{{Short description|Diatonic scale tuning sequence proposed by Ptolemy}} [[Image:Diatonic scale on C.png|thumb|right|330px|Diatonic scale on C, equal tempered {{audio|Diatonic scale on C.mid|Play}} and Ptolemy's intense or just {{audio|Just diatonic scale on C.mid|Play}}.]]
'''Ptolemy's intense diatonic scale''', also known as the '''Ptolemaic sequence''',<ref>Partch, Harry (1979). ''[[Genesis of a Music]]'', pp. 165, 173. {{ISBN|978-0-306-80106-8}}.</ref> '''justly tuned major scale''',<ref> Murray Campbell, Clive Greated (1994). ''The Musician's Guide to Acoustics'', pp. 172–73. {{ISBN|978-0-19-816505-7}}.</ref><ref> Wright, David (2009). ''Mathematics and Music'', pp. 140–41. {{ISBN|978-0-8218-4873-9}}.</ref><ref> Johnston, Ben and Gilmore, Bob (2006). "A Notation System for Extended Just Intonation" (2003), ''"Maximum clarity" and Other Writings on Music'', p. 78. {{ISBN|978-0-252-03098-7}}.</ref> '''Ptolemy's tense diatonic scale''', or the '''syntonous''' (or '''syntonic''') '''diatonic scale''', is a [[musical tuning|tuning]] for the [[diatonic scale]] proposed by [[Ptolemy]],<ref>see {{cite book|first=John|last=Wallis|title=Opera Mathematica, Vol. III|publisher=Oxford|year=1699|page=39}} (Contains ''Harmonics'' by Claudius Ptolemy.)</ref>, created only with intervals from [[5-limit]] [[just intonation]].<ref name="EB">Chisholm, Hugh (1911). ''[https://books.google.com/books?id=vf8tAAAAIAAJ&dq=ptolemy%27s+intense+diatonic+scale&pg=PA961 The Encyclopædia Britannica]'', Vol.28, p. 961. The Encyclopædia Britannica Company.</ref> While Ptolemy is famous for this version of just intonation, it is important to realize this was only one of several genera of just, diatonic intonations he describes. He also describes [[7-limit tuning|7-limit "soft"]] diatonics and an [[11-limit interval|11-limit]] "even" diatonic.
This tuning was declared by [[Gioseffo Zarlino|Zarlino]] to be the only tuning that could be reasonably sung, it was also supported by [[Giuseppe Tartini]],<ref>Dr. Crotch (October 1, 1861). "[https://archive.org/stream/jstor-3355208/3355208#page/n1/mode/2up On the Derivation of the Scale, Tuning, Temperament, the Monochord, etc.]", ''The Musical Times'', p. 115.</ref> and is equivalent to [[Sapta Svara|Indian Gandhar tuning]] which features exactly the same intervals.
It is produced through a [[tetrachord]] consisting of a [[greater tone]] (9:8), [[lesser tone]] (10:9), and [[just diatonic semitone]] (16:15).<ref name="EB"/> This is called Ptolemy's intense diatonic tetrachord (or "tense"), as opposed to Ptolemy's soft diatonic tetrachord (or "relaxed"), which is formed by [[septimal chromatic semitone|21:20]], 10:9 and 8:7 intervals.<ref> Chalmers, John H. Jr. (1993). [http://eamusic.dartmouth.edu/~larry/published_articles/divisions_of_the_tetrachord/ ''Divisions of the Tetrachord'']. Hanover, NH: Frog Peak Music. {{ISBN|0-945996-04-7}} Chapter 2, Page 9</ref>
==Structure== The structure of the intense diatonic scale is shown in the tables below, where T is for greater tone, t is for lesser tone and s is for semitone:
{| class="wikitable" style="text-align: center" ! rowspan="5" | Note ! style="width:6em;" | Name ! colspan="2" style="width: 3em" | C ! colspan="2" style="width: 3em" | D ! colspan="2" style="width: 3em" | E ! colspan="2" style="width: 3em" | F ! colspan="2" style="width: 3em" | G ! colspan="2" style="width: 3em" | A ! colspan="2" style="width: 3em" | B ! style="width: 3em" | C |- ! style="width:6em;" | Solfege | colspan="2" | Do | colspan="2" | Re | colspan="2" | Mi | colspan="2" | Fa | colspan="2" | Sol | colspan="2" | La | colspan="2" | Ti | Do |- ! Ratio from C | colspan="2" | [[unison|1:1]] | colspan="2" | 9:8 | colspan="2" | [[major third|5:4]] | colspan="2" | [[perfect fourth|4:3]] | colspan="2" | [[perfect fifth|3:2]] | colspan="2" | [[major sixth|5:3]] | colspan="2" | [[major seventh|15:8]] | [[octave|2:1]] |- ! Harmonic | colspan="2" | {{audio|Unison on C.mid|24}} | colspan="2" | {{audio|Major tone on C.mid|27}} | colspan="2" | {{audio|Just major third on C.mid|30}} | colspan="2" | {{audio|Just perfect fourth on C.mid|32}} | colspan="2" | {{audio|Just perfect fifth on C.mid|36}} | colspan="2" | {{audio|Just major sixth on C.mid|40}} | colspan="2" | {{audio|Just major seventh on C.mid|45}} | {{audio|Perfect octave on C.mid|48}} |- ! Cents | colspan="2" | 0 | colspan="2" | 204 | colspan="2" | 386 | colspan="2" | 498 | colspan="2" | 702 | colspan="2" | 884 | colspan="2" | 1088 | 1200 |- ! rowspan="3" | Step ! Name | rowspan=3 style="width:1.5em" | | colspan="2" style="width:3em" | T | colspan="2" style="width:3em" | t | colspan="2" style="width:3em" | s | colspan="2" style="width:3em" | T | colspan="2" style="width:3em" | t | colspan="2" style="width:3em" | T | colspan="2" style="width:3em" | s |- ! Ratio | colspan="2" | 9:8 | colspan="2" | 10:9 | colspan="2" | 16:15 | colspan="2" | 9:8 | colspan="2" | 10:9 | colspan="2" | 9:8 | colspan="2" | 16:15 |- ! Cents | colspan="2" | 204 | colspan="2" | 182 | colspan="2" | 112 | colspan="2" | 204 | colspan="2" | 182 | colspan="2" | 204 | colspan="2" | 112 |} {| class="wikitable" style="text-align: center" ! rowspan="4" | Note ! style="width:6em;" | Name ! colspan="2" style="width: 3em" | A ! colspan="2" style="width: 3em" | B ! colspan="2" style="width: 3em" | C ! colspan="2" style="width: 3em" | D ! colspan="2" style="width: 3em" | E ! colspan="2" style="width: 3em" | F ! colspan="2" style="width: 3em" | G ! style="width: 3em" | A |- ! Ratio from A | colspan="2" | 1:1 | colspan="2" | 9:8 | colspan="2" | 6:5 | colspan="2" | 4:3 | colspan="2" | 3:2 | colspan="2" | 8:5 | colspan="2" | 9:5 | 2:1 |- ! Harmonic of Fundamental B{{music|b}} | colspan="2" | 120 | colspan="2" | 135 | colspan="2" | 144 | colspan="2" | 160 | colspan="2" | 180 | colspan="2" | 192 | colspan="2" | 216 | 240 |- !Cents | colspan="2" | 0 | colspan="2" | 204 | colspan="2" | 316 | colspan="2" | 498 | colspan="2" | 702 | colspan="2" | 814 | colspan="2" | 1018 | 1200 |- ! rowspan="3" | Step ! Name | rowspan="3" style="width:1.5em" | | colspan="2" style="width:3em" | T | colspan="2" style="width:3em" | s | colspan="2" style="width:3em" | t | colspan="2" style="width:3em" | T | colspan="2" style="width:3em" | s | colspan="2" style="width:3em" | T | colspan="2" style="width:3em" | t |- ! Ratio | colspan="2" | 9:8 | colspan="2" | 16:15 | colspan="2" | 10:9 | colspan="2" | 9:8 | colspan="2" | 16:15 | colspan="2" | 9:8 | colspan="2" | 10:9 |- ! Cents | colspan="2" | 204 | colspan="2" | 112 | colspan="2" | 182 | colspan="2" | 204 | colspan="2" | 112 | colspan="2" | 204 | colspan="2" | 182 |}
==Comparison with other diatonic scales== Ptolemy's intense diatonic scale can be constructed by lowering the pitches of [[Pythagorean tuning]]'s 3rd, 6th, and 7th [[Degree (music)|degrees]] (in C, the notes E, A, and B) by the [[syntonic comma]], 81:80. This scale may also be considered as derived from the just major chord (ratios 4:5:6, so a major third of 5:4 and fifth of 3:2), and the major chords a fifth below and a fifth above it: FAC–CEG–GBD. This perspective emphasizes the central role of the tonic, dominant, and subdominant in the diatonic scale.
In comparison to [[Pythagorean tuning]], which only uses 3:2 perfect fifths (and fourths), the Ptolemaic provides just thirds (and sixths), both major and minor (5:4 and 6:5; sixths 8:5 and 5:3), which are smoother and more easily tuned than Pythagorean thirds (81:64 and 32:27) and Pythagorean sixths (27:16 and 128/81),<ref> {{cite book |last1 = Johnston |first1 = Ben |last2 = Gilmore |first2 = Bob |year = 2006 |title = 'Maximum Clarity' and Other Writings on Music |page = 100 |ISBN = 978-0-252-03098-7 }} </ref> with one minor third (and one major sixth) left at the Pythagorean interval, at the cost of replacing one fifth (and one fourth) with a wolf interval.
Intervals between notes ([[Wolf interval |wolf intervals]] bolded): {| class="wikitable" style="text-align:center;vertical-align:center" !    || C || D || E || F || G || A || B || C′ || D′ || E′ || F′ || G′ || A′ || B′ || C″ |- ! C | 1:1 || 9:8 || 5:4 || 4:3 || 3:2 || 5:3 || 15:8 || 2:1 || 9:4 || 5:2 || 8:3 || 3:1 || 10:3 || 15:4 || 4:1 |- ! D | 8:9 || 1:1 || 10:9 ||'''32:27'''|| 4:3 ||'''40:27'''|| 5:3 || 16:9 || 2:1 || 20:9 ||'''64:27'''|| 8:3 ||'''80:27'''|| 10:3 || 32:9 |- ! E | 4:5 || 9:10 || 1:1 || 16:15 || 6:5 || 4:3 || 3:2 || 8:5 || 9:5 || 2:1 || 32:15 || 12:5 || 8:3 || 3:1 || 16:5 |- ! F | 3:4 ||'''27:32'''|| 15:16 || 1:1 || 9:8 || 5:4 || '''45:32''' || 3:2 ||'''27:16'''|| 15:8 || 2:1 || 9:4 || 5:2 || '''45:16''' || 3:1 |- ! G | 2:3 || 3:4 || 5:6 || 8:9 || 1:1 || 10:9 || 5:4 || 4:3 || 3:2 || 5:3 || 16:9 || 2:1 || 20:9 || 5:2 || 8:3 |- ! A | 3:5 ||'''27:40'''|| 3:4 || 4:5 || 9:10 || 1:1 || 9:8 || 6:5 ||'''27:20'''|| 3:2 || 8:5 || 9:5 || 2:1 || 9:4 || 12:5 |- ! B | 8:15 || 9:15 || 2:3 || '''32:45''' || 4:5 || 8:9 || 1:1 || 16:15 || 6:5 || 4:3 || '''64:45''' || 8:5 || 16:9 || 2:1 || 32:15 |- ! C′ | 1:2 || 9:16 || 5:8 || 2:3 || 3:4 || 5:6 || 15:16 || 1:1 || 9:8 || 5:4 || 4:3 || 3:2 || 5:3 || 15:8 || 2:1 |}
[[Image:Pythagorean diatonic scale on C.png|thumb|right|330px|Pythagorean diatonic scale on C {{audio|Pythagorean diatonic scale on C.mid|Play}}. Johnston's notation; + indicates the [[syntonic comma]].]]
Note that D–F is a [[Pythagorean minor third]] or semiditone (32:27), its inversion F–D is a [[Pythagorean major sixth]] (27:16); D–A is a [[wolf fifth]] (40:27), and its inversion A–D is a wolf fourth (27:20). All of these differ from their just counterparts by a [[syntonic comma]] (81:80). More concisely, the triad built on the 2nd degree (D) is out-of-tune.
F-B is the [[tritone]] (more precisely, an augmented fourth), here 45:32, while B-F is a diminished fifth, here 64:45.
==References== {{reflist|25em}}
{{Musical tuning}} {{Scales}}
[[Category:5-limit tuning and intervals]] [[Category:Heptatonic scales]] [[Category:Ptolemy]]