In an earlier article on this blog I have written about and explained the "Renold-I" temperament, the first version of the "Scale of Fifths" that Maria Renold made in 1962. If you have not read that article yet, then I suggest you do so, just to be able to place everything in perspective.
After Maria Renold created – what later would be known as the "Renold-I" temperament – she continue experimenting and developing her concept. The result is what is known as the "Renold-II" (also known as "Middle Tuning" temperament.
In this article I will share more information about how the Renold-II temperament works. How this temperament works was explained by Peter Stevens in "A Handbook on the Middle Tuning" (2nd edition). This work is shared (in PDF format) online at www.eurythmy.co.nz. You can download it there to have more information then I will provide in this article.
WHAT IS DIFFERENT ABOUT RENOLD II?
The most important difference between "Renold-I" and "Renold-II" is the type of "Musical Interval System" (the range of notes within the system / the number of actual tones available to use). In the case of "Renold I" a closed system is used, the "Renold-II" uses an "open system".
Well, generally with a "closed system" the octaves are perfect doublings: C4=256Hz thus C5=512Hz, a difference of 1200 cents. In this tuning system there are 12 tones per octave and every octave includes a perfect doubling of those 12 tones.
Let's look at the piano to visualize it. A standard acoustic piano has 88 keys. This covers 7 octaves with 12 tones (=84), +4 semitones.
C1=32Hz, C2=64Hz, C3=128Hz, C4=256Hz, C5=512Hz, C6=1024HZ, C7=2048Hz and C8=4096Hz.
All these "C's" are "identical", they are the same frequencies doubled or halved. When you would play two or more perfect octaves (1200 cents precisely) of for example those "C's" simultaneously, most people would no longer differentiate between the "C's" but hear them as one.
Another reason why I call them "identical" is that they are all part of the same harmonic series. If we take for example C1 as the "fundamental" (the first harmonic), then C2 aligns perfectly with the 2nd harmonic of C1, as do all other perfect octaves of C1: C3=4th harmonic, C4=8th harmonic, C5=16th harmonic, C6=32nd harmonic, C7=64th harmonic and C8=128th harmonic. From C2 we could do exactly the same and those "C's" above C2 would align in the same relative fasion to C2 as the "C's" above C1 did to C1.
To visualize a closed system you could draw it as a circle.
C1=31.815Hz, C2=63.753Hz, C3=127.753Hz, C4=256Hz, C5=512.992Hz, C6=1027.971Hz, C7=2059.923 and C8=4127.927Hz.
The consequence of this is that we end up having 88 "different" tones on the piano. To visualize an open system, you could draw it as a spiral.
Below a part of the Renold-II temperament. To add the complete range (C0-C9) would be too much for this page. You can though download the full range of tones and frequencies in a PDF file.
In the 1st column you see "-1203.35 down" and "1203.35 up". This is the amount of cents up or down from every tone from C4 to B4.
In the 2nd column you see a series of cent corrections for the tones C4 – B4 compared to Equal Temperament.
The 3rd column containes the tone names.
In column 5 you see the corrected tone frequencies (Renold-II).
The recommended concert pitch for this temperament is C3=128Hz (C4=256Hz & A4=432Hz). Reason of it is explained in Maria Renold's book on page 69.
The frequencies in the table on in this article are only a "guideline". For proper results fine tuning should be done by ear (according to scheme included in Maria Renold's book). For tuning by ear three tuning fork are required: C4=256Hz, A4=432Hz and "gelis" (F#/Gb)=362,40Hz.
It might be good to mention that the Renold-II temperament doesn't sound (as well as Renold-I) on synthesizers and other electronic devices! The Renold-II temperament does work well for acoustic instruments.
Special thanks to Sławek Kędra providing additional information.