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Definitions provided from https://www.konicaminolta.eu/fileadmin/content/eu/Measuring_Instruments/4_Learning_Centre/L_D/The_language_of_Light/language_of_light.pdf
The Colour Rendering Index is a quantification of the faithfulness of colour appearance under the test light source compared to the colour appearance under the standard illuminant.
The test compare the appearance of eight color samples (see below) under the light in question and a reference light source. The average differences measured are subtracted from 100 to get the CRI. Small average differences will result in a higher score, while larger differences give a lower number. Of all the colors possible, only these eight are measured. Samples used are pastels, not saturated colors.
CRI is calculated by measuring the difference between the lamp in question and a reference lamp in terms of how they render the eight color samples. If the lamp to be tested has a correlated color temperature (CCT) of less than 5000 Kelvin (K), the reference source is a black body radiator (approximately like an incandescent lamp). For higher CCT sources, the reference is a specifically defined spectrum of daylight. Light sources that mimic incandescent light or daylight for the eight color samples are, by definition, the ones that will score highest on the CRI.
Main 8 patches are pastel colors and 8 more patches provide diversity adding more saturated colors and few other useful ones.
Check good PDF at https://www.konicaminolta.eu/fileadmin/content/eu/Measuring_Instruments/4_Learning_Centre/C_A/What_is_Colour_Rendering_Index/Colour_Rendering_Index_EN.pdf
Some good CRI lights can perform poor on colors, especially red.
And light that produce pleasing accurate, but saturated colors can have low CRI.
CRI was made mostly for fluorescent measurements, and designed to allow companies to get good results optimizing not for overall color accuracy, but for good looking numbers.
The CQS is fidelity metric, producing one number like Ra, but the CQS has improvements including the ability to account for the direction of object color shifts. The CQS thus agrees well with visual ratings for color-enhancing sources, while it works as an accurate fidelity metric for non-color-enhancing sources. The CQS score works to represent the overall color quality of products (perceptual color fidelity) for all types of light sources.
It uses Root Mean Square to aggregate all 15 patches results, also do not penalize chroma only difference, only color shifts.
Usage examples, note that it is very close to CRI for fluos:
See more details at http://www.seleconlight.com/images/stories/files/Color_Quality_Scale_2010.pdf and http://www1.eere.energy.gov/buildings/publications/pdfs/ssl/cqs_rationale_06-10.pdf
Unfortunately, high quality, high CRI leds have less efficiency compared to low quality ones, especially if they are not warm.
Light Measurement Handbook by Alexander D. Ryer
Useful, on point and short.
Check at http://optdesign.narod.ru/book/light_measurement_handbook_1997.pdf or https://www.intl-lighttech.com/sites/default/files/pdf/handbook/ILT-Light-Measurement-Handbook.pdf
Language of light
Illimination fundamentals
Free at https://optics.synopsys.com/lighttools/pdfs/illuminationfund.pdf