Yahoo Tuning Groups Ultimate Backup tuning An improved 79-tone scale

This was my original 79 MOS 159-tET proposition:

1. Take 1/33 of log (4/3) times 1200 divided by (log 2) =

498.045 divided by 33 =

15.0923 cents

2. Multiply the ~15.1 cent comma 79 times:

15.0923 x 1 = 15.0923 cents
15.0923 x 2 = 30.1845 cents
15.0923 x 3 = 45.2768 cents
etc...
15.0923 x 79 = 1192.29 cents

3. When the 79th pitch is completed to 1200 cents, A larger comma appears between the 78th-79th steps:

(1200-1192.29) + 15.0923 =

7.71 + 15.0923 =

22.80273 cents

4. When this larger comma is carried between the 45th-46th steps by key transposing the scale to the 33rd degree, a pure fifth at the 46th step is obtained:

(15.0923 x 45) + 22.80273 =

679.1523 + 22.80273 =

701.955001 cents =

log (3/2) times 1200 divided by (log 2)

5. Thus, a pleasing closed sytem of 79 tones made up of a chain of 33 meantone and 46 pure fifths is produced. Moreover, the ~709 cent Super-Pythagorean fifths allow for non-interrupted modulations from Rast to Suz-i Dilara at every key.

However, my argument with Paul Erlich proved that the above-mentioned meantone fifth was worse than the 19-tET fifth by half a cent:

1200 bölü 19 çarpı 11 = 694.737 sent >

15.0923 x 46 = 694.244 sent

This argument occupied me until recently, where I came up with an intriguing solution without compromising the structure of my proposed scale. Here it is:

1. Multiply a 2/7 comma meantone fifth by 33:

2/7 of log (81/80) times 1200 divided by (log 2) =

21.50629 divided by 7 and multiplied by 2 =

6.1446542 cents

log (3/2) times 1200 divided by (log 2) - 6.1446542 =

701.955001 - 6.1446542 =

695.8103467 cents

times 33 =

22961.741441 cents

2. Add to this number 46 times the pure fifth:

log (3/2) times 1200 divided by (log 2) and multiplied by 46 =

46 x 701.955001 =

32289.93004 cents

plus 22961.741441 =

55251.6715 cents

3. Originally, this number was 55200 cents, or in other words, 46 octaves above the reference tone. The error is now as large as a quarter-tone. To temper out the error, we need to temper the octaves thusly:

55251.6715 divided by 46 =

1201.1233 cents

4. Following the same pattern in the original proposal, and reducing all tones into one octave by the amount given above, we achieve a similar system where the meantone fifths are finally satisfactory. If A is the pure, B is the meantone fifth, the chain should follow the scheme given below:

ab ab ab aab

ab ab aab
ab aab

ab ab aab
ab ab

5. Lastly, we transpose all tones within an octave (where all octaves are larger by ~1 cents) and re-order them:

79 MOS 159tET improved
|
0: 1/1 C unison, perfect prime
1: 15.126 cents C/
2: 30.253 cents C//
3: 45.379 cents C^ Db(
4: 60.505 cents C) Dbv
5: 75.632 cents C#\ Db\\
6: 90.758 cents C# Db\
7: 105.884 cents C#/ Db
8: 121.010 cents C#// Db/
9: 136.137 cents C#^ D(
10: 151.263 cents C#) Dv
11: 166.389 cents D\\
12: 181.516 cents D\
13: 196.642 cents D
14: 211.768 cents D/
15: 226.895 cents D//
16: 242.021 cents D^ Eb(
17: 257.147 cents D) Ebv
18: 272.274 cents D#\ Eb\\
19: 287.400 cents D# Eb\
20: 302.526 cents D#/ Eb
21: 317.653 cents D#// Eb/
22: 332.779 cents D#^ E(
23: 347.905 cents D#) Ev
24: 363.031 cents E\\
25: 378.158 cents E\
26: 393.284 cents E
27: 408.410 cents E/ Fb
28: 423.537 cents E// Fb/
29: 438.663 cents E^ F(
30: 453.789 cents E) Fv
31: 468.916 cents E#\ F\\
32: 484.042 cents E# F\
33: 499.168 cents F
34: 514.295 cents F/
35: 529.421 cents F//
36: 544.547 cents F^ Gb(
37: 559.674 cents F) Gbv
38: 574.800 cents F#\ Gb\\
39: 589.926 cents F# Gb\
40: 605.052 cents F#/ Gb
41: 620.179 cents F#// Gb/
42: 635.305 cents F#^ G(
43: 650.431 cents F#) Gv
44: 665.558 cents G\\
45: 680.684 cents G\
46: 701.955 cents G
47: 717.081 cents G/
48: 732.208 cents G//
49: 747.334 cents G^ Ab(
50: 762.460 cents G) Abv
51: 777.587 cents G#\ Ab\\
52: 792.713 cents G# Ab\
53: 807.839 cents G#/ Ab
54: 822.965 cents G#// Ab/
55: 838.092 cents G#^ A(
56: 853.218 cents G#) Av
57: 868.344 cents A\\
58: 883.471 cents A\
59: 898.597 cents A
60: 913.723 cents A/
61: 928.850 cents A//
62: 943.976 cents A^ Bb(
63: 959.102 cents A) Bbv
64: 974.229 cents A#\ Bb\\
65: 989.355 cents A# Bb\
66: 1004.481 cents A#/ Bb
67: 1019.608 cents A#// Bb/
68: 1034.734 cents A#^ B(
69: 1049.860 cents A#) Bv
70: 1064.986 cents B\\
71: 1080.113 cents B\
72: 1095.239 cents B
73: 1110.365 cents B/ Cb
74: 1125.492 cents B// Cb/
75: 1140.618 cents B^ C(
76: 1155.744 cents B) Cv
77: 1170.871 cents B#\ C\\
78: 1185.997 cents B# C\
79: 1201.123 cents C

Attached are the original, improved and Lucy-improved versions for scrutiny with SCALA. Just type SET NOTA E79 to play with the chromatic klavier. The Lucy-improved will likely interest Charles, as it contains 33 instances of 300/PI + 600 cent fifths AKA the Lucy-Harrison fifth.

Cordially,
Ozan

An improved 79-tone scale

🔗Ozan Yarman <ozanyarman@ozanyarman.com>

🔗Gene Ward Smith <genewardsmith@coolgoose.com>

🔗Ozan Yarman <ozanyarman@ozanyarman.com>

🔗Ozan Yarman <ozanyarman@ozanyarman.com>