Schismic: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
Xenwolf (talk | contribs)
autopatrolled, Bureaucrats, confirmaccount-notify, Interface administrators, Sysops
8,705 edits
Changed redirect target from en:Schismatic to Schismatic family
Tag: Redirect target changed
mNo edit summary
 
(8 intermediate revisions by 3 users not shown)
Line 1: Line 1:
#REDIRECT [[Schismatic family]]
: ''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]]

Latest revision as of 20:11, 26 June 2025

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 temperament which takes a roughly justly tuned 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 commas 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 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.

# 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

* In 5-limit CWE tuning

Tunings

Target tunings

Target tunings
Target Minimax Least squares
Generator Eigenmonzo* Generator Eigenmonzo*
5-odd-limit ~3/2 = 701.7379 ¢ 5/3 ~3/2 = 701.728 ¢ [0 -10 17⟩
Retrieved from "https://en.xen.wiki/index.php?title=Schismic&oldid=203711"