I agree! Amazing how irrationality is part of the system. Our heart is not in the center of the chest, so to speak.BOB Cooper wrote:in some cases (or, I guess, most if not all cases) perceptions "prefer" tunings or pulses that are not exact (at some point). Irregularity is much richer in information /more pleasurable (again, at some point, and I mean very fine-grained irregularities).
biological systems, brains, experiences of perception... are dynamic systems : they are not mechanistically tuned to some sort of abstract predefined / fixed values. it's far more chaotic and (then) contextual !
Tone Deft wrote:kind of off topic, tuning guitars...
I've always preferred to trust a tuner over my ears, tune it and forget it. lately I've used a tuner to get close then finish it off by ear and I seem to be getting better tunings. am I crazy in thinking this?
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This is a technical article that may be boring to many, so feel free to skip it.
There was a time that I was quite interested in alternate tunings and I made all sorts of unusual scales. I had an Ensoniq VFX which allowed one to make tuning tables.
I was particularly interested in making scales based on mathematical constants and their ratios.
I devised a method to make them more easily manageable by easily finding simple fractional intervals that are indiscernible to the exact, often irrational or transcendental, mathematical values. I used a rule of thumb value of about 3 cents difference as indiscernible in pitch. That corresponds to a frequency ratio of 501:500.
[ For a frequency ratio R: Cents(R) = 3986.313713865 log R ]
It involves something called expansion by continued fractions. It is easiest to show by example.
Suppose we have a frequency interval whose ratio is 1.265625... What is a small fraction equivalent ratio?
Let's start by truncating the value to 1.265625.
Then we expand it into a rational fraction:
1.265625 = 1 + 265625/1000000
The last fraction can be written into a similar form:
265625/1000000 = 1/(1000000/265625)
= 1/(3 + 203125/265625)
and this last fraction similarly is transformed:
203125/265625 = 1/(265625/203125)
= 1/(1 + 62500/203125)
Here it is shown in complete form:
1.265625 = 1 + 265625/1000000 =
1 + 1/(1000000/265625) =
1 + 1/(3 + 203125/265625) =
1 + 1/(3 + 1/(265625/203125)) =
1 + 1/(3 + 1/(1 + 62500/203125)) =
1 + 1/(3 + 1/(1 + 4/13)) =
1 + 1/(3 + 1/(1 + 1/(13/4))) =
1 + 1/(3 + 1/(1 + 1/(3 + 1/4)))
Now we form first, second, and third approximations until there is no discernible difference:
first approx: 1 + 1/3 = 4:3 = 1.3333.. = 498 cents
second: 1 + 1/(3+1) = 5:4 = 1.25 = 386 cents
third: 1 + 1/(3 + 1/(1+1/3) = 1 + 4/15 = 19:15 = 1.2666 = 409 cents
fourth: complete term 81:64 = 1.265625 = 408 cents
Since there is no discernible different between 408 and 409 cents, to the ear 81:64 = 19:15.
So we can use 19:15 intervals in tuning a scale based on that number.Code: Select all
And here is a named table of common western intervals in ascending order:
Interval name interval value cents in interval
--------------- --------------- -----------------
unison 1:1 0.000
jnd 501:500 3.459 just noticeable difference
savart ~436:435 3.986
a skilled player can stretch a note on an instrument
by as much as 20 cents to be in tune with other instruments
syntonic comma 80:81 21.506
Pythagorean comma 531441:524288 23.460 3**12/2**19
diesis 128:125 41.059
smaller chromatic semitone
25:24 70.672
Pythagorean diatonic semitone = limma
256:243 90.225 2**8/3**5
larger chromatic semitone (small limma)
135:128 92.179
minor second 18:17 98.955
semitone 16:15 111.731
Pythagorean chromatic semitone
2187:2048 113.685 3**7/2**11
meantone semitone 8:(5*fourth_root(5)) 117.108
13:12 138.573
12:11 150.637
11:10 165.004
65536:59049 180.450
minor tone 10:9 182.404
meantone whole tone sqrt(5):2 193.157
449:400 200.059
major tone (Pythagorean whole tone) ---
9:8 203.910 3**2/2**3 (two fifths)
256:225 223.462 (two semitones)
8:7 231.174
pi:e 250.561 3.14159../2.71828...
7:6 266.871
13:11 289.210
32:27 294.135
minor third 6:5 315.641
19683:16384 317.595
11:9 347.408
100:81 364.807 (two minor tones)
8192:6561 384.360
major third 5:4 386.314
81:64 407.820 (two major tones)
9:7 435.084
13:10 454.214
perfect fourth 4:3 498.045
11:8 551.318
7:5 582.512
1024:729 588.270
augmented fourth 45:32 590.224
(also called a tritone = two major tones + 1 minor tone)
diminished fifth 64:45 609.777
729:512 611.730
10:7 617.488
36:25 631.283 (two minor thirds)
13:9 636.618
262144:177147 678.495
perfect fifth 3:2 701.955
25:16 772.628 (two major thirds)
11:7 782.492
128:81 792.180
minor sixth 8:5 813.686
6561:4096 815.640
phi:1 833.090 phi = 1.608033988749
13:8 840.528
32768:19683 882.405
major sixth 5:3 884.359
27:16 905.865
12:7 933.129
harmonic minor seventh 7:4 968.826
grave minor seventh 16:9 996.090 (two perfect fourths)
minor seventh 9:5 1017.596
59049:32768 1019.550
11:6 1049.363
13:7 1071.702
4096:2187 1086.315
major seventh 15:8 1088.269
243:128 1109.775
1048576:531441 1176.540
octave 2:1 1200.000
for the first few minutes. Oddly I found the same. I friend who is a guitarist (and plays with synths alot) regularly tunes his guitar just from his head - ie no tuner, just from his memory of the right pitch. So randomly I tried doing the same with mine one day when I was telling a mate about him doing this without ever expecting to get even vaguely close - much to my suprise - I was exactly on - really exactly on as close as could be determined with a good digital tuner (ie within a +/- a cent), and repeatably after detuning it.Tone Deft wrote:kind of off topic, tuning guitars...
I've always preferred to trust a tuner over my ears, tune it and forget it. lately I've used a tuner to get close then finish it off by ear and I seem to be getting better tunings. am I crazy in thinking this?
Perhaps a bit more relaxing, so probably great for Trance/Ambient etc?