*Previously, I wrote about a method for comparing cyclical tunings using the 'contour' of each note interval's deviation from Equal Temperament, arranged with highest first to create a 'tuning signature' which can be easily sorted to compare to others. While this can be applied to any list of tunings, I'll stay true to my original intention and report my findings in analysing the wealth of 12-tone tunings found in Murray J Barbour's 'Tuning and Temperament'*

The analysis turned up some obvious matches such as

Calculations were all done to the nearest cent. While monochord string lengths were translated to the nearest cent value, those that appear equal may, in actuality differ slightly, but this level of matching was deemed insignificant for these purposes. It's worth pointing out that arriving at a determining figure of significance (a difference limen) is a useful pursuit in being able to make analyses such as these. If the analysis was carried out using a resolution of 2 cents (a difference that may not actually be significantly detectable), there would be more matches to report.

Starting with a technical, very obvious and easy match will help explain the methods of analysis used.

The two are acoustically the same, differing only in the offset of their starting notes. The highest positive deviation from an ET step of 100 cents from the previous step is marked in red (this is an arbritary starting point, having no statistical bearing). When this note is taken as the starting point, the similarity between the two tunings can easily be seen from their signature:

**22.****Aaron 1/4-comma**&**25. Rossi's Meantone**(which are exactly the same in the book),**Galilei's Linear Corrections 1**and**2**and various other attempts at divining*Equal Temperament*, but these are too obvious to include as 'hidden' matches.Calculations were all done to the nearest cent. While monochord string lengths were translated to the nearest cent value, those that appear equal may, in actuality differ slightly, but this level of matching was deemed insignificant for these purposes. It's worth pointing out that arriving at a determining figure of significance (a difference limen) is a useful pursuit in being able to make analyses such as these. If the analysis was carried out using a resolution of 2 cents (a difference that may not actually be significantly detectable), there would be more matches to report.

**Matching Principles**Starting with a technical, very obvious and easy match will help explain the methods of analysis used.

**Difference Column 3**(Barbour's Table 61) was built by "using the lengths of**Difference Column 2**as differences". The 'adjusted' version (Barbour's Table 63) was an amendment made to 61 by Barbour, so isn't counted as one of the seven matches discussed here.The two are acoustically the same, differing only in the offset of their starting notes. The highest positive deviation from an ET step of 100 cents from the previous step is marked in red (this is an arbritary starting point, having no statistical bearing). When this note is taken as the starting point, the similarity between the two tunings can easily be seen from their signature:

**+1 / 0 / 0 / 0 / 0 / 0 / 0 / -1 / 0 / 0 / 0 / 0**.Simple Contour Match - 61 & 63 | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

61. Difference Column 3 | 100 | 200 | 301 | 401 | 501 | 601 | 701 | 801 | 901 | 1000 | 1100 | 1200 |

Contour | 0 | 0 | +1 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 |

63. Monochord 3 Adjusted | 100 | 200 | 300 | 399 | 499 | 599 | 699 | 799 | 900 | 1000 | 1100 | 1200 |

Contour | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | +1 | 0 | 0 | 0 |

**Two JI Matches and Non-matches**

**Mersenne's second Spinet Tuning (93)**and

**Mersenne's first Lute Tuning (94)**match acoustically as shown below, despite being derived in different ways - the former as having a more largely reduced comma exponent of -2, the latter as having smaller variations (-1 and +1). Barbour notes that these two tunings, together with three others, one of which is

**Mersenne's Spinet Tuning 1 (92)**(see twice below),

*"are worth including here as evidence of the variety that is possible in a type of tuning that is ordinarily thought to be fixed and uniform"*

Two Matching Just Intonation Tunings - 93 & 94 | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

93. Mersenne's Spinet 2 | 70 | 204 | 274 | 386 | 498 | 568 | 702 | 772 | 884 | 996 | 1088 | 1200 |

Contour | -30 | +34 | -30 | +12 | +12 | -30 | +34 | -30 | +12 | +12 | -8 | +12 |

94. Mersenne's Lute 1 | 112 | 182 | 316 | 386 | 498 | 610 | 702 | 814 | 884 | 1018 | 1088 | 1200 |

Contour | +12 | -30 | +34 | -30 | +12 | +12 | -8 | +12 | -30 | +34 | -30 | +12 |

**Mean and Standard Deviations**

To Barbour's credit, (as well as the fact that he didn't have access to spreadsheet software) he does note that 93 & 94 have the same statistical deviation (Mean Deviation 21.3, Standard Deviation 23.6). However, these methods of analysis also point to

**Mersenne's Lute 2**

**and Spinet 1**tunings (92 & 95) as being equal, having as they do, a Mean Deviation of 17.7 and a Standard Deviation of 20.1. Our method of analysis, however, deems 92 & 95 acoustically different.

Mersenne's Ostensibly Matching Tunings | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

92. Spinet 1 | 112 | 182 | 316 | 386 | 498 | 610 | 702 | 814 | 884 | 996 | 1088 | 1200 |

Contour | +12 | -30 | +34 | -30 | +12 | +12 | -8 | +12 | -30 | +12 | -8 | +12 |

95. Lute 2 | 112 | 204 | 316 | 386 | 498 | 610 | 702 | 814 | 884 | 1018 | 1088 | 1200 |

Contour | +12 | -8 | +12 | -30 | +12 | +12 | -8 | +12 | -30 | +34 | -30 | +12 |

Standard and Mean deviations from Equal Temperament are, it seems, not fit for analysing tunings with the exactness of this method.

**Kepler and Malcolm's JI Match**

Barbour mentions that 92 & 95 are as good as Kepler's JI monochords. While they're certainly not the same as it, we can take a deeper look in comparing one of Kepler's to Malcolm's - Barbour points this out, mentioning that Malcolm's Monochord is "a fifth lower".

Two More Matching Just Intonation Tunings - 91 & 99 | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

91. Kepler's Monochord | 92 | 204 | 316 | 386 | 498 | 590 | 702 | 814 | 906 | 1018 | 1088 | 1200 |

Contour | -8 | +12 | +12 | -30 | +12 | -8 | +12 | +12 | -8 | +12 | -30 | +12 |

99. Malcolm's Monochord | 112 | 204 | 316 | 386 | 498 | 590 | 702 | 814 | 884 | 996 | 1088 | 1200 |

Contour | +12 | -8 | +12 | -30 | +12 | -8 | +12 | +12 | -30 | +12 | -8 | +12 |

**Three Matching Just Intonations**

Barbour does mention that Mersenne's Spinet 1 (92) is "constructed exactly the same as De Caus's Monochord" (89).

Again, while 89 & 92 share the same Mean and Standard Deviations,

**Euler's Monochord**(101) differs in its Mean Deviation. As shown below, though, all three of the tunings are the same.

Three Matching Just Intonation Tunings | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

89. De Caus's Monochord | 70 | 182 | 274 | 386 | 498 | 568 | 702 | 772 | 884 | 996 | 1088 | 1200 |

Contour | -30 | +12 | -8 | +12 | +12 | -30 | +34 | -30 | +12 | +12 | -8 | +12 |

92. Mersenne's Spinet 1 | 112 | 182 | 316 | 386 | 498 | 610 | 702 | 814 | 884 | 996 | 1088 | 1200 |

Contour | +12 | -30 | +34 | -30 | +12 | +12 | -8 | +12 | -30 | +12 | -8 | +12 |

101. Euler's Monochord | 70 | 204 | 274 | 386 | 498 | 590 | 702 | 772 | 884 | 976 | 1088 | 1200 |

Contour | -30 | +34 | -30 | +12 | +12 | -8 | +12 | -30 | +12 | -8 | +12 | +12 |

**Further Reading**

**Matt Grenfell: The Development of The Equal Temperament Scale - Evolution or Radical Change?**- a survey (using, in part, Barbour's work) of whether the evolution of tunings tended towards ET, and therefore, whether we have reached a conclusion in it.

**Murray J Barbour: Tuning and Temperament - A Historical Survey**

**Scala**- Thousands of tunings available for easy integration with software, and a nice comparison

**Bradley Lehman: Larips**- In which, in amongst the chaos, I swear I once found what Bradley called a 'Recipe Book' - a spreadsheet with which a bank of tunings could be analysed but not compared. I have never found it since!