Just intonation intervals really sweeter than 12et?

[Lorenzo]

Sorry, I had to step out for a while.

It would happen with equal temperament too, then. An equal octave is the same as a just octave. If the distance between octaves is the issue, then the problem is octave stretch. If you're trying to argue that an adjustment needs to be made to accomodate octave stretch, well and good, but it has nothing to do with the relationship between just and equal tuning.

No, an Equal octave is not the same as a Just octave if you keep the 3rds pure in the demonstration I gave earlier (F4, G#4, B4, D4, F5 / G#5). A Just Intoned octave, containing 12 semitones, ends up about 42 cents sharp by the time it reaches the last upper octave note (F4 to F5) using pure 3rds. We wish this wasn't true. But, that's why we've been forced to temper the central octave. You can keep the octaves pure in JI, within a 12 semitone chromatic scale, but the last interval is way off unless you stretch the octave by about ¼ of a step. That's unthinkable.

The stretch, beyond this central zone of our hearing range, is needed for our fallible ears/brains. We trick it into thinking what we are hearing is right, when it really isn't...according to the math or electronic measuring machines--which are far more accurate.

All you really need to do, for a practical conclusion, is hit a bass note on a pump organ. While sustaining that note (pretend this is a man humming the note in a choir), hit a harmony note, or two harmony notes, up two or three octaves from this bass drone. If it/they sound in tune, it's because the octaves have been tempered. If they sound ¼ or ½ step higher in pitch its because they haven't been tempered. Voices, strings, horns, bells and whistles not withstanding. This is called the temperament scale. A Just Intonation scale, conainting 12 semitones, would pull the upper harmony notes north by an unacceptable margin as they cross each octave. Of course an ensemble has not problem with this because it doesn't need all the combinations of 12 semitones. It finds JI easier as it's reduced to only a few intervals at a time, and ever adjusting. No responsibility coordinating with an ever-steady basic note.

Sure, a trio or choir could adjust as they go, keeping their intervals pure, but what about that bass singer who maintains the steady pitch? They can't stray from him. They have a responsibility to temper their notes accodingly and stay the right distance.

And sure, I'm talking about full scales here, rather than isolated cases, because as you maintain the base drone and check all other higher notes against it, the 3rd intervals can be pure in the first octave, whether human voice, reeds, horns, bells and whistles, or strings, but keep that bass note steady, and expect the higher notes in other octaves to be pure against it--well, that ain't going to happen unless you temper the first octave. Once you temper the first octave, you tune each note above and below it to a pure octave. This way, all octaves are always pure throughout. Not so with JI. Just intervals are too wide to fit between pure octaves. This ¼ of a step excess that accumulates within JI octaves containing, 12 semitones, is different than the almost undetectable octave stretch we talk about in ET octaves.

If you need the math to show where pure 3rds end up as you maintain the bass note and hit higher intervals, I'll get them for you tomorrow if I can locate my book.

[BoneQuint]

I have to agree with Jerry again. He's not talking about creating scales with intact perfect thirds everywhere. He's talking about creating chords from a root note, with every note in the chord in a "just intonation" relationship to that root note -- NOT necessarily to a note a third below or above it.

The goal isn't to harmonize every note with every other possible note, but to the root of the chord.

We all agree you can't make a scale where all the fifths are perfect, all the thirds are perfect, etc. But you can always pick a perfect third or a perfect fifth (or fourth, or whatever) relationship to a single note, no matter what octave it's in. Which means the E you sing in an A major chord (a fifth above the E) will be different from the E you sing in a C major chord (a third above the C).

This also means that in a "just" C major chord, there will be a perfect third between the C and the E, but in a "just" A minor chord, there won't be a perfect third between the C and the E (because the E being sung will be a fifth above the A, not a third above the C). That's because the goal isn't perfect thirds everywhere, it's being in tune with the root.

[Jerry Freeman]

Seriously, Lorenzo.

You're a smart guy. Please try harder. For years now, you've been conflating the issues in piano tuning with the relationship between just intonation and equal temperament, and you've been missing the central concept.

For the record, just intonation simply does NOT attempt to build scales by stacking pure thirds on top of pure thirds (or pure anything on top of pure anything else). You're accusing just intonation of a crime it doesn't commit. It's the crime piano tuners commit, but just intonation doesn't work that way.

That's what piano tuners start with and then adjust to compensate for the fact that it doesn't work, in making the compromise that allows the use of a single pitch in any given chord or key where several just intonation pitches would be needed to do the same jobs. Until you get that through your head, you're going to continue with your fallacy.

Several of us have posted trying to explain in terms you will understand, where you've gone off the rails with this, but you seem to be filtering the information out.

Best wishes,
Jerry

[Jerry Freeman]

I have to agree with Jerry again. He's not talking about creating scales with intact perfect thirds everywhere. He's talking about creating chords from a root note, with every note in the chord in a "just intonation" relationship to that root note -- NOT necessarily to a note a third below or above it.

The goal isn't to harmonize every note with every other possible note, but to the root of the chord.

We all agree you can't make a scale where all the fifths are perfect, all the thirds are perfect, etc. But you can always pick a perfect third or a perfect fifth (or fourth, or whatever) relationship to a single note, no matter what octave it's in. Which means the E you sing in an A major chord (a fifth above the E) will be different from the E you sing in a C major chord (a third above the C).

This also means that in a "just" C major chord, there will be a perfect third between the C and the E, but in a "just" A minor chord, there won't be a perfect third between the C and the E (because the E being sung will be a fifth above the A, not a third above the C). That's because the goal isn't perfect thirds everywhere, it's being in tune with the root.

To elaborate and clarify a little ...

In a major triad, the major third is a ratio of 5/4 to the root frequency. The fifth is a ratio of 3/2 to the root frequency.

There will also be a whole number ratio relationship between the major third and the fifth: 6/5.

So even though the notes in the chord are built on the root note, and not on each other, there will still be a harmonious, just intonation relationship among the various notes of each chord.

As the whole numbers in the ratios become larger, the harmonies will be weaker and there will be more tension, but there will always be just relationships among all the notes. This is not the case with equal temperment. The physical structure of just intonation is fundamentally stronger because you will always have components of the accoustical wave form in phase among the various notes. This, in my opinion, results in a more expressive, more powerful musicality.

And, because the physical structure of just intonation is fundamentally stronger because of the wave forms of the various notes being in phase, just intonation is easier to perform with variable pitch instruments and voices. That's one of the reasons why just intonation is not just an intellectual idea. It's actually the way music is performed by competent musicians, even if the musicians themselves know little about the theory behind it and believe they're using an equal tempered scale. If all they do is listen to their pitches and strike the pitch that sounds in tune with the other voices, they're performing in just intonation, not equal temperament.

Best wishes,
Jerry

[Jerry Freeman]

Here, by the way, is a list of some of the pitches that can be used to construct an octave. Please note that although there are numerous pitches to choose from for any given interval, there is only one octave, regardless of whether you're using just intervals or equal temperament. There is not an "equal temperament octave," a "Pythagorean octave," a "just intonationation octave," etc. Only one octave, period:

Ratio: Cents Name (if any)
1/1 0.000 tonic
32805/32768 1.954 schisma (3 to the 8th/2 to the 12th x 5/8)
126/125 13.795
121/120 14.367
100/99 17.399
99/98 17.576
81/80 21.506 syntonic comma
531441/524288 23.460 Pythagorean comma (3 to the 12th/2 to the 19th)
65/64 26.841 65th harmonic
64/63 27.264
63/62 27.700
58/57 30.109
57/56 30.642
56/55 31.194 Ptolemy's enharmonic
55/54 31.767
52/51 33.617
51/50 34.283
50/49 34.976
49/48 35.697
46/45 38.051 inferior quarter-tone (Ptolemy)
45/44 38.906
128/125 41.059 diminished second (16/15 x 24/25)
525/512 43.408 enharmonic diesis (Avicenna)
40/39 43.831
39/38 44.970 superior quarter-tone (Eratosthenes)
77/75 45.561
36/35 48.770 superior quarter-tone (Archytas)
250/243 49.166
35/34 50.184 E.T. 1/4-tone approximation
34/33 51.682
33/32 53.273 33rd harmonic
32/31 54.964 inferior quarter-tone (Didymus)
125/121 56.305
31/30 56.767 superior quarter-tone (Didymus)
30/29 58.692
29/28 60.751
57/55 61.836
28/27 62.961 inferior quarter-tone (Archytas)
80/77 66.170
27/26 65.337
26/25 67.900 1/3-tone (Avicenna)
51/49 69.259
126/121 70.100
25/24 70.672 minor 5-limit half-step
24/23 73.681
117/112 75.612
23/22 76.956
67/64 79.307 67th harmonic
22/21 80.537 hard 1/2-step (Ptolemy, Avicenna, Safiud)
21/20 84.467
81/77 87.676
20/19 88.801
256/243 90.225 Pythagorean half-step
58/55 91.946
135/128 92.179 limma ascendant
96/91 92.601
19/18 93.603
55/52 97.104
128/121 97.364
18/17 98.955 E.T. half-step approximation
2 to the 1/12th 100.000 equal-tempered half-step
35/33 101.867
52/49 102.876
86/81 103.698
17/16 104.955 overtone half-step
33/31 108.237
49/46 109.377
16/15 111.731 major 5-limit half-step
31/29 115.458
77/72 116.234
15/14 119.443 Cowell just half-step
29/27 123.712
14/13 128.298
69/64 130.229 69th harmonic
55/51 130.721
27/25 133.238 alternate Renaissance half-step
121/112 133.810
13/12 138.573 3/4-tone (Avicenna)
64/59 140.828
38/35 142.373
63/58 143.159
88/81 143.498
25/23 144.353
62/57 145.568
135/124 147.143
49/45 147.428
12/11 150.637 undecimal "median" 1/2-step
59/54 153.307
35/32 155.140 35th harmonic
23/21 157.493
57/52 158.940
34/31 159.920
800/729 160.897
56/51 161.915
11/10 165.004
54/49 168.213
32/29 170.423
21/19 173.268
31/28 176.210
567/512 176.646
51/46 178.636
71/64 179.697 71st harmonic
10/9 182.404 minor whole-tone
49/44 186.334
39/35 187.343
29/26 189.050
125/112 190.115
48/43 190.437
19/17 192.558
160/143 194.468
28/25 196.198
121/108 196.771
55/49 199.980
2 to the 1/6th 200.000 equal-tempered whole-tone
64/57 200.532
9/8 203.910 major whole-tone
62/55 207.404
44/39 208.835
35/31 210.104
26/23 212.253
112/99 213.598
17/15 216.687
25/22 221.309
58/51 222.667
256/225 223.463
33/29 223.696
729/640 225.416
57/50 226.841
73/64 227.789 73rd harmonic
8/7 231.174 septimal whole-tone
63/55 235.104
55/48 235.677
39/34 237.527
225/196 238.886
31/27 239.171
147/128 239.607
169/147 241.449
23/20 241.961
2187/1900 243.545
38/33 244.240
144/125 244.969 diminished third (6/5 x 24/25)
121/105 245.541
15/13 247.741
52/45 250.304
37/32 251.344 37th harmonic
81/70 252.680
125/108 253.076
22/19 253.805
51/44 255.592
196/169 256.596 consonant interval (Avicenna)
29/25 256.950
36/31 258.874
93/80 260.677
57/49 261.816
64/55 262.368
7/6 266.871 septimal minor third
90/77 270.080
75/64 274.582 augmented second (9/8 x 25/24)
34/29 275.378
88/75 276.736
27/23 277.591
20/17 281.358
33/28 284.447
46/39 285.792
13/11 289.210
58/49 291.925
45/38 292.711
32/27 294.135 Pythagorean minor third
19/16 297.513 overtone minor third
2 to the 1/4th 300.000 equal-tempered minor third
25/21 301.847
31/26 304.508
105/88 305.777
55/46 309.357
6/5 315.641 5-limit minor third
77/64 320.144 77th harmonic
35/29 325.562
29/24 327.622
75/62 329.547
98/81 329.832
121/100 330.008
23/19 330.761
63/52 332.208
40/33 333.041
17/14 336.130
243/200 337.148
62/51 338.125
28/23 340.552
39/32 342.483 39th harmonic
128/105 342.905
8000/6561 343.301
11/9 347.408 undecimal "median" third
60/49 350.617
49/40 351.338
38/31 352.477
27/22 354.547
16/13 359.472
79/64 364.537 79th harmonic
100/81 364.807
121/98 364.984
21/17 365.825
99/80 368.914
26/21 369.747
57/46 371.194
31/25 372.408
36/29 374.333
56/45 378.602
96/77 381.811
8192/6561 384.360 Pythagorean "schismatic" third
5/4 386.314 5-limit major third
64/51 393.090
49/39 395.169
44/35 396.178
39/31 397.447
34/27 399.090
2 to the 1/3rd 400.000 equal-tempered major third
63/50 400.108
121/96 400.681
29/23 401.303
125/99 403.713
24/19 404.442
512/405 405.866
62/49 407.384
81/64 407.820 Pythagorean major third
19/15 409.244
33/26 412.745
80/63 413.578
14/11 417.508
51/40 420.597
125/98 421.289
23/18 424.364
32/25 427.373 diminished fourth
41/32 429.062 41st harmonic
50/39 430.145
77/60 431.875
9/7 435.084 septimal major third
58/45 439.353
49/38 440.139
40/31 441.278
31/24 443.081
1323/1024 443.517
128/99 444.772
22/17 446.363
57/44 448.150
162/125 448.879
35/27 449.275
83/64 450.047 83rd harmonic
100/77 452.484
13/10 454.214
125/96 456.986 augmented third (5/4 x 25/24)
30/23 459.994
64/49 462.348
98/75 463.069
17/13 464.428
72/55 466.278
55/42 466.851
38/29 467.936
21/16 470.781 septimal fourth
46/35 473.135
25/19 475.114
320/243 476.539
29/22 478.259
675/512 478.492
33/25 480.646
45/34 485.286
85/64 491.269 85th harmonic
4/3 498.045 perfect fourth
2 to the 5/12ths 500.000 equal-tempered perfect fourth
75/56 505.757
51/38 509.397
43/32 511.518 43rd harmonic
121/90 512.412
39/29 512.905
35/26 514.612
66/49 515.621
31/23 516.761
27/20 519.551
23/17 523.319
42/31 525.745
19/14 528.687
110/81 529.812
87/64 531.532 87th harmonic
34/25 532.328
49/36 533.742
15/11 536.951
512/375 539.104
26/19 543.015
63/46 544.462
48/35 546.815
1000/729 547.211
11/8 551.318 undecimal tritone (11th harmonic)
62/45 554.812
40/29 556.737
29/21 558.796
112/81 561.006
18/13 563.382
25/18 568.717 augmented fourth (4/3 x 25/24)
89/64 570.880 89th harmonic
32/23 571.726
39/28 573.657
46/33 575.001
88/63 578.582
7/5 582.512 septimal tritone
108/77 585.721
1024/729 588.270 low Pythagorean tritone
45/32 590.224 high 5-limit tritone
38/27 591.648
31/22 593.718
55/39 595.149
24/17 597.000
Square root of 2 600.000 equal-tempered tritone
99/70 600.088
17/12 603.000
44/31 606.282
125/88 607.623
27/19 608.352
91/64 609.354 91st harmonic
64/45 609.776 low 5-limit tritone
729/512 611.730 high Pythagorean tritone
57/40 613.154
77/54 614.279
10/7 617.488 septimal tritone
63/44 621.418
33/23 624.999
56/39 626.343
23/16 628.274 23rd harmonic
36/25 631.283 diminished fifth (3/2 x 24/25)
121/84 631.855
49/34 632.696
13/9 636.618
81/56 638.994
55/38 640.119
42/29 641.204
29/20 643.263
45/31 645.188
93/64 646.991 93rd harmonic
16/11 648.682
51/35 651.771
729/500 652.789
35/24 653.185
19/13 656.985
375/256 660.896
22/15 663.049
47/32 665.507 47th harmonic
72/49 666.258
25/17 667.672
81/55 670.188
28/19 671.313
31/21 674.255
189/128 674.691
34/23 676.681
40/27 680.449 dissonant "wolf" 5-limit fifth
46/31 683.239
95/64 683.827 95th harmonic
49/33 684.379
52/35 685.388
58/39 687.095
125/84 688.160
112/75 694.243
121/81 694.816
2 to the 7/12ths 700.000 equal-tempered perfect fifth
3/2 701.955 perfect fifth
121/80 716.322
50/33 719.354
97/64 719.895 97th harmonic
1024/675 721.508
44/29 721.741
243/160 723.461
38/25 724.886
35/23 726.865
32/21 729.219
29/19 732.064
84/55 733.149
55/36 733.722
26/17 735.572
75/49 736.931
49/32 737.652 49th harmonic
23/15 740.006
192/125 743.014 diminished sixth (8/5 x 24/25)
20/13 745.786
77/50 747.516
54/35 750.725
125/81 751.121
17/11 753.637
99/64 755.228 99th harmonic
48/31 756.919
31/20 758.722
45/29 760.674
14/9 764.916 septimal minor sixth
120/77 768.125
39/25 769.855
25/16 772.627 augmented fifth
36/23 775.636
11/7 782.492 undecimal minor sixth
63/40 786.422
52/33 787.255
101/64 789.854 101st harmonic
30/19 790.756
128/81 792.180 Pythagorean minor sixth
49/31 792.616
405/256 794.134
19/12 795.558
46/29 798.697
100/63 799.892
2 to the 2/3rds 800.000 equal-tempered minor sixth
27/17 800.910
62/39 802.553
35/22 803.822
51/32 806.910 51st harmonic
8/5 813.686 5-limit minor sixth
6561/4096 815.640 Pythagorean "schismatic" sixth
77/48 818.189
45/28 821.398
103/64 823.801 103rd harmonic
29/18 825.667
50/31 827.592
121/75 828.053
21/13 830.253
55/34 832.676
34/21 834.175
81/50 835.193
125/77 838.797
13/8 840.528 overtone sixth
57/35 844.328
44/27 845.453
31/19 847.523
80/49 848.662
49/30 849.383
18/11 852.592 undecimal "median" sixth
105/64 857.095 105th harmonic
64/39 857.517
23/14 859.448
51/31 861.875
400/243 862.852
28/17 863.870
33/20 866.959
38/23 869.239
81/49 870.168
48/29 872.378
53/32 873.505 53rd harmonic
58/35 874.438
63/38 875.223
128/77 879.856
107/64 889.760 107th harmonic
5/3 884.359 5-limit major sixth
57/34 894.513
52/31 895.492
42/25 898.153
121/72 898.726
2 to the 3/4ths 900.000 equal-tempered major sixth
32/19 902.487
27/16 905.865 Pythagorean major sixth
49/29 908.075
22/13 910.790
39/23 914.208
56/33 915.553
17/10 918.642
109/64 921.821 109th harmonic
46/27 922.409
75/44 923.264
29/17 924.622
128/75 925.418 diminished seventh (16/9 x 24/25)
77/45 929.920
12/7 933.129 septimal major sixth
55/32 937.632 55th harmonic
31/18 941.126
441/256 941.562
50/29 943.050
19/11 946.195
216/125 946.924
121/70 947.496
45/26 949.696
26/15 952.259
111/64 953.299 111th harmonic
125/72 955.031 augmented sixth (5/3 x 25/24)
33/19 955.760
40/23 958.039
54/31 960.829
96/55 964.323
110/63 964.896
7/4 968.826 septimal minor seventh
58/33 976.304
225/128 976.537
51/29 977.333
44/25 978.691
30/17 983.313
113/64 984.215 113th harmonic
99/56 986.402
23/13 987.747
62/35 989.896
39/22 991.165
55/31 992.596
16/9 996.090 Pythagorean small min. seventh
57/32 999.468 57th harmonic
2 to the 5/6ths 1000.000 equal-tempered minor seventh
98/55 1000.020
25/14 1003.802
34/19 1007.442
52/29 1010.950
88/49 1013.666
115/64 1014.588 115th harmonic
9/5 1017.596 5-limit large minor seventh
56/31 1023.790
38/21 1026.732
29/16 1029.577 29th harmonic
49/27 1031.787
20/11 1034.996
51/28 1038.085
729/400 1039.103
31/17 1040.080
42/23 1042.507
117/64 1044.438 117th harmonic
64/35 1044.860
4000/2187 1045.256
11/6 1049.363 undecimal "median" seventh
90/49 1052.572
57/31 1054.432
46/25 1055.647
81/44 1056.502
35/19 1057.627
59/32 1059.172 59th harmonic
24/13 1061.427
50/27 1066.762
63/34 1067.780
13/7 1071.702
119/64 1073.781 119th harmonic
54/29 1076.288
28/15 1080.557
58/31 1084.542
15/8 1088.269 5-limit major seventh
62/33 1091.763
32/17 1095.045
49/26 1097.124
66/35 1098.133
2 to the 11/12ths 1100.000 equal-tempered major seventh
17/9 1101.045
121/64 1102.636 121st harmonic
125/66 1105.668
36/19 1106.397
256/135 1107.821
55/29 1108.054
243/128 1109.775 Pythagorean major seventh
19/10 1111.199
40/21 1115.533
61/32 1116.885 61st harmonic
21/11 1119.463
44/23 1123.044
23/12 1126.319
48/25 1129.328
121/63 1129.900
123/64 1131.017 123rd harmonic
25/13 1132.100
77/40 1133.830
52/27 1134.663
27/14 1137.039 septimal major seventh
56/29 1139.249
29/15 1141.308
60/31 1143.233
31/16 1145.036 31st harmonic
64/33 1146.727
33/17 1148.318
243/125 1150.834
35/18 1151.230
39/20 1156.169
125/64 1158.941 augmented seventh (15/8 x 25/24)
88/45 1161.094
45/23 1161.949
96/49 1164.303
49/25 1165.024
51/26 1166.383
108/55 1168.233
55/28 1168.806
57/29 1169.891
63/32 1172.736 63rd harmonic
160/81 1178.494
99/50 1182.601
125/63 1186.205
127/64 1186.422 127th harmonic
2/1 1200.000 octave


Source: http://www.kylegann.com/Octave.html

Best wishes,
Jerry

[Lorenzo]

I have to agree with Jerry again. He's not talking about creating scales with intact perfect thirds everywhere. He's talking about creating chords from a root note, with every note in the chord in a "just intonation" relationship to that root note -- NOT necessarily to a note a third below or above it.

...

But you can always pick a perfect third or a perfect fifth (or fourth, or whatever) relationship to a single note, no matter what octave it's in.

Sometimes Jerry is talking about JI as related to one interval, or two intervals as with a 3-note chord, or 3 intervals including the octave note--which BTW is two 3rds and one 4th. Other times he claims a piano can be tuned to JI, although he did finally concede that fixed chromatic instruments producing chords don't lend themselves to JI. Other times it's a choir or ensemble--not specifying how many intervals are needed.

As I mentioned above, if you isolate an interval, or a triad with 2 intervals, they can always be kept pure. A trio can sing all over the place, wandering from key to key, or even ending up a ¼ step above where they started, while maintaining JI throughout. There's no responsibility to a sustained basic when that is done. No particular distance is required to be maintained except betweeen the intervals of these three notes.

But, if the 5-note diminished chord I mentioned above (F4, G#4, B4, D4, F5) is sung, or played, the intervals cannot all be kept pure. Something has to give. That many pure 3rd intervals are too wide to fit inside pure octaves, so therefore JI doesn't always work within an octave. And this is a legitimate chord used all the time with both ensembles and pianos.

[Jerry Freeman]

I have to agree with Jerry again. He's not talking about creating scales with intact perfect thirds everywhere. He's talking about creating chords from a root note, with every note in the chord in a "just intonation" relationship to that root note -- NOT necessarily to a note a third below or above it.

...

But you can always pick a perfect third or a perfect fifth (or fourth, or whatever) relationship to a single note, no matter what octave it's in.

Sometimes Jerry is talking about JI as related to one interval, or two intervals as with a 3-note chord, or 3 intervals including the octave note--which BTW is two 3rds and one 4th. Other times he claims a piano can be tuned to JI, although he did finally concede that fixed chromatic instruments producing chords don't lend themselves to JI. Other times it's a choir or ensemble--not specifying how many intervals are needed.

As I mentioned above, if you isolate an interval, or a triad with 2 intervals, they can always be kept pure. A trio can sing all over the place, wandering from key to key, or even ending up a ¼ step above where they started, while maintaining JI throughout. There's no responsibility to a sustained basic when that is done. No particular distance is required to be maintained except betweeen the intervals of these three notes.

But, if the 5-note diminished chord I mentioned above (F4, G#4, B4, D4, F5) is sung, or played, the intervals cannot all be kept pure. Something has to give. That many pure 3rd intervals are too wide to fit inside pure octaves, so therefore JI doesn't always work within an octave. And this is a legitimate chord used all the time with both ensembles and pianos.

No, JI always works within an octave or across octaves or whatever.

Your example is misleading. You can create any chord, using just intonation intervals, such that by stacking pure intervals the way you propose, the further away you go from the root note, the larger the whole numbers in the whole number ratios for the pitch created.

Eventually, the whole number ratio will be constructed of such large integers that the note will sound more dissonant than consonant. If you stack a pure third on a pure third on a pure third, the ratio is 625/384, by the way, which is by far comprised of higher intergers than any but a few Pythagorean intervals. You could call it a "wolf" note if you like, except that wolf notes are unintentional and unavoidable without making compromises away from pure intervals. Your wolf note is both intentional and avoidable without making compromises away from pure intervals.

Your highest note in the chord you've invented isn't out of tune at all, and there's nothing about it that doesn't fit within an octave. All you've done is manufactured a perfectly in tune pitch that hardly anyone would use because there's so much dissonance. But it isn't out of tune. It's just a microtonal pitch that's unfamiliar and that most people wouldn't consider musical sounding. The fact that you can create a pitch like that doesn't prove your point.

You're still not getting the basic concept. Please try harder to get it, and please consider the possibility that you really have got this wrong.

Best wishes,
Jerry

[Jerry Freeman]

But, if the 5-note diminished chord I mentioned above (F4, G#4, B4, D4, F5) is sung, or played, the intervals cannot all be kept pure. Something has to give. That many pure 3rd intervals are too wide to fit inside pure octaves, so therefore JI doesn't always work within an octave. And this is a legitimate chord used all the time with both ensembles and pianos.

Most certainly, the intervals can be kept pure.

Please pick the five limit just intonation pitches that correspond to the intervals you want, and you will produce a five note diminished chord that will work across any number of octaves, and without anything having to give. I think you'll find, that just intoned five note diminished chord will be closer to what competent ensembles and choirs will perform, while pianos will play the approximated pitches necessitated by the fact that they are limited in the pitches they can produce.

The problem, still, is that you're trying to derive pitches by stacking intervals in a way that's artificial and only applies to the exercise of trying to tune fixed pitch instruments. The fact that you can produce an acceptable diminished chord by making the compromises you're accustomed to doesn't mean you can't construct a better sounding diminished chord using intervals that maintain a precise, rather than approximate, harmonic relationship.

Best wishes,
Jerry

[Jerry Freeman]

This is an interesting, although extremely frustrating exercise.

It would be much easier to simply explain how just intonation works, have you understand it on the first try, and then leave it at that.

However, in this instance, because pianos is your starting point and frame of reference, it seems we have to work in reverse.

We have to untangle the whole, artificial process of tuning a piano that compensates for the fact that a piano can't adjust pitches to maintain pure intervals across chords and keys, before we can begin to get across to you that we're talking about a relatively simple set of principles.

Just intonation is simple, and it works. But it doesn't work with a fixed pitch instrument like a piano. Tuning pianos is complicated, but the complications you're describing don't arise from the principles of just intonation; they arise from what happens when you can't follow the principles of just intonation because you can't produce an unlimited selection of pitches and are forced, as you are with pianos, to select a limited, and therefore arbitrary set of pitches.

Best wishes,
Jerry

[Lorenzo]

If I remember correctly, in order to keep a pure octave, the minor thirds have to be narrowed by about 16 cents, and the major thirds widened by about 14 cents. With the diminished chord F4, G#4, B4, D4, F5 you have four minor 3rd intervals within an octave, including the two octave notes themselves. In this case, it doesn't matter if it's a piano or human voices producing the tones.

I'll have to find my book to make sure about these 3rds and get the precise ratios.

[Jerry Freeman]

If I remember correctly, in order to keep a pure octave, the minor thirds have to be narrowed by about 16 cents, and the major thirds widened by about 14 cents. With the diminished chord F4, G#4, B4, D4, F5 you have four minor 3rd intervals within an octave, including the two octave notes themselves. In this case, it doesn't matter if it's a piano or human voices producing the tones.

I'll have to find my book to make sure about these 3rds and get the precise ratios.

Again, you're stacking intervals, rather than building the chord based on the root note. Give me a few minutes, and I'll give you some more information.

Best wishes,
Jerry

[Nanohedron]

This is interesting: it seems that equal temperament was sought after and measured (among theorists, at least) in Tang China in the matter of tuning bells that were meant to be played in ensembles!

http://www.cechinatrans.demon.co.uk/ctm-psm.html

From the footnotes:

"3. In the time of the Northern and Southern Dynasties 南北朝 Qian Lezhi went to the 360th pitch, for example. The point soon comes however when any difference from the octave is too small to be audible. The argument was one among theoreticians, not performers."

An impressive and beautiful chime of bronze and stone bells from the Warring States period, the reproductions tuned to 12-tone temperament (53et was apparently known, but it is unclear from the article as to what temperament was employed on the originals):

http://www.china.org.cn/english/feature ... /78688.htm

[Lorenzo]

You may call it "stacking" intervals, Jerrry, but the only difference between stacking four intervals and stacking two intervals in a triad is simply adding two more necessary intervals to make up the Fdim chord. There's nothing artificial or "invented" about this common chord. This basic F chord has been around since the foundations of the world were laid.

[Jerry Freeman]

Here you go.

F4 (root note) 1/1
G#4 6/5 (minor third)
B4 25/18 (augmented fourth)
D4 5/3 (major sixth)
F5 2/1 (octave)

You can strike any of those notes in any octave you like by multiplying or dividing the frequency by any multiple of 2. The chord will remain pure, and nothing will need to be adjusted or approximated. Nothing will have to give.

Best wishes,
Jerry

[Jerry Freeman]

This basic F chord has been around since the foundations of the world were laid.

Not true at all. The basic chord has been around since the foundations of the world were laid, but your equal tempered approximation of it is a fairly recent invention. The version I've posted is based in the laws of physics, and certainly has been around forever. It would have been discovered at some point in history, whereas your version can only have been invented, and at a much later point in history.

Best wishes,
Jerry