Yahoo Tuning Groups Ultimate Backup tuning Two pianos; 24 notes tuned to lie on a spiral of 2/9 comma flattened fifths

Hello -

I've just heard a recording of a vocal group's performance of a work in just intonation by Henk Badings where 11-ratio harmonies were sensitively and accurately employed. Although I had played around for nearly 30 years with computer synthesized 11-ratio chords and small compositions employing them, I had not until then heard an excellent sung performance of a composition by a master in which higher ratio harmonies played an essential role. The song I heard sounded beautiful to me and contained harmonies outside of anything offered by "traditional" music not having 11-ratio harmonies. There were harmonies based on integers higher than 11 which added to the richness of the music but I'm focussing here on those based on 11 and lower integers.

Surprisingly, although a few composers like Badings were composing music built upon 7-, 9-, 11- and higher integer ratio harmonies by the middle of the 20th century, the adoption of these harmonies into the mainstream of modern classical music has hardly yet taken place. While limitations of many kinds of musical instrument make the performance of music employing these harmonies using them difficult, skilled singers have demonstrated that vocal performance of music with these harmonies is entirely possible and they add significantly to its content and variety.

I suspect the hegemony which 12 equal temperament enjoyed over the last century has had something to do with this.

Performing serious complex music having within it these harmonies based on 7, 9 and 11 on the piano with good intonation worthy of what a piano can truly produce (such as when in 1/4 comma mean tone tuning) is problematic if one has only twelve pitches to work with on a single piano. However with two very similar pianos upon which two (or more) practiced pianists may perform, a world can be opened up in which a great deal of serious music freely employing harmonies based on ratios of 7, 9, and 11 (as well as 3 and 5) can be performed.

Let the 24 notes of the two pianos be tuned having fifths uniformly tempered flat by 2/9 comma (4.8 cents) beginning say with E-flat on the higher pitched piano going through all the fifths on that piano to G-sharp and then starting with D-sharp on the second piano tuned a 2/9 comma flattened fifth above the G-sharp on the first piano, A-sharp a 2/9 comma flattened fifth above the D-sharp, right on to the final "low G-sharp". This scheme permits modulation between 20 "good" major triads, 20 such minor triads with both major and minor triads contiguously on 17 of the keys. There are good harmonic sevenths on 14 of the notes contiguously from E-flat on the first piano to A-sharp on the second piano. Of especial interest there are six contiguous good 11th chords whose notes lie in approximate frequency ratio 4:5:6:7:9:11 from E-flat on the first piano to D on the first piano. Some of the tempering departures from just are: fifth -4.8 cents, major third +2.4 cents, minor third -7.2 cents, harmonic 7th i.e. 7/4 +2.9 cents, 9th i.e. 9/8 -9.6 cents, and harmonic 11th i.e. 11/8 -2.2 cents.

A listener could detect several tempering departures from just here - of course the flattening of the fifths, then what I suspect is the most troubling is the narrowing of the 9/8 and along with that the narrowing of the 9/7 equivalent interval by 12.5 cents. A high rising upper note in the 9/7 carries powerful emotional expression, but that interval is more often than not narrowed by a lot more than 12.5 cents in performances generally felt to be excellent by nearly everyone.

In my listening experience with pianos in different tunings, 1/4 comma mean tone tuning gives the piano a beautiful flowing resonant sound and a harmoniousness which seem to turn it into a different instrument than the 12 equal tempered piano which many of us once thought was "the whole story" regarding the piano. Although to my ears the sound of just harmonies on the piano can have a clarity and beauty even greater than the 1/4 comma mean tone harmonies, I find good music on a 1/4 comma mean tone piano to be delightful to listen to and it's so much easier to produce music having variety and complexity with the mean tone tuning. I find that harmonies sounded on a piano in 1/6 comma mean tone tuning are rougher sounding than in 1/4 comma and generally less pleasing, although still sweeter sounding than in equal temperament.

For a short period I played a piano in 5/24 comma mean tone and as I remember it, its sound was as harmonious as the 1/4 comma mean tone sound - very smooth and harmonious.

I believe the two-piano 2/9 comma studio scheme proposed here could be used to perform much very beautiful and strikingly different music. By keeping the system and keyboard layouts relatively straightforward and familiar it would be "doable" for many musicians and the pianos would not suffer the fate which the excellent but complex Bosanquet 53 per octave piano suffered spending most of its life sitting unplayed in a museum.

I hope some non-twelve enthusiasts here in San Diego will give the proposal offered here serious thought.

Dave Hill Borrego Springs, CA Tue. 8/18/09

Two pianos; 24 notes tuned to lie on a spiral of 2/9 comma flattened fifths

🔗Ralph Hill <ASCEND11@...>

🔗Carl Lumma <carl@...>

🔗Mike Battaglia <battaglia01@...>