# DIMI temperament

DIMI-A and DIMI-O are generating individual notes with counter circuits. The idea is to divide frequencies down from a single ultra sonic oscillator. When the divider ratios are large enough, it is possible to approximate an equal temperament scale.

Bigger ratios would demand more TTL chips, so the as compromise DIMIs use ratios that can be presented with less than 12 bits, preferable less than 8. In addition to this, ratios where you are counting should not have more than eight ‘1’s, so they can be detected with a single 8 input NAND-port. This limits usable ratios to be around 300, which is not enough to achieve perfect estimation for equally tempered scale.

The divider circuit has two 7493 TTL counter circuits in series. They count numbers up every time clock pulse is received at N11 pin 14. Selected outputs are wired to a 7430 NAND gate. This programs the value the circuit is counting to. This case it is 1100 0100 which in 10 decimal system corresponds to 196. When this number is reached the feedback connection from the output will reset counters and counting will restart from zero.

The ratios (confirmed from hardware) on DIMI-A and DIMI-O are following:

- C# 350
- D 330
- D# 312
- E 294
- F 277
- F# 261
- G 247
- G# 233
- A 220
- A# 208
- B 196
- C 185

DIMI-O uses same ratios, but with one note offset, so ratio 350 is used for C while 185 is used for B. Not fully sure where this set of ratios come from. It is worth noting that ratios 185/196 are mentioned in other sources being the lowest ratios giving more or less perfect semitone ( 1.0594595 vs ‘perfect ‘ 1.059463094) [1,2].

Ratios giving more or less perfect approximation of an equal temperament are a bit larger, between 508 and 959 [1]. Ratios this high would require significant amount of additional gates compared to smaller numbers used in DIMIs.

As the temperament given by DIMI ratios is not optimal, some intervals are more off than others. Therefore when instrument is used together with other instruments some ratios are more usable than others to be used when tuning the main oscillator. If I’m reading my excel sheets correctly tuning main oscillator ratio 247 dead on will keep other notes as close as possible to the true equal temperament as possible. Worst idea would be using ratio 261 as starting point.

B | A# | A | G# | G | F# | F | E | D# | D | C# | C | |||

185 | 196 | 208 | 220 | 233 | 247 | 261 | 277 | 294 | 312 | 330 | 350 | SUM of DIFFs | ||

C | 350 | 0.00414 | 0.00392 | 0.00090 | 0.00351 | 0.00384 | 0.00279 | 0.00616 | 0.00362 | 0.00127 | -0.00067 | 0.00114 | 1.00000 | 0.03195 |

C# | 330 | 0.00199 | 0.00188 | -0.00086 | 0.00169 | 0.00210 | 0.00119 | 0.00445 | 0.00213 | -0.00001 | -0.00177 | 1.00000 | 0.00114 | 0.01921 |

D | 312 | 0.00469 | 0.00444 | 0.00169 | 0.00397 | 0.00422 | 0.00324 | 0.00620 | 0.00389 | 0.00176 | 1.00000 | -0.00177 | -0.00067 | 0.03653 |

D# | 294 | 0.00179 | 0.00169 | -0.00075 | 0.00152 | 0.00188 | 0.00108 | 0.00397 | 0.00191 | 1.00000 | 0.00176 | -0.00001 | 0.00127 | 0.01764 |

E | 277 | -0.00101 | -0.00095 | -0.00311 | -0.00083 | -0.00037 | -0.00100 | 0.00184 | 1.00000 | 0.00191 | 0.00389 | 0.00213 | 0.00362 | 0.02065 |

F | 261 | -0.00340 | -0.00321 | -0.00511 | -0.00284 | -0.00229 | -0.00278 | 1.00000 | 0.00184 | 0.00397 | 0.00620 | 0.00445 | 0.00616 | 0.04225 |

F# | 247 | 0.00030 | 0.00028 | -0.00171 | 0.00027 | 0.00062 | 1.00000 | -0.00278 | -0.00100 | 0.00108 | 0.00324 | 0.00119 | 0.00279 | 0.01526 |

G | 233 | -0.00046 | -0.00043 | -0.00227 | -0.00037 | 1.00000 | 0.00062 | -0.00229 | -0.00037 | 0.00188 | 0.00422 | 0.00210 | 0.00384 | 0.01885 |

G# | 220 | -0.00002 | -0.00001 | -0.00177 | 1.00000 | -0.00037 | 0.00027 | -0.00284 | -0.00083 | 0.00152 | 0.00397 | 0.00169 | 0.00351 | 0.01681 |

A | 208 | 0.00186 | 0.00176 | 1.00000 | -0.00177 | -0.00227 | -0.00171 | -0.00511 | -0.00311 | -0.00075 | 0.00169 | -0.00086 | 0.00090 | 0.02180 |

A# | 196 | 0.00000 | 1.00000 | 0.00176 | -0.00001 | -0.00043 | 0.00028 | -0.00321 | -0.00095 | 0.00169 | 0.00444 | 0.00188 | 0.00392 | 0.01857 |

B | 185 | 1.00000 | 0.00000 | 0.00186 | -0.00002 | -0.00046 | 0.00030 | -0.00340 | -0.00101 | 0.00179 | 0.00469 | 0.00199 | 0.00414 | 0.01966 |

It seems to me that couple ratios could be tweaked to give better temperament. Changing 261 to 262 and 277 to 278 would in my mind give better results.

DIMIs will generate all necessary clock frequencies from the same main oscillator that is used to generate note frequencies. Therefore main oscillator is tuned much higher than note generators require. On DIMI-A main oscillator frequency is around ~12.4Mhz, which is divided down by four before being fed to the note generator divider circuit. Octaves of different notes are generated by dividing note generator outputs with 2^n.

## References

- https://electro-music.com/forum/phpbb-files/harmonic_logs_781.pdf
- https://electro-music.com/forum/topic-65094.html