Editing Tetrachord (section)

===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}}