Re: Well-temperament interval table (Neo-Gothic 17)

--------------------------------------------------------
A 17-note well-temperament in the neo-Gothic manner:
Table of interval categories and sizes
--------------------------------------------------------

As promised, here is a table of interval categories and sizes for the
Neo-Gothic well-temperament in 17 notes which I introduced in a recent
post. For a bit of helpful context, I am reproducing a keyboard tuning
chart and a diagram of the tuning circle of fifths from the earlier
post.

Duly bidding pardon for the length of this post, and also for various
"bugs" which may well be lurking in this first draft, I nevertheless
hope that the table of interval sizes may give enjoyment to readers
curious about the structure and range of just and tempered intervals
encountered in this kind of tuning.

196:121
130.639 286.869 626.262 835.016 991.246
c#' eb' f#' g#' bb'
_130.6|78.1_78.1|130.6_ _130.6|78.1_130.6|78.1_78.1|130.6_
c' d' e' f' g' a' b' c''
1:1 14:11
0 208.754 417.508 495.623 704.377 913.131 1121.885 1200
208.754 208.754 78.115 208.754 208.754 208.754 78.115
db' d#' gb' ab' a#'
_70.1|138.7_135.03|73.7_ _65.7|143.0_74.5|134.3_139.4|69.3_
70.100 343.786 561.329 778.871 1052.558
126:121

This tuning circle may be helpful in following the generation of
different interval categories and sizes, as indicated in the table,
from the three different sizes of fifths, here specified in cents and
explained at the beginning of the table:

D
704.377 704.377
G A
704.377 704.377
C E
704.377 704.377
F B
704.377 704.377
Bb F#
704.377 704.377
Eb C#
707.988 704.377
Ab G#
708.771 708.771
Db D#
708.771 708.771
Gb A#
708.771

======================================================================
Neo-Gothic Well-Temperament for 17 notes
combining 14:11 and 9:7 hypermeantone tunings
----------------------------------------------------------------------
Note: This well-temperament uses three varieties of fifths:
11 Regular fifths (R): 4R = pure 14:11
5 Supplementary fifths (S): 4S = pure 9:7
1 Odd fifth (O): O = 10 octaves - 11R - 5S

In the "Fifths" column of the table, each interval size within a
category is derived from a combination of these three types of fifths
in its generating chain. For all intervals in a given category, this
chain conventionally follows the _shorter_ direction around the tuning
circle. This shorter chain may actually define either the given
interval itself or its octave complement.

For example, both diminished fourths (8 fifths down) and augmented
seconds (9 fifths up) are derived from a chain of 8 fifths in the
"down" or "counterclockwise" direction. For augmented seconds,
e.g. Db-E at a pure 11:9, a "Fifths" chain such as "4R-4S" (here
Db-Gb-A#-D#-G#-C#-F#-B-E) actually defines a diminished seventh,
e.g. E-Db at 18:11, i.e. 14:11 (4R) + 9:7 (4S).

Despite this complication, the convention of following the shorter
chain of fifths for all intervals within a listed category has the
advantage of permitting readier comparisons across the category. For
example, for diminished fourths and augmented seconds alike, we find
that these intervals grow smaller as the fifths in the specified
chain become larger.

----------------------------------------------------------------------
Fifths (1 fifth up) -- 16
Doubly diminished sevenths (16 fifths down) -- 1
----------------------------------------------------------------------
Number Size cents Ratio/variance Fifths Example(s)
----------------------------------------------------------------------
11 ~704.377 ~3:2 (+~2.422) 1R Eb-Bb, Bb-F, F-C, C-G,
(4R = pure 14:11) G-D, D-A, A-E, E-B,
B-F#, F#-C#, C#-G#
......................................................................
5 ~708.771 ~3:2 (+~6.816) 1S Db-Ab, Gb-Db, A#-Gb
(4S = pure 9:7) D#-A#, G#-D#
......................................................................
1 ~707.998 ~3:2 (+~6.043) 1O Ab-Eb
----------------------------------------------------------------------
Fourths (1 fifth down) -- 16
Doubly augmented seconds (16 fifths up) -- 1
----------------------------------------------------------------------
Number Size cents Ratio/variance Fifths Example(s)
----------------------------------------------------------------------
11 ~495.623 ~4:3 (-~2.422) 1R Bb-Eb, F-Bb, C-F, G-C,
D-G, A-D, E-A, B-E,
F#-B, C#-F#, G#-C#
......................................................................
5 ~491.229 ~4:3 (-~6.816) 1S Ab-Db, Db-Gb, Gb-A#
A#-D#, D#-G#
......................................................................
1 ~492.002 ~4:3 (-~6.043) 1O Eb-Ab
----------------------------------------------------------------------
Whole-tones or major seconds (2 fifths up) -- 15
Doubly diminished fourths (15 fifths down) -- 2
----------------------------------------------------------------------
10 ~208.754 ~9:8 (~+4.844) 2R C-D, D-E, F-G, G-A,
A-B, Eb-F, Bb-C, E-F#,
B-C#, F#-G#
......................................................................
1 ~212.375 ~9:8 (~+8.465) 1R-10 Ab-Bb
......................................................................
1 ~213.148 ~9:8 (~+9.238) 1R-1S C#-D#
......................................................................
1 ~216.769 ~9:8 (~+12.859) 1O-1S Db-Eb
~17:15 (~+ 0.082)
~8:7 (~-14.405)
......................................................................
4 ~217.542 ~9:8 (~+13.632) 2S Gb-Ab, A#-Db,
~17:15 (~+ 0.855) D#-Gb, G#-A#
~8:7 (~-13.632)
----------------------------------------------------------------------
Minor sevenths (2 fifths down) -- 15
Doubly augmented fifths (15 fifths up) -- 2
----------------------------------------------------------------------
10 ~991.246 ~16:9 (~-4.844) 2R D-C, E-D, G-F, A-G,
B-A, F-Eb, C-Bb, F#-E,
C#-B, G#-F#
......................................................................
1 ~987.625 ~23:13 (~-0.122) 1R-10 Bb-Ab
~16:9 (~-8.465)
......................................................................
1 ~986.852 ~23:13 (~-0.895) 1R-1S D#-C#
~16:9 (~+9.238)
......................................................................
1 ~983.231 ~7:4 (~+14.405) 1O-1S Eb-Db
~30:17 (~- 0.082)
~16:9 (~-12.859)
......................................................................
4 ~982.458 ~7:4 (~+13.632) 2S Ab-Gb, Db-A#,
~30:17 (~-0.855) D#-Gb, G#-A#
~16:9 (~+13.632)
----------------------------------------------------------------------
Minor thirds (3 fifths down) -- 14
Doubly augmented primes (14 fifths up) -- 3
----------------------------------------------------------------------
9 ~286.869 ~13:11 (~-2.341) 3R D-F, E-G, A-C, B-D,
C-Eb, G-Bb, F#-A,
C#-E, G#-B,
......................................................................
1 ~283.248 ~20:17 (~+1.890) 2R-1O Ab-C
~13:11 (~-5.962)
......................................................................
1 ~282.475 ~20:17 (~+1.117) 2R-1S D#-F#
~13:11 (~-6.735)
......................................................................
1 ~278.854 ~7:6 (~+11.983) 1R-1O-1S Gb-Bb
~20:17 (~-2.504)
......................................................................
1 ~278.081 ~7:6 (~+11.210) 1R-2S A#-C#
~27:23 (~+0.490)
~20:17 (~-3.277)
......................................................................
1 ~274.460 ~7:6 (~+7.589) 1O-2S Eb-Gb
......................................................................
3 ~273.687 ~7:6 (~+6.816) 3S Ab-A#, Db-D#, Gb-G#
----------------------------------------------------------------------
Major sixths (3 fifths up) -- 14
Doubly diminished octaves (14 fifths down) -- 3
----------------------------------------------------------------------
9 ~913.131 ~22:13 (~+2.341) 3R F-D, G-E, C-A, D-B,
Eb-C, Bb-G, A-F#,
E-C#, B-G#
......................................................................
1 ~916.752 ~22:13 (~+5.962) 2R-1O C-Ab
~17:10 (~-1.890)
......................................................................
1 ~917.525 ~13:11 (~+6.735) 2R-1S F#-D#
~17:10 (~-1.117)
......................................................................
1 ~921.146 ~17:10 (~+2.504) 1R-1O-1S Bb-Gb
~12:7 (~+11.983)
......................................................................
1 ~921.919 ~17:10 (~+3.277) 1R-2S C#-A#
~12:7 (~+11.210)
......................................................................
1 ~925.540 ~12:7 (~-7.589) 1O-2S Gb-Eb
......................................................................
3 ~926.313 ~12:7 (~+6.816) 3S A#-Ab, D#-Db, G#-Gb
----------------------------------------------------------------------
Major thirds (4 fifths up) -- 13
Doubly diminished fifths (13 fifths down) -- 4
----------------------------------------------------------------------
8 ~417.508 14:11 pure 4R C-E, F-A, G-B, Bb-D
Eb-G, D-F#, A-C#, E-G#
......................................................................
1 ~421.129 ~14:11 (~+3.621) 3R-1O Ab-C
......................................................................
1 ~421.902 ~14:11 (~+4.394) 3R-1S B-D#
......................................................................
1 ~425.523 ~14:11 (~+8.015) 2R-1O-1S Db-F
~23:18 (~+1.159)
~9:7 (~-9.561)
......................................................................
1 ~426.296 ~23:18 (~+1.932) 2R-2S F#-A#
~32:25 (~-1.077)
~9:7 (~-8.788)
......................................................................
1 ~429.917 ~32:25 (~+2.544) 1R-1O-2S Gb-Bb
~9:7 (~-5.167)
......................................................................
1 ~430.690 ~9:7 (~-4.394) 1R-3S C#-Gb
......................................................................
1 ~434.311 ~9:7 (~-0.773) 1O-3S A#-Eb
......................................................................
2 ~435.084 9:7 pure 4S G#-Db, D#-Ab
----------------------------------------------------------------------
Minor sixths (4 fifths down) -- 13
Doubly augmented fourths (13 fifths up) -- 4
----------------------------------------------------------------------
8 ~782.492 11:7 pure 4R E-C, A-F, B-G, D-Bb
G-Eb, F#-D, C#-A, G#-E
......................................................................
1 ~778.871 ~11:7 (~-3.621) 3R-1O C-Ab
......................................................................
1 ~778.098 ~11:7 (~-4.394) 3R-1S D#-B
......................................................................
1 ~774.477 ~14:9 (~+9.561) 2R-1O-1S F-Db
~11:7 (~-8.015)
......................................................................
1 ~773.704 ~14:9 (~+8.788) 2R-2S F#-A#
~25:16 (~+1.077)
......................................................................
1 ~770.083 ~14:9 (~+5.167) 1R-1O-2S Bb-Gb
~25:16 (~-2.544)
......................................................................
1 ~769.310 ~14:9 (~+4.394) 1R-3S Gb-C#
......................................................................
1 ~765.689 ~14:9 (~+0.773) 1O-3S Eb-A#
......................................................................
2 ~764.916 14:9 pure 4S Db-G#, Ab-D#
----------------------------------------------------------------------
Diatonic semitones or limmas (5 fifths down) -- 12
Commas or dieses (12 fifths up) -- 5
----------------------------------------------------------------------
7 ~78.115 ~23:22 (~+1.159) 5R E-F, B-C, A-Bb, D-Eb,
~22:21 (~-2.422) F#-G, C#-D, G#-A
......................................................................
1 ~74.494 ~24:23 (~+0.813)
~23:22 (~-2.462) 4R-1O G-Ab
......................................................................
1 ~73.721 ~24:23 (~+0.040) 4R-1S D#-E
......................................................................
1 ~70.100 126:121 pure 3R-1O-1S C-Db
~25:24 (~-0.572)
......................................................................
1 ~69.327 ~26:25 (~+1.427) 3R-2S A#-B
~25:24 (~-1.345)
......................................................................
1 ~65.706 ~27:26 (~+0.369) 2R-1O-2S F-Gb
~26.25 (~-2.194)
......................................................................
1 ~64.933 ~28:27 (~+1.972) 2R-3S Gb-F#
~27:26 (~-0.404)
......................................................................
1 ~61.312 ~29:28 (~+0.561) 1R-1O-3S Bb-A#
~28:27 (~-1.649)
......................................................................
1 ~60.539 ~30:29 (~+1.847) 1R-4S Db-C#
~29.28 (~-0.212)
......................................................................
1 ~56.918 ~31:30 (~+0.151) 10-4S Eb-D#
......................................................................
1 ~56.145 ~32:31 (~+1.180) 5S Ab-G#
~31:30 (~-0.622)
----------------------------------------------------------------------
Major sevenths (5 fifths up) -- 12
Doubly diminished ninths (12 fifths down) -- 5
----------------------------------------------------------------------
7 ~1121.885 ~21:11 (~+2.422) 5R F-E, C-B, Bb-A, Eb-D,
G-F#, D-C#, A-G#
......................................................................
1 ~1125.506 ~23:12 (~-0.813) 4R-1O Ab-G

......................................................................
1 ~1126.279 ~23:12 (~-0.040) 4R-1S E-D#
......................................................................
1 ~1129.900 121:63 pure 3R-1O-1S Db-C
~48:25 (~+0.572)
......................................................................
1 ~1130.673 ~48:25 (~+1.345) 3R-2S B-A#
~25:13 (~-1.427)
......................................................................
1 ~1134.294 ~25:13 (~-2.194) 2R-1O-2S Gb-F
......................................................................
1 ~1135.067 ~27:14 (~-1.972) 2R-3S F#-Gb
......................................................................
1 ~1138.688 ~27:14 (~+1.649) 1R-1O-3S A#-Bb
......................................................................
1 ~1139.461 ~29:15 (~-1.847) 1R-4S C#-Db
......................................................................
1 ~1143.082 ~60:31 (~-0.151) 10-4S D#-Eb
......................................................................
1 ~1143.855 ~60:31 (~+0.622) 5S G#-Ab
~31:16 (~-1.180)
----------------------------------------------------------------------
Tritones or augmented fourths (6 fifths up) -- 11
Diminished sixths (11 fifths down) -- 6
----------------------------------------------------------------------
6 ~626.262 ~10:7 (~+8.774) 6R F-B, Bb-E, Eb-A,
~23:16 (~-2.012) C-F#, D-G#, G-C#
......................................................................
1 ~629.883 ~10:7 (~+12.395) 5R-1O Ab-D
~23:16 (~+1.609)
~13:9 (~-6.735)
......................................................................
1 ~630.656 ~10:7 (~+13.168) 5R-1S A-D#
~23:16 (~+2.382)
~13:9 (~-5.962)
......................................................................
1 ~634.277 ~13:9 (~-2.341) 4R-1O-1S Db-G
......................................................................
1 ~635.050 ~13:9 (~-1.568) 4R-2S E-A#
......................................................................
1 ~638.671 ~13:9 (~+2.053) 3R-1O-2S Gb-C
......................................................................
1 ~639.444 ~13:9 (~+2.826) 3R-3S B-Gb
~16:11 (~-9.238)
......................................................................
1 ~643.065 ~13:9 (~+6.447) 2R-10-3S A#-F
~16:11 (~-5.617)
......................................................................
1 ~643.838 ~13:9 (~+7.220) 2R-4S F#-Db
~16:11 (~-4.844)
......................................................................
1 ~647.459 ~16:11 (~-1.223) 1R-1O-4S D#-Bb
......................................................................
1 ~648.232 ~16:11 (~-0.450) 1R-5S C#-Ab
......................................................................
1 ~651.853 ~16:11 (~+3.171) 1O-5S G#-Eb
(11R down)
----------------------------------------------------------------------
Diminished fifths (6 fifths down) -- 11
Augmented thirds (11 fifths up) -- 6
----------------------------------------------------------------------
6 ~573.738 ~32:23 (~+2.012) 6R B-F, E-Bb, A-Eb,
~7:5 (~-8.774) F#-C, G#-D, C#-G
......................................................................
1 ~570.117 ~18:13 (~+6.735) 5R-1O D-Ab
~32:23 (~-1.609)
~7:5 (~-12.395)
......................................................................
1 ~569.344 ~18:13 (~+5.962) 5R-1S A-D#
~32:23 (~-2.382)
~7:5 (~-13.168)
......................................................................
1 ~565.723 ~18:13 (~+2.341) 4R-1O-1S G-Db
......................................................................
1 ~564.950 ~18:13 (~+1.568) 4R-2S A#-E
......................................................................
1 ~561.329 ~18:13 (~-2.053) 3R-1O-2S C-Gb
......................................................................
1 ~560.556 ~11:8 (~+9.238) 3R-3S Gb-B
~18:13 (~-2.826)
......................................................................
1 ~556.935 ~11:8 (~+5.617) 2R-10-3S F-A#
~18:13 (~-6.447)
......................................................................
1 ~556.162 ~11:8 (~+4.844) 2R-4S Db-F#
~18:13 (~-7.220)
......................................................................
1 ~552.541 ~11:8 (~+1.223) 1R-1O-4S Bb-D#
......................................................................
1 ~551.768 ~11:8 (~+0.450) 1R-5S Ab-C#
......................................................................
1 ~548.147 ~11:8 (~+3.171) 1O-5S Eb-G#
(11R up)
----------------------------------------------------------------------
Apotomes or chromatic semitones (7 fifths up) -- 10
Diminished thirds (10 fifths down) -- 7
----------------------------------------------------------------------
5 ~130.639 ~14:13 (~+2.341) 7R Bb-B, Eb-E, F-F#,
C-C#, G-G#
......................................................................
1 ~134.260 ~14:13 (~+5.962) 6R-1O Ab-A
~27:25 (~+1.022)
~13:12 (~-4.313)
......................................................................
1 ~135.033 ~27:25 (~+1.795) 6R-1S D-D#
~13:12 (~-3.540)
......................................................................
1 ~138.654 ~13:12 (~+0.081) 5R-1O-1S Db-D
......................................................................
1 ~139.427 ~13:12 (~+0.854) 5R-2S A-A#
......................................................................
1 ~143.048 ~13:12 (~+4.475) 4R-1O-2S Gb-G
~12:11 (~-7.589)
......................................................................
1 ~143.821 ~13:12 (~+5.248) 4R-3S E-Gb
~25:23 (~-0.532)
~12:11 (~-6.816)
......................................................................
1 ~147.442 ~12:11 (~-3.195) 3R-1O-3S A#-C
......................................................................
1 ~148.215 ~12:11 (~-2.422) 3R-4S B-Db
......................................................................
1 ~151.836 ~12:11 (~+1.199) 2R-1O-4S D#-F
......................................................................
1 ~152.609 ~12:11 (~+1.972) 2R-5S F#-Ab
......................................................................
2 ~156.230 ~12:11 (~+5.593) 1R-1O-5S C#-Eb, G#-Bb
~11:10 (~-8.774) (10R down)
----------------------------------------------------------------------
Diminished octaves (7 fifths down) -- 10
Augmented sixths (10 fifths up) -- 7
----------------------------------------------------------------------
5 ~1069.361 ~13:7 (~-2.341) 7R B-Bb, E-Eb, F#-F,
C#-C, G#-G
......................................................................
1 ~1065.740 ~24:13 (~+4.313) 6R-1O A-Ab
~50:27 (~-1.022)
~13:7 (~-5.962)
......................................................................
1 ~1064.967 ~50:27 (~-1.795) 6R-1S D#-D
~24:13 (~-3.540) (~-3.540)
......................................................................
1 ~1061.346 ~24:13 (~-0.081) 5R-1O-1S D-Db
......................................................................
1 ~1060.573 ~24:13 (~-0.854) 5R-2S A#-A
......................................................................
1 ~1056.952 ~11:6 (~+7.589) 4R-1O-2S G-Gb
~24:13 (~-4.475)
......................................................................
1 ~1056.179 ~11:6 (~+6.816) 4R-3S Gb-E
~46:25 (~+0.532)
~24:13 (~-5.248)
......................................................................
1 ~1052.558 ~11:6 (~+3.195) 3R-1O-3S C-A#
......................................................................
1 ~1051.785 ~11:6 (~+2.422) 3R-4S Db-B
......................................................................
1 ~1048.164 ~11:6 (~-1.199) 2R-1O-4S F-D#
......................................................................
1 ~1047.391 ~11:6 (~-1.972) 2R-5S Ab-F#
......................................................................
2 ~1043.770 ~20:11 (~+8.774) 1R-1O-5S Eb-C#, Bb-G#
~11:6 (~-5.593) (10R up)
----------------------------------------------------------------------
Diminished fourths (8 fifths down) -- 9
Augmented seconds (9 fifths up) -- 8
----------------------------------------------------------------------
4 ~364.984 121:98 pure 8R C#-F, F#-Bb,
~21:17 (~-0.841) G#-C, B-Eb
(Note: 121:98 varies from 21:17 by a ratio of 2058:2057)
......................................................................
1 ~361.363 ~16:13 (~+1.891) 7R-1O E-Ab
~21:17 (~-4.462)
......................................................................
1 ~360.590 ~16:13 (~+1.118) 7R-1S D#-G
......................................................................
1 ~356.969 ~11:9 (~+9.561) 6R-1O-1S A-Db
~16:13 (~-2.503)
......................................................................
1 ~356.196 ~11:9 (~+8.788) 6R-2S A#-D
~16:13 (~-3.276)
......................................................................
1 ~352.575 ~11:9 (~+5.167) 5R-1O-2S D-Gb
~49:40 (~+1.237)
~16:13 (~-6.897)
......................................................................
1 ~351.802 ~11:9 (~+4.394) 5R-3S Gb-A
~49:40 (~+0.464)
......................................................................
1 ~348.181 ~11:9 (~+0.773) 4R-1O-3S G-A#
......................................................................
1 ~347.408 11:9 pure 4R-4S Db-E
(Note: 11:9 is the octave complement of 4R + 4S or 14:11 + 9:7)
......................................................................
1 ~343.787 ~11:9 (~-3.621) 3R-1O-4S C-D#
......................................................................
1 ~343.014 ~11:9 (~-4.394) 3R-5S Ab-B
......................................................................
3 ~339.393 ~17:14 (~+3.263) 2R-1O-5S Eb-F#, F-G#, Bb-C#
~28:23 (~-1.159) (9R up)
~11:9 (~-8.015)
----------------------------------------------------------------------
Augmented fifths (8 fifths up) -- 9
Diminished sevenths (9 fifths down) -- 8
----------------------------------------------------------------------
4 ~835.016 196:121 pure 8R F-C#, Bb-F#,
~Phi (~+1.926) C-G#, Eb-B
~34:21 (~+0.841)
(Note: 196:121 varies from 34:21 by a ratio of 2058:2057;
Phi is the "Golden Ratio" at ~1.61803398875, ~833.090 cents.)
......................................................................
1 ~838.637 ~13:8 (~-1.891) 7R-1O Ab-E
~34:21 (~+4.462)
......................................................................
1 ~839.410 ~13:8 (~-1.118) 7R-1S G-D#
......................................................................
1 ~843.031 ~13:8 (~+2.503) 6R-1O-1S Db-A
~18:11 (~-9.561)
......................................................................
1 ~843.804 ~13:8 (~+3.276) 6R-2S D-A#
~18:11 (~-8.788)
......................................................................
1 ~847.425 ~13:8 (~+6.897) 5R-1O-2S Gb-D
~18:11 (~-5.167)
......................................................................
1 ~848.198 ~18:11 (~-4.394) 5R-3S A-Gb

......................................................................
1 ~851.819 ~18:11 (~-0.773) 4R-1O-3S A#-G
......................................................................
1 ~852.592 18:11 pure 4R-4S E-Db
(Note: 18:11 is equal to 4R + 4S or 14:11 + 9:7)
......................................................................
1 ~856.213 ~18:11 (~+3.621) 3R-1O-4S D#-C
......................................................................
1 ~856.986 ~18:11 (~+4.394) 3R-5S B-Ab
......................................................................
3 ~860.607 ~18:11 (~+8.015) 2R-1O-5S F#-Eb, G#-F, C#-Bb
~23:14 (~+1.159) (9R down)
~28:17 (~-3.263)
----------------------------------------------------------------------
----------------------------------------------------------------------