Schismic: Difference between revisions

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-#REDIRECT [[Schismic and garibaldi]]
+: ''This page is about a regular temperament sometimes known as "helmholtz". For the music theorist, see [[Hermann von Helmholtz]].''
+'''Schismic''', '''schismatic''', or '''helmholtz''' is a [[5-limit]] [[regular temperament|temperament]] which takes a roughly justly tuned [[3/2|perfect fifth]] and stacks it eight times to reach [[8/5]], thus finding the 5th harmonic at the diminished fourth (e.g. C–F♭) and [[tempering out]] the [[schisma]], 32805/32768. [[5/4]] can be respelled as a major third flattened by one [[Pythagorean comma]], and thus, the Pythagorean and [[syntonic comma]]s are equated into a generalized "comma", and the octave can be split into two diatonic major thirds and one downmajor third representing 5/4. It is one of the most basic examples of a [[microtemperament]], as the fifth generator can be detuned by a fraction of a cent from just, or left untouched entirely (as the difference between [[8192/6561]] and 5/4, the schisma being tempered out, is approximately 2 cents, which is [[just-noticeable difference|unnoticeable]] to most people). Technically, the best tuning in the 5-limit is to flatten the fifth by a fraction of a cent, though tunings on both sides of the just interval work fine.
+Extensions to schismic include [[garibaldi]], which equates the generalized comma further to [[64/63]] and [[50/49]] (tempering out [[225/224]] and [[5120/5103]]) to provide an efficient framework for [[7-limit]] harmony, and unlike 5-limit schismic performs best with a fifth tuned slightly sharp of just; [[pontiac]], which tempers out [[4375/4374]] to induce very little damage on schismic harmonies, at the cost of 7 being quite complex; and the 2.3.5.19 [[subgroup]] extension [[nestoria]], which equates the minor third to [[19/16]], major third to [[19/15]] and [[24/19]], and the minor second to [[19/18]] and [[20/19]] (tempering out [[513/512]] and [[361/360]]). This page, however, focuses on the basic 5-limit temperament.
+See [[Schismatic family #Schismic, schismatic, a.k.a. helmholtz]] for technical data.
+== Interval chain ==
+In the following table, odd harmonics 1–9 and their inverses are in '''bold'''.
+{| class="wikitable center-1 right-2"
+|-
+! #
+! Cents*
+! Approximate ratios
+|-
+| 0
+| 0.00
+| '''1/1'''
+|-
+| 1
+| 701.73
+| '''3/2'''
+|-
+| 2
+| 203.46
+| '''9/8'''
+|-
+| 3
+| 905.19
+| 27/16
+|-
+| 4
+| 406.92
+| 81/64
+|-
+| 5
+| 1108.65
+| 243/128, 256/135
+|-
+| 6
+| 610.38
+| 64/45
+|-
+| 7
+| 112.12
+| 16/15
+|-
+| 8
+| 813.85
+| '''8/5'''
+|-
+| 9
+| 315.58
+| 6/5
+|-
+| 10
+| 1017.31
+| 9/5
+|-
+| 11
+| 519.04
+| 27/20
+|-
+| 12
+| 20.77
+| 81/80
+|}
+<nowiki/>* In 5-limit CWE tuning
+== Tunings ==
+=== Target tunings ===
+{| class="wikitable center-all left-5 mw-collapsible mw-collapsed"
+|+ style="white-space: nowrap;" | Target tunings
+! rowspan="2" | Target
+! colspan="2" | Minimax
+! colspan="2" | Least squares
+|-
+! Generator
+! Eigenmonzo*
+! Generator
+! Eigenmonzo*
+|-
+| 5-odd-limit
+| ~3/2 = 701.7379{{c}}
+| 5/3
+| ~3/2 = 701.728{{c}}
+| {{Monzo| 0 -10 17 }}
+|}
+[[Category:Schismatic| ]] <!-- main article -->
+[[Category:Rank-2 temperaments]]
+[[Category:Schismatic family]]