Editing Tetrachord (section)

==History==
The name comes from ''tetra'' (from Greek—"four of something") and ''chord'' (from Greek ''chordon''—"string" or "note"). In ancient Greek music theory, ''tetrachord'' signified a segment of the [[Musical system of ancient Greece#Systema ametabolon, an overview of the tone system|greater and lesser perfect systems]] bounded by ''immovable'' notes ({{math|{{langx|el|ἑστῶτες}}}}); the notes between these were ''movable'' ({{math|{{langx|el|κινούμενοι}}}}). It literally means ''four strings'', originally in reference to harp-like instruments such as the [[lyre]] or the kithara, with the implicit understanding that the four strings produced adjacent (i.e., conjunct) notes.

Modern music theory uses the [[octave]] as the basic unit for determining tuning, where ancient Greeks used the tetrachord. Ancient Greek theorists recognized that the octave is a fundamental interval but saw it as built from two tetrachords and a [[major second|whole tone]].<ref>{{cite book |first=Thomas J. |last=Mathiesen |section=Greece&nbsp;§I: Ancient |title=The New Grove Dictionary of Music and Musicians |title-link=The New Grove Dictionary of Music and Musicians |edition=second |editor1-first=S. |editor1-last=Sadie |editor1-link=Stanley Sadie |editor2-link=John Tyrrell (musicologist) |editor2-first=J. |editor2-last=Tyrrell |place=London, UK |publisher=Macmillan |year=2001 |at=6&nbsp;Music Theory, (iii)&nbsp;Aristoxenian Tradition, (d)&nbsp;Scales}}</ref>

===Ancient Greek music theory===
{{Main|Genus (music)}}
[[Music of ancient Greece|Ancient Greek music]] theory distinguishes three ''genera'' (singular: ''genus'') of tetrachords. These genera are characterized by the largest of the three intervals of the tetrachord:

;[[Genus (music)#Diatonic|Diatonic]]
: A diatonic tetrachord has a characteristic interval that is less than or equal to half the total interval of the tetrachord (or approximately 249&nbsp;[[cent (music)|cent]]s). This characteristic interval is usually slightly smaller (approximately 200&nbsp;cents), becoming a [[major second|whole tone]]. Classically, the diatonic tetrachord consists of two intervals of a tone and one of a [[semitone]], e.g. A–G–F–E.
;[[Genus (music)#Chromatic|Chromatic]]
: A chromatic tetrachord has a characteristic interval that is greater than about half the total interval of the tetrachord, yet not as great as four-fifths of the interval (between about 249 and 398 cents). Classically, the characteristic interval is a [[minor third]] (approximately 300 cents), and the two smaller intervals are equal semitones, e.g. A–G{{Music|flat}}–F–E.
;[[Genus (music)#Enharmonic|Enharmonic]]
[[File:Greek Dorian enharmonic genus.png|thumb|Two Greek tetrachords in the enharmonic genus, forming an enharmonic Dorian scale]]
: An enharmonic tetrachord has a characteristic interval that is greater than about four-fifths the total tetrachord interval. Classically, the characteristic interval is a [[ditone]] or a [[major third]],{{sfn|Chalmers|1993|p=8}} and the two smaller intervals are variable, but ''approximately'' [[quarter tone]]s, e.g. {{nobr|A–G{{Music|double flat}}–F{{Music|half flat}}–E.}} 

When the composite of the two smaller intervals is less than the remaining ([[incomposite interval|incomposite]]) interval, the three-note group is called the ''[[pyknon|pyknón]]'' (from ''pyknós'', meaning "compressed"). This is the case for the chromatic and enharmonic tetrachords, but not the diatonic (meaning "stretched out") tetrachord.

Whatever the tuning of the tetrachord, its four degrees are named, in ascending order, ''hypate'', ''parhypate'', ''lichanos'' (or ''hypermese''), and ''mese'' and, for the second tetrachord in the construction of the system, ''paramese'', ''trite'', ''paranete'', and ''nete''. The ''hypate'' and ''mese'', and the ''paramese'' and ''nete'' are fixed, and a perfect fourth apart, while the position of the ''parhypate'' and ''lichanos'', or ''trite'' and ''paranete'', are movable.

As the three genera simply represent ranges of possible intervals within the tetrachord, various ''shades'' (''chroai'') with specific tunings were specified. Once the genus and shade of tetrachord are specified, their arrangement can produce three main types of scales, depending on which note of the tetrachord is taken as the first note of the scale. The tetrachords themselves remain independent of the scales that they produce, and were never named after these scales by Greek theorists.{{sfn|Chalmers|1993|p=103}}

;Dorian scale : The first note of the tetrachord is also the first note of the scale.
:Diatonic: E–D–C–B | A–G–F–E
:Chromatic: E–D{{Music|flat}}–C–B | A–G{{Music|flat}}–F–E
:Enharmonic: E–D{{Music|double flat}}–C{{Music|half flat}}–B │ A–G{{Music|double flat}}–F{{Music|half flat}}–E

;Phrygian scale: The second note of the tetrachord (in descending order) is the first of the scale.

:Diatonic: D–C–B | A–G–F–E | D
:Chromatic: D{{Music|flat}}–C–B  | A–G{{Music|flat}}–F–E | D{{Music|flat}}
:Enharmonic: D{{Music|double flat}}–C{{Music|half flat}}–B | A–G{{Music|double flat}}–F{{Music|half flat}}–E | D{{Music|double flat}}

; Lydian scale: The third note of the tetrachord (in descending order) is the first of the scale.
:Diatonic: C–B | A–G–F–E | D–C
:Chromatic: C–B | A–G{{Music|flat}}–F–E | D{{Music|flat}}–C
:Enharmonic: C{{Music|half flat}}–B | A–G{{Music|double flat}}–F{{Music|half flat}}–E | D{{Music|double flat}}–C{{Music|half flat}}

In all cases, the extreme notes of the tetrachords, E – B, and A – E, remain fixed, while the notes in between are different depending on the genus.

===Pythagorean tunings===
Here are the traditional [[Pythagorean tuning]]s of the diatonic and chromatic tetrachords:

{| style="text-align:center; margin-left: 4em;"
 |-
 !colspan=7| Diatonic
 |-
 !  hypate || &emsp;  &emsp; || parhypate || &emsp; &emsp; &emsp; &emsp; &nbsp; || lichanos || &emsp; &emsp; &emsp; &emsp; &nbsp; || mese
 |- 
 | {{sfrac|4|3}} ||   || {{sfrac|81|64}} ||  || {{sfrac|9|8}} ||      || {{sfrac|1|1}}
 |- 
 |  │  || {{sfrac|256|243}} || │  || {{sfrac|9|8}}  || │ || {{sfrac|9|8}} ||  │
 |- 
 | −498[[cents (music)|¢]] ||  || −408[[cents (music)|¢]] ||  || −204[[cents (music)|¢]] ||  || 0[[cents (music)|¢]]
 |-
 |colspan=7| {{Listen|filename=Diatonic tetrachord pythagorean tuning.mid|title=Diatonic tetrachord pythagorean tuning}}
 |}

{| style="text-align:center; margin-left: 4em;"
 |-
 !colspan=7| Chromatic
 |-
 ! hypate  ||   &emsp;  &emsp; ||  parhypate ||  &emsp; &emsp; || lichanos || &emsp; &emsp; &emsp; &emsp; &emsp; &emsp; &emsp; || mese
 |-
 | {{sfrac|4|3}} || || {{sfrac|81|64}}   || || {{sfrac|32|27}} ||  || {{sfrac|1|1}}
 |-
 |   │  || {{sfrac|256|243}} ||  │ || {{sfrac|2187|2048}}  ||  │  || {{sfrac|32|27}} || │
 |-
 | −498[[cents (music)|¢]] ||  || −408[[cents (music)|¢]] ||  ||  −294[[cents (music)|¢]] ||         ||  0[[cents (music)|¢]]
 |-
 |colspan=7| {{Listen|help=no|filename=Chromatic tetrachord pythagorean tuning.mid|title=Chromatic tetrachord pythagorean tuning}}
 |}
Here is a representative Pythagorean tuning of the enharmonic genus attributed to [[Archytas]]:

{| style="text-align:center; margin-left: 4em;"
 |- 
 !colspan=7| Enharmonic
 |- 
 !  hypate ||       || parhypate ||       || lichanos || &emsp; &emsp; &emsp; &emsp; &emsp; &emsp; &emsp; &emsp; &nbsp; || mese
 |- 
 | {{sfrac|4|3}} || || {{sfrac|9|7}} || || {{sfrac|5|4}} ||           || {{sfrac|1|1}}
 |- 
 | │ || {{sfrac|28|27}} || │ || {{sfrac|36|35}} || │ || {{sfrac|5|4}} ||  │
 |- 
 |  −498[[cents (music)|¢]] ||  || −435[[cents (music)|¢]] ||   || −386[[cents (music)|¢]]   ||    || 0[[cents (music)|¢]]
 |-
 |colspan=7| {{Listen|help=no|filename=Enharmonic tetrachord pythagorean tuning.mid|title=Enharmonic tetrachord pythagorean tuning}}
 |}

The number of strings on the classical lyre varied at different epochs, and possibly in different localities – four, seven and ten having been favorite numbers. Larger scales are constructed from conjunct or disjunct tetrachords. Conjunct tetrachords share a note, while disjunct tetrachords are separated by a ''disjunctive tone'' of 9/8 (a Pythagorean major second). Alternating conjunct and disjunct tetrachords form a scale that repeats in octaves (as in the familiar [[diatonic scale]], created in such a manner from the diatonic genus), but this was not the only arrangement.

The Greeks analyzed genera using various terms, including diatonic, enharmonic, and chromatic. Scales are constructed from conjunct or disjunct tetrachords.

{| class="wikitable" style="margin-left: 4em;"
 | Didymos’ chromatic tetrachord
 | 4:3 || (6:5) || 10:9 || (25:24) || 16:15 || (16:15) || 1:1
 | [[File:Didymos chromatic tetrachord.mid]]
 |-
 | Eratosthenes’ chromatic tetrachord
 | 4:3 || (6:5) || 10:9 || (19:18) || 20:19 || (20:19) || 1:1
 | [[File:Eratosthenes chromatic tetrachord.mid]]
 |-
 | Ptolemy’s soft chromatic
 | 4:3 || (6:5) || 10:9 || (15:14) || 28:27 || (28:27) || 1:1
 | [[File:Ptolemy soft chromatic tetrachord.mid]]
 |-
 | Ptolemy’s intense chromatic
 | 4:3 || (7:6) || 8:7 || (12:11) || 22:21 || (22:21) || 1:1
 | [[File:Ptolemy intense chromatic tetrachord.mid]]
 |-
 |Archytas’ enharmonic
 | 4:3 || (5:4) || 9:7 || (36:35) || 28:27 || (28:27) || 1:1
 | [[File:Enharmonic tetrachord pythagorean tuning.mid]]
 |}

This is a partial table of the [[superparticular interval|superparticular]] divisions by Chalmers after Hofmann.{{Who|date=July 2013}}<!-- The cited online publication says "I.E. Hofmann (Vogel 1975)", but fails to identify this mysterious source. -->{{sfn|Chalmers|1993|p=11}}