Clavier tuning optimization: Difference between revisions

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The minor adjusments of the syntonic comma resulted in a slightly stretched octave to accentuate the 3rd and 5th harmonics and to avoid the 7th and up. The 7th can be clearly heard in less mainstream music like jazz and blues and is often referred to as "the blue note". Wind instruments easily accomodate higher harmonics because without tuning they play notes from the harmonic series. Other instruments like claviers are not as accomodating, e.g. most piano's are not tuned to play the blue note.
The minor adjusments of the syntonic comma resulted in a slightly stretched octave to accentuate the 3rd and 5th harmonics and to avoid the 7th and up. The 7th can be clearly heard in less mainstream music like jazz and blues and is often referred to as "the blue note". Wind instruments easily accomodate higher harmonics because without tuning they play notes from the harmonic series. Other instruments like claviers are not as accomodating, e.g. most piano's are not tuned to play the blue note.


This article presents a tuning scale based on the microtonal 28/27 ratio, originating from a 19-keyed piano. The scale uses this ratio to create distances of two (E-F, B-C) or three (C-D, D-E, F-G, G-A, A-B) quarter tones, rather than the usual semitones. The 3 quartertone intervals leave room for the black keys to be used for the 18/17 interval relative to the key before it. C#/Db is a pure 18/17 tone, D#/Eb becomes (28/27)^3 * (18/17). Note how the factor 9 is present in both 27 and 18, creating optimal consonance despite using the high prime number 17.
This article presents a tuning scale based on the microtonal 28/27 ratio, originating from a 19-keyed piano. The scale uses this ratio to create distances of two (E-F, B-C) or three (C-D, D-E, F-G, G-A, A-B) quarter tones, rather than the usual semitones.  


Octaves are very slightly narrowed using this scale, but are closer to an actual 1:2 ratio. All other just ratio's like C:G = 2:3 and C:F = 3:4 are very well maintained. The C and following A are closer together, so the scale as a whole is tuned to A-438.  
Octaves are very slightly narrowed using this scale, but are closer to an actual 1:2 ratio. All other just ratio's like C:G = 2:3 and C:F = 3:4 are very well maintained. The C and following A are closer together, so the scale as a whole is tuned to A-438.  
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The calculated frequencies for a five octave keyboard:
The calculated frequencies for a five octave keyboard:


C2 66.09554941
C2 66,0955494142999
69.98352291
70,4678514975879
73.71485550
73,7148554967589
78.05102347
78,5911840712824
82.21249341
82,2124934138525
88.41508208
88,4150820801927
93.61596926
94,2638487671923
98.60732012
98,6073201150427
104.40775071
105,130315914109
109.97449023
109,974490228391
116.44357789
117,249438345096
122.65203523
122,652035233127
C3 131.90561814
C3 131,905618138233
139.66477215
140,63133740517
147.11132090
147,111320904866
155.76492802
156,842916157004
164.06989364
164,069893639365
176.44828068
176,448280676629
186.82759130
188,120551987352
196.78873431
196,788734309473
208.36454221
209,806551705855
219.47397732
219,473977318577
232.38421128
233,99245150876
244.77431032
244,774310323497
C4 263.24150795
C4 263,241507947355
278.72630253
280,655258249839
293.58723683
293,58723682672
310.85707429
313,008394507975
327.43113462
327,431134624812
352.13444382
352,134443821473
372.84823463
375,428571428571
392.72749636
392,727496355684
415.82911379
418,706904435299
A4 438.00000000
A4 438
463.76470588
466,97424001238
488.49138851
488,491388507849
C5 525.34602001
C5 525,346020013928
556.24872707
560,098306939574
585.90640814
585,906408136244
620.37149097
624,66483940139
653.44802476
653,448024762836
702.74794433
702,747944326561
744.08605870
749,235587265204
783.75871940
783,758719395246
829.86217348
835,605324983272
874.10818514
874,10818514274
925.52631368
931,931519282263
974.87287915
974,872879147151
C6 1048.42295919
C6 1048,42295919255
1110.09489797
1117,77743054902
1169.28216228
1169,2821622819
1238.06346595
1246,63161893066
1304.07366897
1304,07366897386
1402.46057130
1402,46057129699
1484.95825196
1495,23506718891
1564.13221872
1564,13221872232
1656.13999629
1667,60149341721
1744.44091172
1744,44091172039
1847.05508300
1859,83782876058
1945.53507565
1945,53507565341
C7 2092.31755736
C7 2092,31755735565


Note that this is a work in progress. Samples for a 61-key clavier are available at [https://space.nurdspace.nl/~cu64/samples.zip]  
 
Note that this is a work in progress. Samples for a 61-key clavier are available at [https://space.nurdspace.nl/~cu64/samplesv10.zip] (Last updated to v1.0 as of 18-3-2019)
Mathematical equations for each key will be made available.
Mathematical equations for each key will be made available.

Revision as of 19:30, 18 March 2019

An equally tempered 12-key scale for clavier instruments that is optimized for more usable intervals and an overall groovy sound.
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An equally tempered 12-key scale for clavier instruments that is optimized for more usable intervals and an overall groovy sound. Property "Tool Image" (as page type) with input value "File:{{{Picture}}}" contains invalid characters or is incomplete and therefore can cause unexpected results during a query or annotation process. {{{Picture}}} {{#if:No | [[Tool Owner::{{{ProjectParticipants}}} | }} {{#if:No | [[Tool Cost::{{{Cost}}} | }}

Music is all about patterns. From rhythmic patterns to melodies built from harmonic intervals, mathematics are everywhere. This article illustrates a possible tuning scale which is, like all tuning scales, built around prime numbers.

The goal is to better tune any clavier to create better sounding intervals. Current standard tuning methods revolve around the 81/80 ratio, also known as the syntonic comma. The origin lies with Pythagorean scales, which contained only the 2nd and 3rd harmonics. Musicians found that slightly adjusting some intervals created a much more consonant sound with more usable intervals.

The minor adjusments of the syntonic comma resulted in a slightly stretched octave to accentuate the 3rd and 5th harmonics and to avoid the 7th and up. The 7th can be clearly heard in less mainstream music like jazz and blues and is often referred to as "the blue note". Wind instruments easily accomodate higher harmonics because without tuning they play notes from the harmonic series. Other instruments like claviers are not as accomodating, e.g. most piano's are not tuned to play the blue note.

This article presents a tuning scale based on the microtonal 28/27 ratio, originating from a 19-keyed piano. The scale uses this ratio to create distances of two (E-F, B-C) or three (C-D, D-E, F-G, G-A, A-B) quarter tones, rather than the usual semitones.

Octaves are very slightly narrowed using this scale, but are closer to an actual 1:2 ratio. All other just ratio's like C:G = 2:3 and C:F = 3:4 are very well maintained. The C and following A are closer together, so the scale as a whole is tuned to A-438.

The calculated frequencies for a five octave keyboard:

C2 66,0955494142999 70,4678514975879 73,7148554967589 78,5911840712824 82,2124934138525 88,4150820801927 94,2638487671923 98,6073201150427 105,130315914109 109,974490228391 117,249438345096 122,652035233127 C3 131,905618138233 140,63133740517 147,111320904866 156,842916157004 164,069893639365 176,448280676629 188,120551987352 196,788734309473 209,806551705855 219,473977318577 233,99245150876 244,774310323497 C4 263,241507947355 280,655258249839 293,58723682672 313,008394507975 327,431134624812 352,134443821473 375,428571428571 392,727496355684 418,706904435299 A4 438 466,97424001238 488,491388507849 C5 525,346020013928 560,098306939574 585,906408136244 624,66483940139 653,448024762836 702,747944326561 749,235587265204 783,758719395246 835,605324983272 874,10818514274 931,931519282263 974,872879147151 C6 1048,42295919255 1117,77743054902 1169,2821622819 1246,63161893066 1304,07366897386 1402,46057129699 1495,23506718891 1564,13221872232 1667,60149341721 1744,44091172039 1859,83782876058 1945,53507565341 C7 2092,31755735565


Note that this is a work in progress. Samples for a 61-key clavier are available at [1] (Last updated to v1.0 as of 18-3-2019) Mathematical equations for each key will be made available.