In the conclusion to the Factor Density topic at https://dozenal.forumotion.com/t51-factor-density, I concluded:

\[ E_B * {D_B}^k \]

I found twelve to be the best base for \( 1.8686 \lesssim k \lesssim 3.206\). For \(k \lessapprox 1.8685\), the best base was the number six. For \(k \gtrapprox 3.207\), the best base was twelfty. When \(k\) is smaller, there is more emphasis on smaller bases being better, whereas when \(k\) is larger, bigger highly composite bases tend to be better. The exponent \(k\) should not be made too large or else the scores for larger highly composite numbers would keep on increasing, and there would be no final best base. This makes it highly probable to interpret that there is no better base than twelve if the factor density and radix economy are the only two contributions taken into account for the score.

Next, I chose a value for the exponent \(k\) such that the ratio between the scores for the top base twelve and the second best base would be maximised. By coincidence, this value of \(k \simeq 2.66770688\) happened to have the advantage of causing the scores for the second and third best bases to be nearly equal, so that one would not have to decide which was better than the other, and also the score for twelfty happened to be the same as binary. I wonder whether there is any insight implied by these coincidences. A table of results for a selection of bases with this exponent is shown below:

Table of Bases Ranked by Optimality Scores

This measure of base optimality is interesting in that bases as apparently different in size such as binary and twelfty are could have the same score.

If the best base is the number twelve and reasons are chosen so as to make twelve the best base, this has implications on the order or rank for optimality of other bases. This optimality score is not comprehensive in that it does not take into consideration the influence of factors of the base plus or minus one on its ability to accurately approximate non-terminating fractions.

https://en.wikipedia.org/wiki/Radix_economy

https://www.tapatalk.com/groups/dozensonline/base-subdivisions-t1629-s24.html#p40009997

https://dozenal.forumotion.com/t24-dozenal-fifths-better-than-decimal-thirds

In order to furnish a measure of optimality of bases that moves large bases down the league, the factor density \(D_B\) can be multiplied by the measure of another property that decreases with the size of the base. One such property is the radix economy, which is optimal for very small bases near the base of the natural logarithm. I will use a modified form of radix economy generalised for alternating bases, as described previously elsewhere:Phaethon, Tue 19th Jun 2022 at 19:44 wrote:This factor density as a computed measure of base efficiency does not penalise an increase in size of the base enough,

I remove the factor of two from that expression which appeared to scale the efficiency of a binary base to two rather than one, but call it \(E_B\) instead of \( E_B \)/2 nonetheless. If this efficiency be multiplied by the factor density, the result tends to decrease as the base increases and does not provide much information about which bases are better than others. The influence of the factor density needs to be increased or weighted so that it is more important than the radix economy. Initially, I chose to make the factor density twice as important as the radix economy efficiency by raising the factor density to the power of two in the product. This resulted in twelve being the base with the highest score of all bases. Encouraged by this, I investigated variation of the exponent \(k\) to find the range of it for which twelve remains the top base, according to the formula:David Kennedy, 3:25 PM - May 29, 2017 wrote:a modified efficiency function, such as \(E_{B}=\displaystyle\frac{2(\log_{2}∏_{i}^{n}b_{i})^\frac{3}{2}}{\sqrt{n}\sum_{i}^{n}{(b_{i}-1)}}\)

\[ E_B * {D_B}^k \]

I found twelve to be the best base for \( 1.8686 \lesssim k \lesssim 3.206\). For \(k \lessapprox 1.8685\), the best base was the number six. For \(k \gtrapprox 3.207\), the best base was twelfty. When \(k\) is smaller, there is more emphasis on smaller bases being better, whereas when \(k\) is larger, bigger highly composite bases tend to be better. The exponent \(k\) should not be made too large or else the scores for larger highly composite numbers would keep on increasing, and there would be no final best base. This makes it highly probable to interpret that there is no better base than twelve if the factor density and radix economy are the only two contributions taken into account for the score.

Next, I chose a value for the exponent \(k\) such that the ratio between the scores for the top base twelve and the second best base would be maximised. By coincidence, this value of \(k \simeq 2.66770688\) happened to have the advantage of causing the scores for the second and third best bases to be nearly equal, so that one would not have to decide which was better than the other, and also the score for twelfty happened to be the same as binary. I wonder whether there is any insight implied by these coincidences. A table of results for a selection of bases with this exponent is shown below:

Table of Bases Ranked by Optimality Scores

Base | \(E_B *{D_B}^{2.66770688}\) |

12 | 1.264630504 |

6 | 1.113341812 |

24 | 1.113330551 |

60 | 1.062100735 |

120 | 1.000000006 |

2 | 1 |

4 | 0.942809042 |

36 | 0.919522883 |

48 | 0.856010571 |

72 | 0.852782923 |

180 | 0.850009264 |

30 | 0.841843609 |

360 | 0.818695262 |

240 | 0.804982549 |

18 | 0.745171206 |

8 | 0.742307489 |

720 | 0.670640405 |

840 | 0.668366963 |

20 | 0.651182733 |

2.718281828 | 0.591569443 |

40 | 0.589910884 |

1260 | 0.577717466 |

42 | 0.555646281 |

1680 | 0.552970045 |

10 | 0.547218881 |

16 | 0.533333333 |

3 | 0.503212301 |

54 | 0.409054139 |

56 | 0.391419125 |

14 | 0.349933442 |

9 | 0.293077693 |

64 | 0.233284737 |

5 | 0.228484451 |

50 | 0.208483258 |

**Conclusion:**If the best base is the number twelve and reasons are chosen so as to make twelve the best base, this has implications on the order or rank for optimality of other bases. This optimality score is not comprehensive in that it does not take into consideration the influence of factors of the base plus or minus one on its ability to accurately approximate non-terminating fractions.

**References:**https://en.wikipedia.org/wiki/Radix_economy

https://www.tapatalk.com/groups/dozensonline/base-subdivisions-t1629-s24.html#p40009997

https://dozenal.forumotion.com/t24-dozenal-fifths-better-than-decimal-thirds

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