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{{Ref improve|date=June 2015}}

A '''schismatic temperament''' is a [[musical tuning]] system that results from [[musical temperament|tempering]] the [[schisma]] of 32805:32768 (1.9537 cents) to a [[unison]]. It is also called the '''schismic temperament''', '''Helmholtz temperament''', or '''quasi-Pythagorean temperament'''.

==Construction==
{{multiple image|caption_align=center|header_align=center
 | width = 150
 | direction = vertical
 | image1 = Tonnetz Pythagorean.png
 | image2 = Tonnetz schismatic.png
 | footer = [[Tonnetz]] for Pythagorean tuning (above) and schismatic temperament (below)
}}

In [[Pythagorean tuning]] all notes are tuned as a number of perfect fifths (701.96 cents {{audio|Perfect fifth on C.mid|play}}). The major third above C, E, is considered four fifths above C. This causes the Pythagorean major third, E{{music|+}} (407.82 cents {{audio|Pythagorean major third on C.mid|play}}), to differ from the just major third, E{{music|natural}} (386.31 cents {{audio|Just major third on C.mid|play}}): the Pythagorean third is sharper than the just third by 21.51 cents (a syntonic comma {{audio|Syntonic comma on C.mid|play}}).

:'''C''' — G — D — A{{music|+}} — '''E{{music|+}}'''

Ellis's "skhismic temperament"<ref name="Helmholtz">{{citation
 | last1 = Helmholtz | first1 = Hermann
 | last2 = Ellis | first2 = Alexander J.
 | edition = Second English
 | publisher = Dover Publications
 | title = [[Sensations of Tone|On the Sensations of Tone]]
 | year = 1885
 | page = 435}}. {{Internet Archive|onsensationsofto00helmrich|On the Sensations of Tone}}</ref> instead uses the note eight fifths ''below'' C, F{{music|b}}{{music|minus}}{{music|minus}} (384.36 cents {{audio|Pythagorean diminished fourth on C.mid|play}}), the [[Pythagorean diminished fourth]] or schismatic major third. Though spelled "incorrectly" for a major third, this note is only 1.95 cents (a schisma) flat of E{{music|natural}}, and thus more in tune than the Pythagorean major third. As Ellis puts it, "the Fifths should be perfect and the Skhisma should be disregarded [accepted/ignored]."

:E{{music|+}} <math>\approx</math> F{{music|b}}{{music|minus}}{{music|minus}}
:'''F{{music|b}}{{music|minus}}{{music|minus}}''' — C{{music|b}}{{music|minus}}{{music|minus}} — G{{music|b}}{{music|minus}}{{music|minus}} — D{{music|b}}{{music|minus}}{{music|minus}} — A{{music|b}}{{music|minus}} — E{{music|b}}{{music|minus}} — B{{music|b}}{{music|minus}} — F — '''C'''

In his eighth-schisma "Helmholtzian temperament"<ref name="Helmholtz"/> the note eight fifths below C is also used as the major third above C. However, in the "skhismic temperament" pure perfect fifths are used to construct an approximate major third, while in the "Helmholtzian temperament" approximate perfect fifths are used to construct a pure major third. To raise the Pythagorean diminished fourth 1.95 cents to a just major third each fifth must be narrowed, or [[musical temperament|tempered]], by 1.95/8 = 0.24 cents. Thus the fifth becomes 701.71 cents instead of 701.96 cents. As Ellis puts it, "the major Thirds are taken perfect, and the Skhisma is disregarded [tempered out]."

::E{{music|natural}} <math>\approx</math> F{{music|b}}{{music|minus}}{{music|minus}}
:'''E{{music|natural}}''' — <math>\approx</math>C{{music|b}}{{music|minus}}{{music|minus}} — <math>\approx</math>G{{music|b}}{{music|minus}}{{music|minus}} — <math>\approx</math>D{{music|b}}{{music|minus}}{{music|minus}} — <math>\approx</math>A{{music|b}}{{music|minus}} — <math>\approx</math>E{{music|b}}{{music|minus}} — <math>\approx</math>B{{music|b}}{{music|minus}} — <math>\approx</math>F — '''C'''

Compare {{audio|Circle progression Pythagorean tuning I IV V I.mid|Pythagorean}} vs. {{audio|I-IV-V-I skhismic.mid|Skhismic}}.

==Comparison with other tunings==

Whereas schismatic temperaments achieve a ratio with a number of fifths, each tempered by a fraction of the schisma; [[meantone temperament]]s achieve a ratio with fifths, each tempered by a fraction of the [[syntonic comma]] (81:80, 21.51 cents). As meantone temperaments are often described by what fraction of the syntonic comma is used to alter the perfect fifths, schismatic temperaments are often described by what fraction of the schisma is used to alter the perfect fifths (thus quarter-comma meantone temperament, eighth-schisma temperament, etc.).

In both eighth-schisma tuning and quarter-comma meantone the octave and major third are just, but eighth-schisma has much much more accurate perfect fifths and minor thirds (less than a quarter of a cent off from just intonation). However, quarter-comma meantone has a large advantage in that the major third and minor third are spelled as such, whereas in schismatic tunings, they're represented by the diminished fourth and augmented second (if spelled according to their construction in the tuning). This places them well outside the span of a single diatonic scale, and requires both a larger number of pitches and more microtonal pitch-shifting when attempting common-practice Western music.

Various [[equal temperament]]s lead to schismatic tunings which can be described in the same terms. [[53 equal temperament|Dividing the octave by 53]] provides an approximately 1/29-schisma temperament; by 65 a 1/5-schisma temperament, by 118 a 2/15-schisma temperament, and by 171 a 1/10-schisma temperament. The last named, 171, produces very accurate septimal intervals, but they are hard to reach, as to get to a 7/4 requires 39 fifths. The −1/11-schisma temperament of 94, with sharp rather than flat fifths, gets to a less accurate but more available 7:4 by means of 14 fourths. Eduardo Sabat-Garibaldi also had an approximation of 7:4 by means of 14 fourths in mind when he derived his 1/9-schisma tuning.

==History of schismatic temperaments==

Historically significant is the eighth-schisma tuning of [[Hermann von Helmholtz]] and Norwegian composer [[Eivind Groven]]. Helmholtz had a special ''Physharmonica'' (a [[Pump organ|harmonium]] by Schiedmayer) with 24 tones to the octave.{{citation needed|date=June 2015|reason=How do we know he did, and that it was used for this tuning?}} Groven built an organ internally equipped with 36 tones to the octave which had the ability to adjust its tuning automatically during performances; the performer plays a familiar 12-key (per octave) keyboard and in most cases the mechanism will choose from among the three tunings for each key so that the chords played sound virtually in [[just intonation]].{{citation needed|date=June 2015|reason=How do we know he did, and that it was used for this tuning?}} A 1/9-schisma tuning has also been proposed by [[Eduardo Sabat-Garibaldi]], who together with his students uses a 53-tone to the octave guitar with this tuning.{{citation needed|date=June 2015|reason=How do we know he did, and that it was used for this tuning?}}

[[Mark Lindley]] and Ronald Turner-Smith argue that schismatic tuning was briefly in use during the late medieval period.<ref>{{citation
 | last1 = Lindley | first1 = Mark | author1-link = Mark Lindley
 | last2 = Turner-Smith | first2 = Ronald
 | contribution = Chapter 17. Quasi-Pythagorean Temperaments
 | location = Bonn-Bad Godesberg
 | pages = 55–57
 | publisher = Verlag fuer systematische Musikwissenschaft, GmbH
 | series = Orpheus-Schriftenreihe zu Grundfragen der Musik
 | title = Mathematical Models of Musical Scales: A New Approach
 | url = https://www.academia.edu/904704
 | volume = 66
 | year = 1993}}</ref>{{quote needed|date=June 2015|reason=Vague claim, that is was 'argued', is challenged on talk page.}} This was not temperament but merely 12-tone [[Pythagorean tuning]]. Justly tuned fifths and fourths generate a reasonable schismatic tuning and therefore schismatic is in some respects an easier way to introduce approach justly tuned thirds into a Pythagorean harmonic fabric than meantone. However, the result suffers from the same difficulties as just intonation – for example, the [[wolf interval|wolf]] B-G{{music|flat}} here arises all too easily when availing oneself of the concordant schismatic substitutions just outlined – so it is not surprising that meantone temperament became the dominant tuning system by the early Renaissance. Helmholtz's and Groven's systems get around some, but not all, of these difficulties by including multiple tunings for each key on the keyboard, so that a particular note can be tuned as G{{music|flat}} in some contexts and F{{music|sharp}} in others, for example.

==References==
{{reflist}}

==External links==
*[http://x31eq.com/schismic.htm "Schismic Temperaments"], ''Intonation Information''.
*{{Xenharmonic wiki|Schismatic_family|Schismatic family}}
*[http://www.tonalsoft.com/enc/s/schismic.aspx "Schismic"], ''Tonalsoft(R) - Encyclopedia of microtonal music theory''.

{{Musical tuning}}

[[Category:Linear temperaments]]