31 equal temperament

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Topic: 31 equal temperament

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	Paul Rapoport: About 31-tone equal temperament
		In 31-tone equal temperament, the smallest interval is called a diesis, as it is close to the diesis of meantone tuning in size and also is the interval between any tone and the tone twelve perfect fifths away, less seven octaves.
		Furthermore, 31-tone equal temperament alone may receive different treatments in composition, some of which are more tonal than others, some of which may use an equality of the third, fifth and seventh harmonics and their close approximations in 31-tone equal temperament to derive many new types of scales.
		As shown earlier, the chromatic scale familiar from 12-tone equal temperament is `unequal' in 31-tone equal temperament, because the combination of chromatic semitones and diatonic semitones produces a combination of intervals of 2 and 3 dieses respectively.
	www.xs4all.nl /~huygensf/doc/rap31.html (5499 words)

	1 Tone Equal Temperament -
		31-tone equal temperament; 12-tone equal temperament is approximately 1/11-comma meantone,...
		Twelve-tone equal temperament, as this common tuning is called, is a...
		Musical tunings, including the conventional 12-tone equal temperament, may be derived mathematically in a number of different ways.
	tone.fmqg.com /index.php?k=1-tone-equal-temperament (1086 words)

	Temps3.htm
		One of the more amusing claims in favour of equal temperament is that J.P. Kirnberger tried to persuade musicians to adopt it, when in fact his tuning systems, which were published near the end of his life are very different from equal temperament, favour the popular keys, and contain at least one pure major 3rd.
		Already in use in Germany in Bach’s time, was a term for equal temperament, or what was generally thought to reflect the manner of tuning this system: Gleichschwebende temperatur (equal beating temperament); it was considered that equal temperament was obtained by tuning all twelve 5ths flat to beat equally.
		If one listens to a Scarlatti sonata on a harpsichord tuned to unequal temperament, and then listens to the same sonata played on a similar instrument tuned to equal temperament, the difference is most pronounced: in equal temperament, both the instrument, and in consequence the music, lose much of their expression and character.
	www.kirnberger.fsnet.co.uk /Temps3.htm (1972 words)

	Meantone temperaments
		This is an argument for using 31-equal as your standard 11-limit meantone: it simplifies everything.
		All equal tempered meantones represent a closure of the spiral of fifths.
		It follows from this that all singly negative temperaments must have an odd number of steps to the octave.
	www.microtonal.co.uk /meantone.htm (2197 words)

	Playing in the Cracks - Alternate Tuning Software Tools
		By contrast, 12-note-per-octave equal temperament is based on the 12th root of 2, which is an irrational number (it's about 1.059463).
		I've recorded a few MP3 files so you can hear the differences between equal temperament and just intonation.
		Tunings with 19 and 31 notes per octave are especially interesting, but it's difficult to play music in those tunings on a standard MIDI keyboard because one octave of notes will be stretched across an octave and a half or more of keys.
	emusician.com /tutorials/emusic_playing_cracks (2645 words)

	Tuning list FAQ - Working Draft
		Equal temperament can be thought of as a special case of a meantone scale, and that may be a helpful way of thinking of it here.
		Other approaches to tuning can produce different MOS sizes: in early modern Europe of the 16th and 17th centuries, for example, the prevailing meantone temperaments for keyboards offered such sizes at 12, 19, or 31 notes, and instruments of all three sizes were designed and built, and music written taking advantage of the larger systems.
		When you have scales in other temperaments and tunings then the notation of note names and numbers of semitones is no longer adequate to describe a scale.
	tunesmithy.netfirms.com /tree/tuning_faq_draft.html (2645 words)

	Golden Tuning Page
		Closure to equal temperament is roughly approached along a Fibonacci series of {5, 7, 12, 19, 31, 50, 81,.
		Golden Meantone lends itself to scales of various sizes.
		After the first few tones, each additional increment divides the existing intervals in a ratio close to 3:2 or 5:3, efficiently filling gaps.
	www.rev.net /~aloe/music/golden.html (174 words)

	scalesdir.txt
		McLaren 'Microtonal Music', volume 2, track 15 cet24.scl 50 least squares fit primes 2-13 cet258.scl 12 12th root of 6, Moreno's C-12 cet29.scl 95 95th root of 5 cet39.scl 49 49th root of 3 cet39a.scl 31 31-tET with least squares octave; equal weight to 5/4, 3/2, 7/4 and 2/1 cet39b.scl 31 31-tET with l.s.
		scale, TL 29-12-1999 lumma_5151.scl 12 Carl Lumma's 5151 temperament III (1197/709.5/696).
		"Some Numbers", Dec. 2002 bushmen.scl 4 Observed scale of South-African bushmen, almost (4 notes) equal pentatonic cairo.scl 26 P.42, of d'Erlanger, vol.5.
	www.xs4all.nl /~huygensf/doc/scalesdir.txt (174 words)

	Musical note intervals
		A few commonly used major thirds are: pure ratio 5/4 (as used in those clips), equal temperament 400 cents, Pythagorean 81/64, 19-tone major third 378.947 cents, and 31 tone major third 387.097 cents.
		Here is the twelve tone equal temperament dominant seventh by way of comparison, resolving to the twelve tone major chord in twelve equal.
		This stands for "19 tone equal temperament", which means, 19 equally spaced notes per octave.
	tunesmithy.netfirms.com /fts_help/Scales_and_Fractal_Tunes.htm (174 words)

	Stichting Huygens-Fokker: Christiaan Huygens
		He recognised the possible consonant nature of intervals with septimal ratios like 4:7 and 5:7, and showed that they are approximated by the augmented sixth and augmented fourth, respectively, in meantone temperament and 31-tone equal temperament.
		Huygens' name is invariably associated with the division of the octave into 31 equal parts.
		Engraving by Frederik Ottens based on the portrait of Edelinck for the publication by 's-Gravesande of the Opera varia (1724)
	www.xs4all.nl /~huygensf/english/huygens.html (174 words)

	Tonalsoft Encyclopaedia of Tuning -- W. S. B. Woolhouse's 'Essay on musical intervals', (c)1999 by Joe Monzo
		In fact, 31- or 50-tone equal temperaments would be better approximations (though not as convenient from a practical point of view), and neither of those tunings commits any errors larger than 6 cents in any of the classic (5-limit) consonant intervals.
		The scale of equal temperament obviates all such inconveniences, and continues to be universally accepted with unqualified satisfaction by the most eminent vocalists; and equally so by the most renowned and accomplished performers on stringed instruments, although these instruments are capable of an indefinite variety of intonation.
		consonances, namely 7/26- comma meantone), and decided that 19-tone equal tempermant was a close enough approximation, and one which gave to the musician the desirable properties of a closed system which were giving 12-equal its rise to prominence at the time.
	tonalsoft.com /monzo/woolhouse/essay.htm (174 words)

	Margo on Meantone
		Note that while Vicentino considers his tuning to divide the whole-tone into five equal parts, he also appears to consider the major thirds "perfect" or pure -- two characteristics which it was shown by the later 17th century actually define two different tunings, 31-tone equal temperament (31-tET) and 1/4-comma meantone.
		In practice, variations in tuning by ear might be greater than the theoretical difference between these two tunings, both of which circulate nicely in 31 notes.
		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)

	bosanquet
		Bosanquet Keyboard (invented in 1875) tuned to 31-tone Equal Temperament.
		This means the octave is divided into 31 equal parts.
	monxmood.free.fr /bosanquet/bosanquet.html (72 words)

	Microtonal Music
		The intervals between pitches can be equal, creating an equal temperament, or unequal, such as in just intonation or linear temperament.
		Some Western composers have embraced the use of microtonal scales, dividing an octave into 19, 31, 43, 72 and other numbers of pitcheses such as the 24 tone quarter tone scale, rather than the more common 12.
	www.wikiverse.org /microtonal-music (209 words)

	Musical tuning - Wikipedia, the free encyclopedia
		However, the fifth may be flattened to a greater or lesser degree than this and the tuning system will retain the essential qualities of meantone temperament; examples include the 31-equal fifth and Lucy tuning.
		For instance, William Sethares shows that the tunings of Balinese gamelans are related to the inharmonic spectra or timbre of their metallophones and the harmonic spectra of stringed instruments such as the rebab, just as just intonation and twelve tone equal temperament are related to the spectra or timbre of harmonic instruments alone.
		It is impossible to tune the twelve-note chromatic scale so that all intervals are "perfect"; many different methods with their own various compromises have thus been put forward.
	www.wikipedia.org /wiki/Musical_tuning (1293 words)

	Bill Alves - The Just Intonation System of Nicola Vicentino
		This is the statement Barbour interpreted as indicating 1/4-syntonic comma temperament, and he showed how his division of the octave into thirty-one tones very cleverly extends this temperament into practically a closed system closely approximating 31-tone equal temperament [14].
		The comma is not included in the gamut because it is smaller than the smallest step necessary to realize the enharmonic genus; however, it does play a role in the tuning of the archicembalo.
		As can be seen in the table, this difference of 21.5 cents is in fact the syntonic comma and is no more insignificant than Vicentino's comma or many of the steps between the other adjacent intervals.
	www2.hmc.edu /~alves/vicentino.html (2833 words)

	John Starrett's Microtonal Music Page
		Some folks base their music on the harmonic series, some divide the octave into 19 or 31 equal parts, some divide the octave into 43 unequal parts, some don't divide the octave at all....
		Most of the music we hear in based on a system called 12 tone equal temperament (or 12TET for short), where the octave is divided into 12 equal parts.
		Microtonal music is generally defined as any music that is not 12TET.
	infohost.nmt.edu /~jstarret/microtone.html (206 words)

	Paul Rapoport: About 31-tone equal temperament
		The major third of 12-tone equal temperament is thus sharp by 400 - 386.314 = 13.686 cents.
		Of course, any music which uses it extensively will sound different from music in 12-tone equal temperament for all three reasons mentioned in the first paragraph of this section.
		If we temper each of those 31 perfect fifths by adding 6.069/31 = 0.196 cents back onto each, then we will have a closed cycle of 31 identical perfect fifths, each of which is only 0.196 cents sharper than those of meantone tuning.
	www.xs4all.nl /~huygensf/doc/rap31.html (206 words)

	Regular meantone strains - www.larips.com
		In all of these where the fifths are narrower than equal temperament's fifths, the chromatic semitones are smaller than the diatonic semitones.
		This is equivalent in practice to a 31-note equal division of the octave.
		This is equivalent in practice to a 19-note equal division of the octave.
	www-personal.umich.edu /~bpl/larips/meantone.html (1467 words)

	Paul Rapoport: About 31-tone equal temperament
		Correspondingly, the diminished fifth, being the harmonic inversion of the augmented fourth, is 1.867 cents sharper in 31-tone equal temperament than 7:10, while in 12-tone equal temperament, the diminished fifth, which is the same as its augmented fourth, is 17.488 cents flatter.
		As shown earlier, the chromatic scale familiar from 12-tone equal temperament is `unequal' in 31-tone equal temperament, because the combination of chromatic semitones and diatonic semitones produces a combination of intervals of 2 and 3 dieses respectively.
		It should also be noted that in meantone tuning, as in 31-tone and 12-tone equal temperament, the major third is divided exactly in half by the major second: the major third comprises four perfect fifths, and the major second two of them.
	www.xs4all.nl /%7Ehuygensf/doc/rap31.html (1467 words)

	Paul Rapoport: About 31-tone equal temperament
		Correspondingly, the diminished fifth, being the harmonic inversion of the augmented fourth, is 1.867 cents sharper in 31-tone equal temperament than 7:10, while in 12-tone equal temperament, the diminished fifth, which is the same as its augmented fourth, is 17.488 cents flatter.
		As shown earlier, the chromatic scale familiar from 12-tone equal temperament is `unequal' in 31-tone equal temperament, because the combination of chromatic semitones and diatonic semitones produces a combination of intervals of 2 and 3 dieses respectively.
		It may be concluded from Table II that all the familiar intervals of 12-tone equal temperament are present in 31-tone equal temperament and invert harmonically the same way: perfect fifth inverts to perfect fourth, major sixth inverts to minor third, augmented sixth inverts to diminished third, etc.
	www.xs4all.nl /~huygensf/doc/rap31.html (5499 words)

	Paul Rapoport: About 31-tone equal temperament
		Correspondingly, the diminished fifth, being the harmonic inversion of the augmented fourth, is 1.867 cents sharper in 31-tone equal temperament than 7:10, while in 12-tone equal temperament, the diminished fifth, which is the same as its augmented fourth, is 17.488 cents flatter.
		It may be concluded from Table II that all the familiar intervals of 12-tone equal temperament are present in 31-tone equal temperament and invert harmonically the same way: perfect fifth inverts to perfect fourth, major sixth inverts to minor third, augmented sixth inverts to diminished third, etc.
		A table for the other consonances, namely perfect fourth, minor sixth, and major sixth, could of course be easily constructed because these three intervals are simply harmonic inversions of the middle three intervals in Table I. The implications of Table I go far beyond the comparison of a few intervals.
	www.xs4all.nl /%7Ehuygensf/doc/rap31.html (5499 words)

	earlym-l.log9610c
		If equal temperament is envisioned, the circle simply rises up on each side, meeting at the apex, with the twelve keys of the octave stationed exactly where the hours of the clock would be.
		Subject: UPDATE EM IN New England Oct 15-31 (LONG) CALENDAR UPDATE NEW ENGLAND EARLY MUSIC CALENDAR LISTINGS October 15-31, 1996 taken from the pages of Boston Early Music News.
		Events subject to change without notice, please call contact numbers given for events to confirm date, time, location, program and ticket price.
	olymp.wu-wien.ac.at /earlym-l/logfiles/earlym-l.log9610c (5499 words)

	Paul Rapoport: About 31-tone equal temperament
		Correspondingly, the diminished fifth, being the harmonic inversion of the augmented fourth, is 1.867 cents sharper in 31-tone equal temperament than 7:10, while in 12-tone equal temperament, the diminished fifth, which is the same as its augmented fourth, is 17.488 cents flatter.
		As shown earlier, the chromatic scale familiar from 12-tone equal temperament is `unequal' in 31-tone equal temperament, because the combination of chromatic semitones and diatonic semitones produces a combination of intervals of 2 and 3 dieses respectively.
		A table for the other consonances, namely perfect fourth, minor sixth, and major sixth, could of course be easily constructed because these three intervals are simply harmonic inversions of the middle three intervals in Table I. The implications of Table I go far beyond the comparison of a few intervals.
	www.xs4all.nl /%7Ehuygensf/doc/rap31.html (5499 words)

	What Are Makams?
		By the time of J.S. Bach, Western classical music had developed into a system of tuning known as equal temperament, where the musical octave is divided into 12 equally spaced half-tones.
		In theory, there are 24 tones in the Turkish octave, however in practice there are probably 31 and perhaps more.
		However, the Pythagorean comma (approximately one-eighth tone) plays a critical role as it is used to augment or diminish whole-tones (creating 'smaller' and 'larger' whole-tones) as well as half-tones.
	www.hinesmusic.com /What_Are_Makams.html (893 words)

	Just Intonation Explained
		Lou Harrison was the next major figure to abandon equal temperament; he has used many tunings taken from Indonesian gamelans, and also, in his Piano Concerto, returned to an almost-pure tuning called Kirnberger II from the 18th century.
		Here's an excerpt, the first minute, from Ben Johnston's Suite for Microtonal Piano (1977), in which the piano is entirely tuned to overtones of C. You can clearly hear the 11th harmonic halfway between F and F#, and the 31-cent "flat" seventh harmonic.
		The fraction or ratio 5/4 gives us what musicians call a "major third," that is, E in the key of C. (The E string vibrates 5/4 as fast as the C string.) Notes that have these simple arithmetical relationships sound good (consonant) together; the ear registers their harmoniousness.
	www.kylegann.com /tuning.html (3143 words)

	Equal temperament
		Beta splits a perfect fourth into two equal parts, which creates a scale where each step is almost 64 cents.
		They have been used sporadically since the 16th century, with 31-tET particularly popular in Holland, there advocated by Christiaan Huygens and Adriaan Fokker.
		Alpha does the same to a minor third to create a scale of 78 cent steps.
	www.sciencedaily.com /encyclopedia/equal_temperament (3143 words)

	Musical note intervals
		In fact, if you play a major chord in, say, the twelve tone equal temperament, and then tune the major third of the chord flatter by about 14 cents or so, then you will find it gets much fuller, richer and warmer, more mellow.
		For an introduction to notations for the 31 tone system, see John Allen's Notation for microtonal scales, part 1.
		To give a few examples of whole tones and semtones: in 19-et, the whole tone consists of 3 steps, and the semitone of 2, so the semitone is two thirds of a tone.
	www.tunesmithy.netfirms.com /fts_help/Scales_and_Fractal_Tunes.htm (3143 words)

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In the News (Sun 20 Dec 09)