so as to bring the seventeenth more nearly, or exactly, equal to this ratio.
		The term meantone temperament is sometimes used to refer specifically to 1/4-comma meantone.
		Since (9/8) / (10/9) = (81/80), the syntonic comma, the fundamental character of a meantone tuning is that all intervals are generated from fifths, and the syntonic comma is tempered to a unison.
	www.mp3.fm /Meantone_temperament.htm (1548 words)

	[No title]
		In a quarter-comma meantone, all but one of the fifths are flattened from the pure 3:2 ratio of 701.96 cents by one-fourth of the syntonic comma, or 5.38 cents.
		In a fifth-comma meantone, the fifths are flattened from the pure 3:2 ratio of 701.96 cents by one-fifth of the syntonic comma, or 4.30 cents.
		In a sixth-comma meantone, the fifths are flattened from the pure 3:2 ratio of 701.96 cents by one-sixth of the syntonic comma, or 3.58 cents.
	www.rreid.net /tuning/Midkiff.info (2080 words)

	1/4-comma meantone / quarter-comma meantone - historical musical tuning
		1/4-comma meantone is historically very important to the development of Western music, as the paradigms postulated by "common-practice" music-theory to a great extent depend on the elimination or tempering-out of the syntonic comma, which is the most prominent feature of all meantones.
		Meantone tunings can be and have been extended to as many notes per octave as desired, but when used for fretted strings or, especially, keyboards, they are usually limited to 12 discrete pitches per 'octave'.
		Use of certain meantone chords may have been the result of their resemblance to certain basic JI sonorities, for example, the "augmented-6th" (+10 generators) is a close approximation of the 7th harmonic (7:4 ratio).
	tonalsoft.com /enc/number/1-4cmt.aspx (2551 words)

	meantone
		I would describe meantone [that is, meantone proper] as the temperament whose fifth is equal to the fourth root of 5 and is thus 1/4 of a syntonic comma flatter than 3/2.
		Meantone cycles which are based on tempering by a fraction of a syntonic comma, which is a 5-limit interval, can have some of their pitches graphed on a lattice diagram.
		Below is a 5-limit lattice illustrating partial cycles of the most common meantone tunings; the cell containing the reference pitch "C" is colored fl because all the cycles contain it, and thus all the axes which represent the different cycles pass thru and are centered on that point on the lattice.
	tonalsoft.com /enc/m/meantone.aspx (2580 words)

	The Just Intonation Primer
		The exemplary variety of meantone temperament, called quarter-comma meantone, produced, in a twelve-tone realization, eight good major triads and eight good minor triads, with the remaining four triads of each type being badly mistuned.
		Meantone tunings satisfied the needs of composers for a time, but as instrumental music became more complex and the desire to modulate to more remote keys increased, the bad triads became a barrier to progress.
		Meantone seems still to have been the predominant system in France in the mid-eighteenth century, and in England meantone continued to be the predominant tuning, at least for organs, until the middle of the nineteenth century.
	www.justintonation.net /primer2.html (4881 words)

	Meantone temperament - Wikipedia, the free encyclopedia
		Quarter-comma meantone is the most well known type of meantone temperament, and the term meantone temperament is often used to refer to it specifically.
		Though quarter-comma meantone is the most common, other systems which flatten the fifth by differing amounts but which still equate the major whole tone, which in just intonation is 9/8, with the minor whole tone, tuned justly to 10/9, are also called meantone systems.
		Although Meantone is best known as a tuning environment associated with earlier music of the common practice period, there is evidence of continuous usage of meantone as a keyboard temperament well into the middle of the 19th century.
	en.wikipedia.org /wiki/Meantone_temperament (726 words)

	Margo on Meantone
		Please let me begin by saying that like many other tunings, 1/4-comma meantone can be precisely defined in mathematical terms, but will vary in "real world" realizations, whether on a 16th-century harpsichord tuned by ear or on a modern synthesizer with limited pitch resolution.
		Using the logarithmic measure of cents, logarithms and their application to music developing in the 17th century, we can say that such a fifth is approximately 5.38 cents narrower than pure, with a size of about 696.5784 cents.
		The keyboards differ by Vicentino's "diesis" or fifthtone, often 128:125 or ~41.06 cents in 1/4-comma meantone (the distance between G# on the lower keyboard and Ab on the higher one, for example).
	www.ptg.org /pipermail/pianotech/2001-January/077900.html (524 words)

	Regular Temperaments
		This notation specifies that the prime factor 2 is approximated by 1200.00 cents (which happens to be an exact octave), 3 is approximated by 1896.58 cents (which is a quarter comma flat from the just tuning of 1901.96 cents), and 5 is approximated by 2786.31 cents (the same as the just interval 5:1).
		In the case of meantone, where the generator can be defined as the meantone approximation of a perfect fourth (4:3), this results in the mapping <1, 2, 4] for the octave and <0, -1, -4] for the fourth.
		It is apparent that this <<1, 4, 4]] is related to the monzo notation of the meantone comma 81:80, which is [-4, 4, -1>.
	www.io.com /~hmiller/music/regular-temperaments.html (2082 words)

	How to calculate the cents values for a mean tone scale from the size of the comma
		The reason they are called meantone scales is because all whole tones are the same size, and in particular, the major third is divided into two equal sized whole tones, rather than the unequal sized 9/8 and 10/9 of just intonation.
		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.
		We see that quarter comma meantone scale has fifths that are flattened more than they are for equal temperament.
	tunesmithy.netfirms.com /japplets/mean_tone_in_cents.htm (949 words)

	Temperament: A Beginner's Guide
		Almost all the fourths and fifths are dead in tune, and the entire comma is 'dumped' on one interval (according to Arnaut de Zwolle between F and Bb), which is therefore unusable.
		The wolf in meantone tuning is so horrible and such an obstacle that, by the later seventeenth century, it was being modified substantially in practice.
		Modified meantone is probably the most appropriate temperament for most of the 'early' organ music we now hear — even though Buxtehude and Bach were clearly among those exploring new tuning systems, their compositional technique remains informed by the meantone system.
	www.albany.edu /piporg-l/tmprment.html (1630 words)

	Just Temperament and Others
		In quarter-comma meantone temperament the just major third is divided into two equal whole tones.
		The fifths are made smaller and the fourths larger by a quarter of a syntonic comma.
		Expressed in terms of whole number ratios, the ordinary (syntonic or Ptolemaic) comma is the interval between a just major third (5:4) and a Phythagorean ditonic or major third (81:64).
	hyperphysics.phy-astr.gsu.edu /hbase/music/just.html (418 words)

	MTO 4.4: Scholtz, Algorithms for Tunings and Temperaments
		In Figure 5, the Pythagorean comma is indicated by the upper-case "P" and 1/12 of the Pythagorean comma by the lower-case "p." The direction of the arrow indicates the ascending harmonic interval.
		Dividing the comma into four "quarter commas" and distributing the quarter-commas so that every fifth was tempered by one quarter-comma also smooths out the three narrow fifths created by the undivided comma in the just scale, replacing them with a sequence of tempered fifths that are far more tolerable to the ear.
		A chromatic meantone tuning for a keyboard is shown in Figure 13 for a chain of fifths tempered by a quarter-comma (q).
	www.societymusictheory.org /mto/issues/mto.98.4.4/mto.98.4.4.scholtz_essay.html (6696 words)

	Classic Meantone Tuning
		Quarter-comma meantone tuning was probably the standard tuning for keyboard instruments for 4 centuries, much longer than the equal temperament we think of as normal has lasted.
		there are other 'meantone' tunings, 1/3 comma, 1/6 comma, even 1/12 comma which is what we know as equal temperament; but they produce few pure intervals; getting everything equally out-of-tune is what you have to do for them, whereas in classic quarter-comma meantone the drill is to get some things exactly in-tune.
		In quarter-comma meantone, The distance would be instead approximately 23/22 from the accidental to the note or vice versa (i.
	www.mindspring.com /~judithconrad/id2.html (1277 words)

	MTO 12.3: Duffin, Just Intonation in Renaissance Theory and Practice
		The syntonic comma is a microtonal interval which is the difference between the Pythagorean major third, or ditone, at 81:64--that is, two consecutive 9:8 whole tones--and the pure major third at 5:4, or 80:64.
		The 2/7 comma version sounds similar but slightly better with the improved fifths and major thirds, and the version in 1/4 comma meantone sounds better still in the open fifths (though not actually pleasant) and very euphonious in the triads.
		In 1/5 comma meantone, the fifths and major thirds are about the same distance from pure (though the third is wide and the fifth narrow), which means that they beat at a similar rate and create a kind of "vibrato" effect that some listeners find attractive.
	mto.societymusictheory.org /issues/mto.06.12.3/mto.06.12.3.duffin.html (9232 words)

	Wolf fifth
		The ratio of the flattened fifth to the exact one is -23.46 cents (2^19/3^12), or a little under a quarter of a semitone.
		The tempering is by a quarter of a comma, which is the reason for the name.
		The meantone part of the name refers to the whole tones - you get them by a division of the pure major third into equal parts - all the whole tone intervals except two of them which are a wider.
	www.tunesmithy.netfirms.com /fts_help/wolf_fifth.htm (394 words)

	The Musical Scale
		Essentially, the notes on the organ are tuned on the basis of a perfect fifth that has been flattened by the fourth root of the syntonic comma, so that while the fifths are not accurate, the major thirds are exact.
		This tuning is called the Pythagorean tuning, and the ratio between B sharp (not shown on the diagram) and C on that tuning is called the Pythagorean comma, as distinct from the syntonic comma.
		The syntonic comma is so named because it is the distance between two notes with the same spelling, thus representing the same tone.
	www.quadibloc.com /other/musint.htm (1570 words)

	CARL FRIEDRICH ABEL, A GAINSBOROUGH PAINTING,
		When we distinguish between say, quarter-comma meantone and fifth comma meantone, we are talking about differences between the two scales ranging from about a twelfth of a semitone to less than a ninetieth of a semitone.
		The key restrictions are consistent with those that would be expected had a meantone temperament been in use, and would also be consistent with the use of an unequal temperament that in the usual way favoured the quality of the major thirds in the ‘home’ keys.
		G sharp- E flat is one of the usual positions to have the 'wolf' in meantone tuning, and in the tuning under examination here, would be the most tempered fifth, or a 'wolf', depending on the amount of tempering in the regular fifths.
	www.amarilli.co.uk /academic/consort/abel.asp (8897 words)

	Mailgate: rec.music.theory: Re: interval dissonance
		Thus, your "quarter comma" I figure is what is called ditonic or Pythagorean comma in my dictionary, which is (3/2)^2 / 2^7 (twelve perfect fifths reduced by seven octaves).
		In the musical aspect, I figure that the ditonic comma is not important, because it is too far from any harmonics to be perceived as such by the human ear.
		Then, if it should be generated by the perfect fifth, the perfect fifth should have an error of at most (5 cent)/12 = 0.42 cent in order to be sure that the ditonic coma, after 12 applications is within 5 cents (as octave transpositions do not generate any error at all).
	mailgate.supereva.it /rec/rec.music.theory/msg23733.html (1015 words)

	The Ultimate Musical tuning - American History Information Guide and Reference
		Just intonation, in which the ratios of the frequencies between all notes are based on relatively low whole numbers, such as 3:2, 5:4 or 7:4; or in which all pitches are based on the harmonic series, which are all whole number multiples of a single tone.
		Meantone temperament, a system of tuning which averages out pairs of ratios used for the same interval (such as 9:8 and 10:9), thus making it possible to tune keyboard instruments.
		To do this, eleven perfect fifths in each octave are flattened by a quarter of a syntonic comma, with the remaining fifth being left very sharp (such an unacceptably out-of-tune fifth is known as a wolf interval).
	www.historymania.com /american_history/Musical_tuning (1108 words)

	LucyTuningLucyScaleDevelopmentsLucyTuned LullabiesPi tuningJohn Longitude Harrison
		The unfortunate term Meantone seems to be derived from the traditional integer frequency ratio logic, as producing a spiral interval mapping from average deviations from integer ratios and expressed as commas.
		Many equal temperaments and circular tunings could also be considered meantones, yet after a whole number of steps of fourths or fifths, they arrive at the same position at which they began.
		In negative meantones (like LucyTuning) the sharp of a note will be closer to the starting note than to the Large interval above; whereas in positive meantones it will be nearer to the Larger interval above.
	www.lucytune.com /tuning/mean_tone.html (239 words)

	The Changing Scale (Site not responding. Last check: 2007-09-10)
		This offset from an exact octave is known as the Pythagorean comma.
		Aron developed a quarter comma meantone temperament without realizing it - named this because the offset of one fourth was applied to the scale with a wolf tone offset of seven fourths.
		Here is Aron's quarter comma meantone temperament scale, reproduced chromatically, and below is a table that shows the offset it creates (table 1).
	www.johnminor.net /writings/overview.htm (2191 words)

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