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→Quartal Chords: added modes to symmetric pentads table with links to demos
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{| class="wikitable"
{| class="wikitable"
|+Tertian triads of the Pinetone diatonic on D
|+Tertian triads of the Pinetone diatonic on D (dorian symmetric minor)
!Root note
!Root note
!Triad notes
!Triad notes
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Porcupine tetrads in the table below are named after the third above the tonic and the third above the fifth, apart from tetrads with a diminished fifth.
Porcupine tetrads in the table below are named after the third above the tonic and the third above the fifth, apart from tetrads with a diminished fifth.
{| class="wikitable"
{| class="wikitable"
|+Tertian tetrads of the Pinetone diatonic on D
|+Tertian tetrads of the Pinetone diatonic on D (dorian symmetric minor)
!Root note
!Root note
!Triad notes
!Triad notes
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Also of interest are the quartal triads of the Pinetone diatonic. We describe these as stacked 3-step intervals (fourths) of the scale, with major and minor designating the large and small 3-step intervals (fourths) respectively. This may seem an odd way to describe quartal chords, but it is consistent with the naming scheme I introduce for the Porcupine[7] 3-step (quartal) triads along side the quartal triads of the Pinetone diatonic.
Also of interest are the quartal triads of the Pinetone diatonic. We describe these as stacked 3-step intervals (fourths) of the scale, with major and minor designating the large and small 3-step intervals (fourths) respectively. This may seem an odd way to describe quartal chords, but it is consistent with the naming scheme I introduce for the Porcupine[7] 3-step (quartal) triads along side the quartal triads of the Pinetone diatonic.
{| class="wikitable"
{| class="wikitable"
|+3-step stacked triads of the Pinetone diatonic on D
|+3-step stacked triads of the Pinetone diatonic on D (dorian symmetric minor)
!Root note
!Root note
!Triad notes
!Triad notes
Line 657: Line 657:
|6:8:11
|6:8:11
|}
|}
3-step pentads ala the quartal voicings of modern jazz harmony may also be of interest. Two octaves are spanned by four 3-step intervals and a 2-step remainder. Given that Pinetone cannot be generated by 4/3 perfect fourths like Meantone can these pentads have much more variety in Pinetone than in Meantone.
We can see that compared to tertian (2-step stacked) chords, quartal (3-step stacked) chords are much more evenly spread in the Pinetone diatonic. 3-step pentads ala the quartal voicings of modern jazz harmony may also be of interest. Two octaves are spanned by four 3-step intervals and a 2-step remainder. Given that Pinetone cannot be generated by 4/3 perfect fourths like Meantone can these pentads have much more variety in Pinetone than in Meantone. As in modern jazz, variety is found through utilizing all inversions of the pentad. The root position and 2nd inversion are shown, where the second inversion spans an octave with a symmetric pattern of two 3-step intervals on either side of a 2-step interval. Root position pentads on A and B (2nd inversion on G and A) cover the notes of the [[The Pinetone System#Pinetone pentatonic|Pinetone major and minor pentatonics]] respectively, introduced below.  
{| class="wikitable"
{| class="wikitable"
|+3-step stacked pentads of the Pinetone diatonic on D
|+3-step stacked pentads of the Pinetone diatonic on D
Line 665: Line 665:
!Porcupine intervals
!Porcupine intervals
!Pinetone intervals
!Pinetone intervals
!JI chord approximated by triad
!JI Intervals
!JI Intervals
!JI chord approximated by triad
|-
|-
!D
!D
|D-G-C-F-B
|D-G-C-F-B
|P4, P4, P4, A4
|P4, P4, P4, A4, (m3)
|m4, M4, m4, M4
|m4, M4, m4, M4, (m3)
|m4, Mm4, m4, M4
|m4, mM4, m4, M4, (m3)
|4/3, 27/20, 4/3, 11/8
|15:20:27:36:50
|30:40:54:72:99
|4/3, 27/20, 4/3, 25/18, (6/5)
|-
|-
!E
!E
|E-A-D-G-C
|E-A-D-G-C
|P4, P4, P4, P4
|P4, P4, P4, P4, (M3)
|M4, m4, m4, M4
|M4, m4, m4, M4, (m3)
|Mm4, m4, m4, Mm4
|mM4, m4, m4, mM4, (Mm3)
|27/20, 4/3, 4/3, 11/8
|20:27:36:48:65
|20:27:36:48:66
|27/20, 4/3, 4/3, 65/48, (16/13)
|-
|-
!F
!F
|F-B-E-A-D
|F-B-E-A-D
|A4, P4, P4, P4
|A4, P4, P4, P4, (m3)
|M4, m4, M4, m4
|M4, m4, M4, m4, (m3)
|M4, m4, Mm4, m4
|M4, m4, mM4, m4, (m3)
|11/8, 4/3, 15/11, 4/3
|24:33:44:60:80
|24:33:44:60:80
|11/8, 4/3, 15/11, 4/3, (6/5)
|-
|-
!G
!G
|G-C-F-B-E
|G-C-F-B-E
|P4, P4, A4, P4
|P4, P4, A4, P4, (m3)
|M4, m4, M4, m4
|M4, m4, M4, m4, (m3)
|Mm4, m4, M4, m4
|mM4, m4, M4, m4, (m3)
|27/20, 4/3, 11/8, 4/3
|20:27:36:50:66
|40:54:72:99:132
|27/20, 4/3, 25/18, 33/25, (40/33)
|-
|-
!A
!A
|A-D-G-C-F
|A-D-G-C-F
|P4, P4, P4, P4
|P4, P4, P4, P4, (M3)
|m4, m4, M4, m4
|m4, m4, M4, m4, (M3)
|m4, m4, Mm4, m4
|m4, m4, mM4, m4, (M3)
|4/3, 4/3, 27/20, 4/3
|25:33:44:60:80
|45:60:80:108:144
|33/25, 4/3, 15/11, 4/3, (5/4)
|-
|-
!B
!B
|B-E-A-D-G
|B-E-A-D-G
|P4, P4, P4, P4
|P4, P4, P4, P4, (M3)
|m4, M4, m4, m4
|m4, M4, m4, m4, (M3)
|m4, Mm4, m4, m4
|m4, mM4, m4, m4, (M3)
|4/3, 27/20, 4/3, 4/3
|15:20:27:36:48
|15:20:27:36:48
|4/3, 27/20, 4/3, 4/3, (5/4)
|-
|-
!C
!C
|C-F-B-E-A
|C-F-B-E-A
|P4, A4, P4, P4
|P4, A4, P4, P4, (m3)
|m4, M4, m4, M4
|m4, M4, m4, M4, (m3)
|m4, M4, m4, Mm4
|m4, M4, m4, mM4, (m3)
|4/3, 11/8, 4/3, 15/11
|18:24:33:44:60
|18:24:33:44:60
|4/3, 11/8, 4/3, 15/11, (6/5)
|}
{| class="wikitable"
|+2nd inversion (symmetric) 3-step stacked pentads of the Pinetone diatonic on D
!Root note
!Mode
!Notes
!Meantone intervals
!Porcupine intervals
!Pinetone intervals
!JI chord approximated by triad
!JI Intervals
|-
!D
|[https://xenpaper.com/#%7B1%2F1_174.055c_320.690c_494.745c_704.524c_878.579c_1025.214c_1199.269c'%7D%5B0_3_6%5D------._%5B0_3_6_9_12%5D------._%5B0_3_6_9_11%5D------._%5B0_3_6_8_11_14%5D------._%5B0_3_5_8_11%5D------._%5B0_2_5_8_11%5D------. Dorian symmetric minor]
|D-G-C-E-A-D
|P4, P4, M3, P4, P4
|m4, M4, m3, M4, m4
|m4, mM4, Mm3, mM4, m4
|15:20:27:33:45:60
|4/3, 27/20, 11/9, 15/11, 4/3
|-
!E
|[https://xenpaper.com/#%7B1%2F1_146.635c_320.690c_530.469c_704.524c_851.159c_1025.214c_1199.269c'%7D%5B0_3_6%5D------._%5B0_3_6_9_12%5D------._%5B0_3_6_9_11%5D------._%5B0_3_6_8_11_14%5D------._%5B0_3_5_8_11%5D------._%5B0_2_5_8_11%5D------. Phrygian bright minor]
|E-A-D-F-B-E
|P4, P4, m3, A4, P4
|M4, m4, m3, M4, m4
|mM4, m4, m3, M4, m4
|11:15:20:24:33:44
|15/11, 4/3, 6/5, 11/8, 4/3
|-
!F
|[https://xenpaper.com/#%7B1%2F1_174.055c_383.834c_557.888c_704.524c_878.579c_1052.633c_1199.269c'%7D%5B0_3_6%5D------._%5B0_3_6_9_12%5D------._%5B0_3_6_9_11%5D------._%5B0_3_6_8_11_14%5D------._%5B0_3_5_8_11%5D------._%5B0_2_5_8_11%5D------. Lydian dark major]
|F-B-E-G-C-F
|A4, P4, m3, P4, P4
|M4, m4, m3, M4, m4
|M4, m4, m3, mM4, m4
|18:25:33:40:54:72
|25/18, 33/25, 40/33, 27/20, 4/3
|-
!G
|[https://xenpaper.com/#%7B1%2F1_209.779c_383.834c_530.469c_704.524c_878.579c_1025.214c_1199.269c'%7D%5B0_3_6%5D------._%5B0_3_6_9_12%5D------._%5B0_3_6_9_11%5D------._%5B0_3_6_8_11_14%5D------._%5B0_3_5_8_11%5D------._%5B0_2_5_8_11%5D------. Mixolydian bright major]
|G-C-F-A-D-G
|P4, P4, M3, P4, P4
|M4, m4, M3, m4, m4
|mM4, m4, M3, m4, m4
|11:15:20:25:33:44
|15/11, 4/3, 5/4, 33/25, 4/3
|-
!A
|[https://xenpaper.com/#%7B1%2F1_174.055c_320.690c_494.745c_668.800c_815.435c_989.490c_1199.269c'%7D%5B0_3_6%5D------._%5B0_3_6_9_12%5D------._%5B0_3_6_9_11%5D------._%5B0_3_6_8_11_14%5D------._%5B0_3_5_8_11%5D------._%5B0_2_5_8_11%5D------. Aeolian magical seventh]
|A-D-G-B-E-A
|P4, P4, M3, P4, P4
|m4, m4, M3, m4, M4
|m4, m4, M3, m4, mM4
|27:36:48:60:80:108
|4/3, 4/3, 5/4, 4/3, 27/20
|-
!B
|[https://xenpaper.com/#%7B1%2F1_174.055c_320.690c_494.745c_668.800c_815.435c_989.490c_1199.269c'%7D%5B0_3_6%5D------._%5B0_3_6_9_12%5D------._%5B0_3_6_9_11%5D------._%5B0_3_6_8_11_14%5D------._%5B0_3_5_8_11%5D------._%5B0_2_5_8_11%5D------. Locrian dark diminished]
|B-E-A-C-F-B
|P4, P4, m3, P4, A4
|m4, M4, m3, m4, M4
|m4, mM4, m3, m4, M4
|33:44:60:72:96:132
|4/3, 15/11, 6/5, 4/3, 11/8
|-
!C
|[https://xenpaper.com/#%7B1%2F1_174.055c_348.110c_494.745c_668.800c_878.579c_1052.633c_1199.269c'%7D%5B0_3_6%5D------._%5B0_3_6_9_12%5D------._%5B0_3_6_9_11%5D------._%5B0_3_6_8_11_14%5D------._%5B0_3_5_8_11%5D------._%5B0_2_5_8_11%5D------. Ionian bright diminished]
|C-F-B-D-G-C
|P4, A4, m3, P4, P4
|m4, M4, m3, m4, M4
|m4, M4, m3, m4, mM4
|27:36:50:60:80:108
|4/3, 25/18, 6/5, 4/3, 27/20
|}
|}
The mode names link to sequenced examples of the root position 3-step triads, and to root position, 1st inversion, 2nd inversion (with double octave), 3rd inversion, and 4th inversion 3-step pentads, on the tonic of each mode, tuned to [http://x31eq.com/cgi-bin/rt.cgi?ets=1ce_4p_2ce&limit=2_3_5_11 TE 2.3.5.11 Ptolemismic].


== The Pinetone pentatonic and the Pinetone chromatic ==
== The Pinetone pentatonic and the Pinetone chromatic ==
Retrieved from "https://en.xen.wiki/w/Pinetone"