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Topic: Meantone

	Meantone temperament - Wikipedia, the free encyclopedia
		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 by the same amount (or equivalently, each fourth widened) in order to make the other intervals like the major third closer to their ideal just ratios.
		Most meantone temperaments share this problem, except for the case where the fifth is exactly 700 cents (tempered by approximately 1/11 of a syntonic comma) and the meantone becomes the familiar 12-tone equal temperament.
		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 (696 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).
	www.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)

	Tonalsoft Encyclopaedia of Tuning -- lattices of JI implications of meantone chains, (c)2001 by Joseph L. Monzo
		The meantone chain forms its own linear axis on the lattice, and makes it easy to visualize which JI ratios are acoustically the most closely implied by the meantone pitches.
		The lattice diagrams, however, only give two ratios in cases where the meantone is exactly at the midpoint between them (as with the +2 generator "D" in 1/4-comma meantone, and with the +3 generator "A" in 1/6-comma meantone); otherwise only the nearest JI implied ratio is given.
		Note that because the JI lattice implied by 1/11-comma meantone includes the skhisma, the chain effectively closes at 12 tones; a typical version could be -3 Eb to +8 G#.
	sonic-arts.org /monzo/meantone/lattices/lattices.htm (1164 words)

	NewMusicBox
		Meantone is the name of one of the most elegant tunings in the history of European music, a beautiful tuning that provided near-perfect consonance in a variety of keys.
		Meantone tuning dominated European music until the early 18th century, and continued being used in certain backwaters, especially England and especially among organ tuners, through the late 19th century.
		Meantone is basically a 12-pitch keyboard tuning, a workable compromise that allows eight usable major triads and eight minor triads, the other four of each being real howlers.
	www.newmusicbox.org /article.nmbx?id=909 (1055 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.
		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.
		Also associated with these various meantones were scales with more than 12 notes which the microtonalists have studied and archived in great detail.
	www.ptg.org /pipermail/caut/2001-May/004094.html (585 words)

	Regular Temperaments (Site not responding. Last check: 2007-11-01)
		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.
		Meantone again provides a convenient example; the most common 7-limit extension of meantone has the generator mapping <1, 2, 4, 7] (periods), <0, -1, -4, -10] (generators).
	www.io.com /~hmiller/music/regular-temperaments.html (2082 words)

	Pythagorean Tuning - Just Intonation Context
		Meantone tunings address this discrepancy of 81:80 or 21.51 cents - the syntonic comma (Section 4.4.2) - by narrowing the fifths slightly to optimize the thirds.
		Meantone temperament sets thirds, the choicest concords of the Renaissance, in their most pure or nearly pure aspect, celebrating as it were the new standard of harmonic beauty.
		In equal temperament, or 1/11-comma meantone (Section 5.6.1), all fifths and fourths are an even 700 cents and 500 cents respectively, varying only 2 cents (actually about 1.96 cents) from their just Pythagorean ratios of 3:2 (701.96 cents) and 4:3 (498.04 cents).
	www.medieval.org /emfaq/harmony/pyth5.html (14527 words)

	Musical tuning - Facts, Information, and Encyclopedia Reference article
		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).
		Unlike meantone temperament, the amount of divergence from just ratios varies according to the exact notes being tuned, so that C-E will probably be tuned closer to a 5:4 ratio than, say, Db-F. Because of this, well temperaments have no wolf intervals.
	www.startsurfing.com /encyclopedia/m/u/s/Musical_tuning.html (1291 words)

	Meantone temperaments (Site not responding. Last check: 2007-11-01)
		Although all meantones are 5+2 scales, not all 5+2 scales are meantones.
		All equal tempered meantones represent a closure of the spiral of fifths.
		Such scales are not meantones as all notes cannot be reached by a single spiral of fifths.
	www.microtonal.co.uk /meantone.htm (2197 words)

	Wolf interval
		Major thirds must average 400 cents, and to each pair of meantone thirds of size 400-4e cents we have a sharp third (or diminished fourth) of 400+8e cents, leading to eight thirds of size 400-4e cents and four of size 400+8e cents.
		triads with meantone fifths, and one triad is the wolf major triad, with a wolf fifth and a sharp major third.
		In 1/4-comma meantone, the flat minor thirds are only 2.335 cents sharper than a subminor third of size 7/6, and the sharp major thirds, of size exactly 32/25, are 7.712 cents flatter than the supermajor third of 9/7.
	www.mp3.fm /Wolf_interval.htm (738 words)

	Meantone Tuning
		A defining feature of Renaissance music was the move away from the active 3rds of Pythagorean Tuning to consonant 3rds at their natural pitch ratio of 5/4 relative to the tonic or 386 cents.
		To achieve this Meantone tuning was introduced in the early 1400s and remained the primary method of tuning for the next 250 years.
		Meantone, because every 5th is reduced by the same amount and therefore the 3rd is always made up of two equal tones, as can be seen if the pitches are shown relative to the tonic.
	www.j2b.co.uk /tuning/meantone.htm (382 words)

	www.myspace.com/meantone
		Meantone's unique tuning that covers the bass and guitar registers along with Young Brets maximal playing of his minimalist drum kit, make you think theres at least three to four people in the band.
		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.myspace.com /meantone (1041 words)

	Additions to the beginner's guide to alternative tunings (Site not responding. Last check: 2007-11-01)
		Meantones are defined such that 5t+2s=oct. Algebraic symbols always mean intervals on a pitch scale.
		A meantone major third is four octave reduced steps on the spiral of fifths.
		Quarter-comma meantone is also the optimal minimax meantone in the 7-limit.
	www.microtonal.co.uk /xtra.htm (2838 words)

	SAPAAN: Historic Organs of Oaxaca - Notes
		Meantone tunings became prevalent in the early Renaissance with the emerging emphasis on triadic harmony.
		Meantone tuning was prevalent on keyboards during the 17th century, and is suitable to most keyboard music written during the period, including that employing liturgical chant or chorale melodies; it is especially suitable to Iberian music of the Renaissance and baroque periods.
		Meantone, however, remained in use on organs in some parts of Europe well into the 19th century.
	www.sapaan.com /vol3/notes.html (1051 words)

	Medieval and Renaissance Tuning
		The comma is "hidden" in Meantone systems by distributing it in small pieces, a little bit into each note on the scale, so that each note is a little bit off resonance with the tonic, but no chord that you're going to play is really horribly off.
		It so overshadows other Meantone systems in the modern mind that when someone just says "Meantone" this is usually the particular Meantone that they meant.
		Meantone systems make all the whole steps the same and all half steps the same, with the combination of 5 whole steps (wholetones) and 2 half steps (semitones) adding up to a pure 1:2 resonance (a perfect octave).
	www.whitetreeaz.com /ezharp/tempment.htm (7263 words)

	Meantone and Temperament in Bach's Time
		In meantone, all but one of the fifths are flattened from the pure 3:2 ratio by 1/4 of the syntonic comma.
		To be more specific: a text-book 1/4-comma meantone has for the five fl notes C#, E-flat, F#, G#, and B-flat -- but because they are all part of pure major 3rds, they cannot be used as their enharmonic equivalents.
		Gottfried Silbermann used a milder sort of meantone, in which the 5ths are tempered by 1/6 comma, which is circular.
	www.albany.edu /piporg-l/meantone.html (1334 words)

	F Diatonic Tuning for 12 String Pedal Steel
		Db Ab Eb Bb Meantone is sweeter sounding than equal temperment, but not as pure as just intonation.
		Meantone is the temperament used for many keyboard instruments made from the 1500's through the mid-1800's.
		I especially enjoy the sound of Bach's music when it is played in meantone on the F diatonic pedal steel.
	www.b0b.com /tunings/fscale12.htm (397 words)

	An Introduction to Historical Tunings
		Meantone tuning appeared sometime around the late 15th century, and was used widely through the early 18th century.
		The generating principle behind meantone was that it was more important to preserve the consonance of the major thirds (C to E, F to A, G to B) than it was to preserve the purity of the perfect fifths (C to G, F to C, G to D).
		Meantone encouraged composers to use major and minor triads, to avoid open perfect fifths without thirds, and to not stray more than three or four steps in the circle of fifths away from a central key.
	home.earthlink.net /~kgann/histune.html (3261 words)

	LucyTuning, meantone, equal temperament
		This is a brief comparison among three varieties of meantone temperament: "LucyTuning," 1/4 syntonic comma meantone, and 12-note equal temperament.
		"Meantone" means that any major third (such as C-E) is built of two "mean" (average) tones of the same size; that is, the interval C-D and the interval D-E are the same size.
		In 1/4 syntonic comma meantone, where every fifth is narrow by 1/4 of a syntonic comma, the size of the large interval ("whole step") is (5/4)^(1/2), or about 193.16 cents.
	www-personal.umich.edu /~bpl/templucy.html (600 words)

	MTO 4.4: Scholtz, Algorithms for Tunings and Temperaments
		The name "meantone" is derived from the resulting mid-size interval for the whole tone even though the division of the whole tone was only a by-product of distributing the syntonic comma.
		The earliest generally accepted meantone scale was described by Pietro Aron in the early sixteenth century for a chromatic scale from Eb to G#.
		A chromatic meantone tuning for a keyboard is shown in Figure 13 for a chain of fifths tempered by a quarter-comma (q).
	mto.societymusictheory.org /issues/mto.98.4.4/mto.98.4.4.scholtz.html (8136 words)

	Tonalsoft Encyclopaedia of Tuning - Mozart's tuning: 55-EDO, (c) 2001 by Joseph L. Monzo
		It is known that Leopold Mozart taught his son Wolfgang to use a meantone tuning where "flats" would be higher in pitch than "sharps".
		The mapping uses "D" as the symmetrical pitch center, and renders the 12-tone subset as a meantone "cycle of 5ths" from Eb to G#, with the "wolf" appearing between those two pitches.
		Below is a lattice illustrating the relationship of 1/6-comma meantone with the 5-limit JI pitch-classes it implies.
	sonic-arts.org /monzo/55edo/55edo.htm (2956 words)

	Stichting Huygens-Fokker: Siemen Terpstra: Theory of meantone and 31-ET harmony
		The theoretical language in use, meanwhile, was largely inherited from the meantone era, with its distinction of sharps and flats.
		The 'true' extended meantone fabric is essentially a line of musical fifths each of which is flattened by exactly one quarter of a syntonic comma.
		The 'pure' meantone tuning can be expressed as a line of fifths in which the ratios are all multiples of two and the fourth root of five.
	www.xs4all.nl /~huygensf/doc/terp31.html (5979 words)

	Temperament: A Beginner's Guide
		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.
		Modified meantone temperament was still being used by English organ builders, including Willis, as late as the 1850s.(includes fifth-comma meantone and sixth-comma meantone, the latter sometimes also known as Silbermann temperament).
	www.albany.edu:8080 /faculty/bec/piporg-l/tmprment.html (1630 words)

	[No title]
		This was in retrospect a failed attempt to use the new technology of the modern organ for polyphonic music; meantone tuning did the opposite of Arnaut's procedure, keeping thirds just while shaving bits off each fifth.
		Meantone proved so successful that, according to Alexander J. Ellis and others, it remained the dominant form of tuning through the 1840s.
		Such extended meantone keyboards flourished during the 17th and 18th century, a period when standardization was not the norm, and when musical tuning--like spelling!-- was considered a matter of individual taste within the overall limits of the meantone system.
	infohost.nmt.edu /~jstarret/mclaren1.html (1496 words)

	Peterson Tuners Forum 2006 - Best tuner for both Buzz Feiten Gtr and Clavinet
		Yes, as long your chosen Meantone temperament's cent values don't overstep the StroboStomp's cent adjustment range (-15 cents to + 15 cents in 0.1 cent increments).
		A workable meantone could be had with 15 cents range, though.
		The range is 30 cents, so 1/4 comma Meantone probably won't fit, but there are quite a few others.
	www.petersontuners.com /forum/topic.asp?TOPIC_ID=1437 (830 words)

	(excerpted from another context) (Site not responding. Last check: 2007-11-01)
		The 1/4 comma meantone is a prominent example of this.
		For example, the fifth interval in 1/4 comma meantone temperament is 5.5 cents flat relative to the perfect, Pythagorean fifth interval.
		When using an equal temperament tuner and A440 is +0, the fifth of E would be -3.5 on the tuner, since E in equal temperament is already two cents off the Pythagorean.
	www.autoharpworks.com /pages/Post/doc/tune/temperament_names.htm (625 words)

	Zeke Hoskin: Non-Equal-Tempered Tunings
		In each case, the distance from C to C# is unequal to the distance from C# to D. One way to fix up the meantone scale is to sacrifice perfect major thirds for better fifths.
		This fifth is even worse than in the quarter-comma meantone scale, and the major third is 7.36 cents short of its sweetest value, much better than in twelve-equal but still not good.
		Under these conditions, the best compromise is a meantone scale that gets the thirds and the fifths where you prefer them, somewhere between quarter-comma and eighth-comma.
	www.openaccess.org /~zeke/tuning.htm (2245 words)

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