10:12:15: Difference between revisions

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{{Infobox Chord|ColorName=gu or g}}
{{Infobox Chord|ColorName=gu or g}}


'''10:12:15''' is the classical [[minor triad]], and can also be referred to as the '''Ptolemaic minor triad'''. It is found on the iii ({{Frac|5|4}}) and vi ({{Frac|5|3}}) of Ptolemy's intense diatonic scale ([[Zarlino]]), which is perhaps the most common [[5-limit]] diatonic.  Unlike [[27:32:40]], which appears on the ii of the same scale, 10:12:15 is [[utonal]].
'''10:12:15''' is a utonal [[minor triad]], known as the '''just minor triad''', '''classical minor triad''', or '''Ptolemaic minor triad'''. It is a common chord that appears as a counterpart to the widespread otonal major triad of [[4:5:6]], with the major third replaced by a minor third. It occurs as a minor triad in the [[5L 2s|diatonic]] scale of [[meantone]] temperament, as the minor third formed from stacking three [[4/3|perfect fourths]] octave reduced, [[32/27]], is equated with [[6/5]] due to vanishing of [[81/80]].


However, there are other 5-limit diatonic scales which don't have the Ptolemaic minor triad occurring in all the same places.  For instance, [[User:Aura|Aura]] is known to use a diatonic minor scale in which this chord only occurs on the i and iv scale degrees while using a Pythagorean minor triad (that is, [[54:64:81]]) on the v.  Conversely, in the diatonic major scale that Aura uses, this chord only really appears on the iii.  Compared to its Pythagorean counterpart, the Ptolemaic major triad sounds like it's more consonant.  Because of these properties, the Ptolemaic major triad has earned its status as a bread-and-butter chord in 5-limit harmony.
It is not the only minor triad in [[just intonation]]; other minor triads include [[54:64:81]] (Pythagorean minor triad), [[6:7:9]] (septimal minor triad), and [[16:19:24]].


There are a number of possible tetrads which can be reasonably built off of this triad, such as [[10:12:15:18]] in the 5-limit, as well as [[70:84:105:120]] in the 7-limit and [[110:132:165:192]] in the 11-limit.
== Audio of close voicings ==
[[File:SculpEufaDem10-12-15-onD.mp3|none|thumb|10:12:15 Root position]]
[[File:SculpEufaDem12-15-20-onD.mp3|none|thumb|12:15:20 1st inversion]]
[[File:SculpEufaDem15-20-24-onD.mp3|none|thumb|15:20:24 2nd inversion]]
 
== Notable voicings ==
Voicings are sorted by [[Wilson norm]]. This list is only a brief overview, see [[Voicings of 10:12:15]] for a more comprehensive list.
{| class="wikitable"
|+
!Voices
![[EFR]]
![[Kite's thoughts on hi-lo notation|Hi-lo name]]
!Special properties
|-
| rowspan="6" |3 voices
|3:5:15
|lo3hi5
|AOV ([[Odd limit#Proposed extensions|all-odd voicing]])
|-
|6:10:15
|lo3
|CAOV (condensed AOV)
|-
|10:12:15
|basic
|
|-
|10:15:24
|hi3
|
|-
|12:15:20
|hiR
|1st inversion
|-
|15:20:24
|lo5
|2nd inversion
|-
| rowspan="2" |4 voices
|10:12:15:20
|add8
|
|-
|10:15:20:24
|hi3add8
|
|}
 
== Related chords ==
Melodic inversion: [[4:5:6]].
 
Plausible [[Chord homonym|homonyms]]: 12:15:20 (1/1 -  5/4 - 5/3), a maj6no5 chord.
 
Notable extensions (5-limit except where noted):
 
* [[10:12:15:18]] adds [[9/5]]
* [[30:36:45:50]] adds [[5/3]]
* [[70:84:105:120]] (1/(12:10:8:7)) – adds 12/7 (7-limit)
* [[60:70:84:105]] (1/(7:6:5:4)) – a homonym of the previous chord


[[Category:Minor triads|#@]] <!-- 2-digit first number -->
[[Category:Minor triads|#@]] <!-- 2-digit first number -->

Latest revision as of 09:32, 30 December 2025

Chord information
Harmonics 10:12:15
Subharmonics 1/(6:5:4)
Intervals from root 1/1–6/5–3/2
Cents from root 0¢–316¢–702¢
Step intervals 6/5, 5/4
Step cents 316¢, 386¢
Color name gu or g
Prime limit 5
Genus 3⋅5 (15)
Intervallic odd limit 5
Otonal odd limit 15
Utonal odd limit 5
Consistent edos (d ≥ 2) 3edo*, 12edo*, 15edo*, 19edo**, …

10:12:15 is a utonal minor triad, known as the just minor triad, classical minor triad, or Ptolemaic minor triad. It is a common chord that appears as a counterpart to the widespread otonal major triad of 4:5:6, with the major third replaced by a minor third. It occurs as a minor triad in the diatonic scale of meantone temperament, as the minor third formed from stacking three perfect fourths octave reduced, 32/27, is equated with 6/5 due to vanishing of 81/80.

It is not the only minor triad in just intonation; other minor triads include 54:64:81 (Pythagorean minor triad), 6:7:9 (septimal minor triad), and 16:19:24.

Audio of close voicings

10:12:15 Root position
12:15:20 1st inversion
15:20:24 2nd inversion

Notable voicings

Voicings are sorted by Wilson norm. This list is only a brief overview, see Voicings of 10:12:15 for a more comprehensive list.

Voices EFR Hi-lo name Special properties
3 voices 3:5:15 lo3hi5 AOV (all-odd voicing)
6:10:15 lo3 CAOV (condensed AOV)
10:12:15 basic
10:15:24 hi3
12:15:20 hiR 1st inversion
15:20:24 lo5 2nd inversion
4 voices 10:12:15:20 add8
10:15:20:24 hi3add8

Related chords

Melodic inversion: 4:5:6.

Plausible homonyms: 12:15:20 (1/1 - 5/4 - 5/3), a maj6no5 chord.

Notable extensions (5-limit except where noted):

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